Inhaltsverzeichnis

Development of
Algorithmic Constructions

00:38:18
Deutsch
29.Mar 2024

Polynom = x^2+68x-151

0. Sequence

1. Algorithm

2. Mathematical background

3. Correctness of the algorithm

4. Infinity of the sequence

5. Sequence of the polynom with 1

6. Sequence of the polynom (only primes)

7. Distribution of the primes

8. Check for existing Integer Sequences by OEIS

0. Sequence

f(0) = 151 = 151
f(1) = 41 = 41
f(2) = 11 = 11
f(3) = 31 = 31
f(4) = 137 = 137
f(5) = 107 = 107
f(6) = 293 = 293
f(7) = 187 = 11*17
f(8) = 457 = 457
f(9) = 271 = 271
f(10) = 629 = 17*37
f(11) = 359 = 359
f(12) = 809 = 809
f(13) = 451 = 11*41
f(14) = 997 = 997
f(15) = 547 = 547
f(16) = 1193 = 1193
f(17) = 647 = 647
f(18) = 1397 = 11*127
f(19) = 751 = 751
f(20) = 1609 = 1609
f(21) = 859 = 859
f(22) = 1829 = 31*59
f(23) = 971 = 971
f(24) = 2057 = 11*11*17
f(25) = 1087 = 1087
f(26) = 2293 = 2293
f(27) = 1207 = 17*71
f(28) = 2537 = 43*59
f(29) = 1331 = 11*11*11
f(30) = 2789 = 2789
f(31) = 1459 = 1459
f(32) = 3049 = 3049
f(33) = 1591 = 37*43
f(34) = 3317 = 31*107
f(35) = 1727 = 11*157
f(36) = 3593 = 3593
f(37) = 1867 = 1867
f(38) = 3877 = 3877
f(39) = 2011 = 2011
f(40) = 4169 = 11*379
f(41) = 2159 = 17*127
f(42) = 4469 = 41*109
f(43) = 2311 = 2311
f(44) = 4777 = 17*281
f(45) = 2467 = 2467
f(46) = 5093 = 11*463
f(47) = 2627 = 37*71
f(48) = 5417 = 5417
f(49) = 2791 = 2791
f(50) = 5749 = 5749
f(51) = 2959 = 11*269
f(52) = 6089 = 6089
f(53) = 3131 = 31*101
f(54) = 6437 = 41*157
f(55) = 3307 = 3307
f(56) = 6793 = 6793
f(57) = 3487 = 11*317
f(58) = 7157 = 17*421
f(59) = 3671 = 3671
f(60) = 7529 = 7529
f(61) = 3859 = 17*227
f(62) = 7909 = 11*719
f(63) = 4051 = 4051
f(64) = 8297 = 8297
f(65) = 4247 = 31*137
f(66) = 8693 = 8693
f(67) = 4447 = 4447
f(68) = 9097 = 11*827
f(69) = 4651 = 4651
f(70) = 9509 = 37*257
f(71) = 4859 = 43*113
f(72) = 9929 = 9929
f(73) = 5071 = 11*461
f(74) = 10357 = 10357
f(75) = 5287 = 17*311
f(76) = 10793 = 43*251
f(77) = 5507 = 5507
f(78) = 11237 = 17*661
f(79) = 5731 = 11*521
f(80) = 11689 = 11689
f(81) = 5959 = 59*101
f(82) = 12149 = 12149
f(83) = 6191 = 41*151
f(84) = 12617 = 11*31*37
f(85) = 6427 = 6427
f(86) = 13093 = 13093
f(87) = 6667 = 59*113
f(88) = 13577 = 13577
f(89) = 6911 = 6911
f(90) = 14069 = 11*1279
f(91) = 7159 = 7159
f(92) = 14569 = 17*857
f(93) = 7411 = 7411
f(94) = 15077 = 15077
f(95) = 7667 = 11*17*41
f(96) = 15593 = 31*503
f(97) = 7927 = 7927
f(98) = 16117 = 71*227
f(99) = 8191 = 8191
f(100) = 16649 = 16649

1. Algorithm

If you are interested in some better algorithms have a look at quadr_Sieb_x^2+1.php.

2. Mathematical background

Lemma: If p | f(x) then also p | f(x+p) and p | f(-x-b/a) a) p | f(x) <=> ax^2 + bx + c = 0 mod p p | f(x+p) <=> a(x+p)^2 + b(x+p) + c = 0 mod p <=> ax^2 + 2axp + ap^2 + bx + bp + c = 0 mod p <=> ax^2 + bx + c = 0 mod p Thus if p | f(x) then p | f(x+p) b) if b = 0 mod a p | f(x) <=> ax^2 + bx + c = 0 mod p p | f(-x-b/a) <=> a(-x-b/a)^2 + b(-x-b/a) + c = 0 mod p <=> ax^2 + 2bx + b^2/a - bx - b^2/a + c = 0 mod p <=> ax^2 + bx + c = 0 mod p Thus if p | f(x) then p | f(-x-b/a)

3. Correctness of the algorithm

The proof for this polynom is similar to the proof for the polynom f(x)=x^2-4x+1. a) First terms for the polynom f(x) = x^2+68x-151

f(0)=151
f(1)=41
f(2)=11
f(3)=31
f(4)=137
f(5)=107
f(6)=293
f(7)=17
f(8)=457
f(9)=271
f(10)=37
f(11)=359
f(12)=809
f(13)=1
f(14)=997
f(15)=547
f(16)=1193
f(17)=647
f(18)=127
f(19)=751
f(20)=1609
f(21)=859
f(22)=59
f(23)=971
f(24)=1
f(25)=1087
f(26)=2293
f(27)=71
f(28)=43
f(29)=1
f(30)=2789
f(31)=1459
f(32)=3049
f(33)=1
f(34)=1
f(35)=157
f(36)=3593
f(37)=1867
f(38)=3877
f(39)=2011
f(40)=379
f(41)=1
f(42)=109
f(43)=2311
f(44)=281
f(45)=2467
f(46)=463
f(47)=1
f(48)=5417
f(49)=2791
f(50)=5749
f(51)=269
f(52)=6089
f(53)=101
f(54)=1
f(55)=3307
f(56)=6793
f(57)=317
f(58)=421
f(59)=3671
f(60)=7529
f(61)=227
f(62)=719
f(63)=4051
f(64)=8297
f(65)=1
f(66)=8693
f(67)=4447
f(68)=827
f(69)=4651
f(70)=257
f(71)=113
f(72)=9929
f(73)=461
f(74)=10357
f(75)=311
f(76)=251
f(77)=5507
f(78)=661
f(79)=521
f(80)=11689
f(81)=1
f(82)=12149
f(83)=1
f(84)=1
f(85)=6427
f(86)=13093
f(87)=1
f(88)=13577
f(89)=6911
f(90)=1279
f(91)=7159
f(92)=857
f(93)=7411
f(94)=15077
f(95)=1
f(96)=503
f(97)=7927
f(98)=1
f(99)=8191

b) Substitution of the polynom
The polynom f(x)=x^2+68x-151 could be written as f(y)= y^2-1307 with x=y-34

c) Backsubstitution Beside by backsubstitution you get an estimation for the huge of the primes with p | f(x) and p < f(x) f'(y)>(2y-1) with with y=x+34
f'(x)>2x+67

4. Infinity of the sequence

The mathematical proof is analogue to the proof for the polynom f(x)=x^2+1

5. Sequence of the polynom with 1

151, 41, 11, 31, 137, 107, 293, 17, 457, 271, 37, 359, 809, 1, 997, 547, 1193, 647, 127, 751, 1609, 859, 59, 971, 1, 1087, 2293, 71, 43, 1, 2789, 1459, 3049, 1, 1, 157, 3593, 1867, 3877, 2011, 379, 1, 109, 2311, 281, 2467, 463, 1, 5417, 2791, 5749, 269, 6089, 101, 1, 3307, 6793, 317, 421, 3671, 7529, 227, 719, 4051, 8297, 1, 8693, 4447, 827, 4651, 257, 113, 9929, 461, 10357, 311, 251, 5507, 661, 521, 11689, 1, 12149, 1, 1, 6427, 13093, 1, 13577, 6911, 1279, 7159, 857, 7411, 15077, 1, 503, 7927, 1, 8191, 16649, 769, 17189, 8731, 17737, 9007, 1663, 1, 173, 563, 19429, 9859, 1, 10151, 479, 337, 21193, 977, 307, 1, 22409, 1, 23029, 1061, 577, 11987, 1429, 397, 2267, 743, 25589, 12959, 26249, 13291, 2447, 13627, 673, 13967, 28277, 1301, 491, 1, 29669, 883, 821, 1, 1, 15727, 31817, 16091, 1, 1, 33289, 16831, 1, 17207, 3163, 409, 1, 17971, 2137, 1669, 863, 1103, 37897, 467, 38693, 1777, 1, 1, 233, 20359, 3739, 20771, 41957, 21187, 42793, 1, 3967, 22031, 2617, 607, 449, 2081, 1, 23327, 47093, 23767, 47977, 1, 48869, 24659, 1, 25111, 1, 691, 51593, 1531, 52517, 1, 1, 26959, 1, 27431, 55337, 1, 1373, 28387, 1847, 28871, 58229, 1, 59209, 29851, 3541, 30347, 5563, 283, 1, 1, 2039, 31859, 5839, 32371, 65257, 32887, 66293, 3037, 1, 33931, 68389, 2027, 1877, 3181, 1, 35527, 71593, 36067, 6607, 1181, 1039, 37159, 74869, 877, 6907, 2251, 77093, 947, 1, 3581, 79349, 1289, 80489, 40531, 81637, 1, 82793, 41687, 1423, 1031, 1, 42859, 5077, 43451, 1483, 2591, 733, 44647, 89897, 1223, 2939, 1, 727, 46471, 93557, 47087, 94793, 4337, 701, 2843, 1, 1, 1, 1, 99817, 50227, 2351, 50867, 1, 51511, 103669, 1213, 2837, 4801, 106277, 1, 6329, 1, 108917, 1, 2689, 1789, 739, 56131, 10267, 56807, 3089, 57487, 115657, 58171, 10639, 829, 118409, 1, 119797, 5477, 7129, 1033, 122597, 61651, 1, 5669, 125429, 1069, 126857, 1, 1, 1, 129737, 1, 7717, 929, 389, 3923, 3119, 67427, 135593, 6197, 137077, 1, 138569, 1699, 140069, 1, 4567, 71167, 143093, 4231, 13147, 1, 8597, 73459, 147689, 74231, 13567, 1, 1493, 75787, 1, 6961, 153929, 77359, 155509, 2521, 9241, 7177, 4289, 4691, 1021, 1, 1, 81359, 3803, 82171, 1, 2677, 1, 1949, 433, 84631, 4597, 1, 1, 86291, 1, 1, 1, 1, 176777, 88811, 178469, 89659, 1489, 90511, 5867, 91367, 183593, 92227, 991, 1, 187049, 5527, 2659, 1, 190537, 95707, 6203, 96587, 194057, 8861, 1733, 2399, 197609, 99251, 18127, 1, 201193, 2731, 11941, 1, 1, 102859, 1627, 2531, 208457, 1, 1861, 105607, 1, 106531, 213989, 9769, 12697, 1013, 217717, 1, 19963, 3557, 221477, 111211, 6037, 1901, 20479, 113111, 227177, 114067, 229093, 10457, 231017, 6823, 1, 116959, 1, 1, 1, 118907, 7703, 1187, 509, 120871, 242729, 523, 2287, 2857, 1, 1, 14629, 124847, 8087, 1, 252709, 126859, 254729, 1801, 256757, 11717, 1889, 3511, 2393, 1, 23899, 131959, 264949, 7823, 267017, 1327, 1439, 4357, 271177, 136111, 273269, 1, 541, 3371, 4703, 139267, 279593, 12757, 281717, 4561, 1, 3313, 25999, 8443, 288137, 3527, 1, 145687, 2417, 1, 294629, 147859, 1, 1231, 8081, 1, 1741, 1, 9787, 13841, 17977, 153359, 569, 1367, 1, 1, 2459, 156707, 8501, 157831, 1, 158959, 319049, 160091, 1, 14657, 2143, 9551, 325877, 1447, 3067, 14969, 330469, 165811, 1, 166967, 1, 168127, 593, 1, 339749, 1, 1, 2909, 20261, 172807, 346793, 15817, 3457, 2969, 351529, 5689, 1, 16141, 1259, 4831, 1, 179947, 1931, 1693, 613, 631, 365929, 183571, 33487, 4507, 370793, 186007, 1487, 17021, 12119, 188459, 378149, 189691, 1481, 1, 3793, 1, 1, 1, 35279, 194659, 1, 1, 393077, 2777, 35963, 198427, 398117, 1, 1, 18269, 641, 202231, 2687, 11971, 1453, 18617, 410857, 1, 413429, 6689, 1, 1523, 418597, 2957, 10273, 211247, 653, 12503, 1, 213859, 25237, 1, 431657, 5851, 434293, 1, 10657, 1811, 10223, 1, 1, 1, 3677, 5189, 1, 224467, 450277, 1, 41179, 227159, 14699, 228511, 458377, 20897, 461093, 2161, 1, 232591, 1, 21269, 4153, 1, 15227, 236707, 2539, 5807, 477557, 2371, 480329, 240859, 1, 1543, 1, 243647, 6883, 22277, 491497, 6011, 29077, 4201, 1, 1, 499957, 250687, 1, 4273, 1, 253531, 508489, 254959, 3257, 256391, 46747, 8317, 7283, 1, 1933, 1, 30757, 1, 525769, 263611, 2269, 24097, 4877, 1, 13037, 7243, 48859, 1, 540389, 270931, 1, 272407, 49663, 16111, 549257, 275371, 13469, 25169, 555209, 7523, 558197, 279847, 1, 25577, 564197, 282851, 567209, 1, 51839, 285871, 33721, 287387, 576293, 288907, 52667, 1, 9871, 1, 1907, 26681, 588517, 1, 1, 4177, 34981, 1, 597769, 17627, 2647, 301211, 54907, 302767, 607093, 9817, 610217, 2707, 1, 7499, 616489, 309031, 5791, 1, 622793, 4397, 36821, 7297, 6229, 28669, 5801, 316951, 14779, 2819, 1873, 823, 641897, 321751, 1, 323359, 3467, 8783, 1, 19211, 1, 29837, 21227, 1, 1, 1, 664549, 1, 667817, 334727, 671093, 9091, 1, 1, 2179, 339659, 1, 341311, 62207, 5813, 687593, 11117, 1823, 31481, 4013, 347959, 16223, 2753, 700937, 1, 2437, 8209, 9967, 1, 1747, 1, 1, 358051, 717797, 359747, 65563, 361447, 724597, 363151, 4637, 1, 2699, 21563, 19861, 368287, 1, 33637, 1, 1, 745189, 373459, 68059, 9151, 752117, 376927, 755593, 378667, 1, 2423, 1, 1, 1, 2053, 769577, 8969, 773093, 1, 13163, 35381, 18143, 390959, 783689, 907, 1, 394507, 790793, 23311, 2953, 12841, 1, 1, 1, 401651, 805097, 36677, 808693, 405247, 812297, 1, 1, 1, 1, 410671, 1, 412487, 6833, 24371, 7349, 5861, 1, 417959, 76159, 419791, 22741, 1, 1, 1, 19739, 4211, 1, 25127, 856169, 1, 50581, 1, 863593, 432727, 2131, 6121, 2069, 436459, 28219, 10691, 79867, 440207, 5843, 1, 52121, 40361, 889829, 26227, 893609, 7589, 8233, 1, 901193, 451547, 904997, 453451, 82619, 1, 1997, 457271, 916457, 27011, 83663, 1979, 54361, 463031, 7307, 983, 931849, 15061, 935717, 12671, 21851, 42797, 943477, 472711, 947369, 474659, 5087, 476611, 23297, 28151, 959093, 480527, 87547, 1, 13619, 484459, 31319, 44221, 974837, 488407, 23873, 1, 26561, 2633, 986729, 494359, 1, 3623, 2917, 1, 16927, 1, 9371, 2213, 1, 504359, 1, 2927, 23599, 1, 59929, 3251, 1022837, 1, 9421, 46769, 1030949, 16661, 1, 12647, 94463, 520567, 1043177, 3461, 10369, 4643, 8689, 30983, 1, 1, 1, 1, 2963, 1,

6. Sequence of the polynom (only primes)

151, 41, 11, 31, 137, 107, 293, 17, 457, 271, 37, 359, 809, 997, 547, 1193, 647, 127, 751, 1609, 859, 59, 971, 1087, 2293, 71, 43, 2789, 1459, 3049, 157, 3593, 1867, 3877, 2011, 379, 109, 2311, 281, 2467, 463, 5417, 2791, 5749, 269, 6089, 101, 3307, 6793, 317, 421, 3671, 7529, 227, 719, 4051, 8297, 8693, 4447, 827, 4651, 257, 113, 9929, 461, 10357, 311, 251, 5507, 661, 521, 11689, 12149, 6427, 13093, 13577, 6911, 1279, 7159, 857, 7411, 15077, 503, 7927, 8191, 16649, 769, 17189, 8731, 17737, 9007, 1663, 173, 563, 19429, 9859, 10151, 479, 337, 21193, 977, 307, 22409, 23029, 1061, 577, 11987, 1429, 397, 2267, 743, 25589, 12959, 26249, 13291, 2447, 13627, 673, 13967, 28277, 1301, 491, 29669, 883, 821, 15727, 31817, 16091, 33289, 16831, 17207, 3163, 409, 17971, 2137, 1669, 863, 1103, 37897, 467, 38693, 1777, 233, 20359, 3739, 20771, 41957, 21187, 42793, 3967, 22031, 2617, 607, 449, 2081, 23327, 47093, 23767, 47977, 48869, 24659, 25111, 691, 51593, 1531, 52517, 26959, 27431, 55337, 1373, 28387, 1847, 28871, 58229, 59209, 29851, 3541, 30347, 5563, 283, 2039, 31859, 5839, 32371, 65257, 32887, 66293, 3037, 33931, 68389, 2027, 1877, 3181, 35527, 71593, 36067, 6607, 1181, 1039, 37159, 74869, 877, 6907, 2251, 77093, 947, 3581, 79349, 1289, 80489, 40531, 81637, 82793, 41687, 1423, 1031, 42859, 5077, 43451, 1483, 2591, 733, 44647, 89897, 1223, 2939, 727, 46471, 93557, 47087, 94793, 4337, 701, 2843, 99817, 50227, 2351, 50867, 51511, 103669, 1213, 2837, 4801, 106277, 6329, 108917, 2689, 1789, 739, 56131, 10267, 56807, 3089, 57487, 115657, 58171, 10639, 829, 118409, 119797, 5477, 7129, 1033, 122597, 61651, 5669, 125429, 1069, 126857, 129737, 7717, 929, 389, 3923, 3119, 67427, 135593, 6197, 137077, 138569, 1699, 140069, 4567, 71167, 143093, 4231, 13147, 8597, 73459, 147689, 74231, 13567, 1493, 75787, 6961, 153929, 77359, 155509, 2521, 9241, 7177, 4289, 4691, 1021, 81359, 3803, 82171, 2677, 1949, 433, 84631, 4597, 86291, 176777, 88811, 178469, 89659, 1489, 90511, 5867, 91367, 183593, 92227, 991, 187049, 5527, 2659, 190537, 95707, 6203, 96587, 194057, 8861, 1733, 2399, 197609, 99251, 18127, 201193, 2731, 11941, 102859, 1627, 2531, 208457, 1861, 105607, 106531, 213989, 9769, 12697, 1013, 217717, 19963, 3557, 221477, 111211, 6037, 1901, 20479, 113111, 227177, 114067, 229093, 10457, 231017, 6823, 116959, 118907, 7703, 1187, 509, 120871, 242729, 523, 2287, 2857, 14629, 124847, 8087, 252709, 126859, 254729, 1801, 256757, 11717, 1889, 3511, 2393, 23899, 131959, 264949, 7823, 267017, 1327, 1439, 4357, 271177, 136111, 273269, 541, 3371, 4703, 139267, 279593, 12757, 281717, 4561, 3313, 25999, 8443, 288137, 3527, 145687, 2417, 294629, 147859, 1231, 8081, 1741, 9787, 13841, 17977, 153359, 569, 1367, 2459, 156707, 8501, 157831, 158959, 319049, 160091, 14657, 2143, 9551, 325877, 1447, 3067, 14969, 330469, 165811, 166967, 168127, 593, 339749, 2909, 20261, 172807, 346793, 15817, 3457, 2969, 351529, 5689, 16141, 1259, 4831, 179947, 1931, 1693, 613, 631, 365929, 183571, 33487, 4507, 370793, 186007, 1487, 17021, 12119, 188459, 378149, 189691, 1481, 3793, 35279, 194659, 393077, 2777, 35963, 198427, 398117, 18269, 641, 202231, 2687, 11971, 1453, 18617, 410857, 413429, 6689, 1523, 418597, 2957, 10273, 211247, 653, 12503, 213859, 25237, 431657, 5851, 434293, 10657, 1811, 10223, 3677, 5189, 224467, 450277, 41179, 227159, 14699, 228511, 458377, 20897, 461093, 2161, 232591, 21269, 4153, 15227, 236707, 2539, 5807, 477557, 2371, 480329, 240859, 1543, 243647, 6883, 22277, 491497, 6011, 29077, 4201, 499957, 250687, 4273, 253531, 508489, 254959, 3257, 256391, 46747, 8317, 7283, 1933, 30757, 525769, 263611, 2269, 24097, 4877, 13037, 7243, 48859, 540389, 270931, 272407, 49663, 16111, 549257, 275371, 13469, 25169, 555209, 7523, 558197, 279847, 25577, 564197, 282851, 567209, 51839, 285871, 33721, 287387, 576293, 288907, 52667, 9871, 1907, 26681, 588517, 4177, 34981, 597769, 17627, 2647, 301211, 54907, 302767, 607093, 9817, 610217, 2707, 7499, 616489, 309031, 5791, 622793, 4397, 36821, 7297, 6229, 28669, 5801, 316951, 14779, 2819, 1873, 823, 641897, 321751, 323359, 3467, 8783, 19211, 29837, 21227, 664549, 667817, 334727, 671093, 9091, 2179, 339659, 341311, 62207, 5813, 687593, 11117, 1823, 31481, 4013, 347959, 16223, 2753, 700937, 2437, 8209, 9967, 1747, 358051, 717797, 359747, 65563, 361447, 724597, 363151, 4637, 2699, 21563, 19861, 368287, 33637, 745189, 373459, 68059, 9151, 752117, 376927, 755593, 378667, 2423, 2053, 769577, 8969, 773093, 13163, 35381, 18143, 390959, 783689, 907, 394507, 790793, 23311, 2953, 12841, 401651, 805097, 36677, 808693, 405247, 812297, 410671, 412487, 6833, 24371, 7349, 5861, 417959, 76159, 419791, 22741, 19739, 4211, 25127, 856169, 50581, 863593, 432727, 2131, 6121, 2069, 436459, 28219, 10691, 79867, 440207, 5843, 52121, 40361, 889829, 26227, 893609, 7589, 8233, 901193, 451547, 904997, 453451, 82619, 1997, 457271, 916457, 27011, 83663, 1979, 54361, 463031, 7307, 983, 931849, 15061, 935717, 12671, 21851, 42797, 943477, 472711, 947369, 474659, 5087, 476611, 23297, 28151, 959093, 480527, 87547, 13619, 484459, 31319, 44221, 974837, 488407, 23873, 26561, 2633, 986729, 494359, 3623, 2917, 16927, 9371, 2213, 504359, 2927, 23599, 59929, 3251, 1022837, 9421, 46769, 1030949, 16661, 12647, 94463, 520567, 1043177, 3461, 10369, 4643, 8689, 30983, 2963,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+68x-151 and
the reducible primes which appear as divisor for the first time
p | x^2+68x-151 and p < x^2+68x-151

To avoid confusion with the number of primes:
I did not count the primes <= A
but I counted the primes appending the x and therefore the x <= A

ABCDEFGHIJK
exponent =log10 (x)<=xnumber of all primesnumber of primes p = f(x) number of primes p | f(x) C/xD/xE/xC(n) / C(n-1)D(n) / D(n-1)E(n) / E(n-1)
11011921.1000000.9000000.2000000.0000000.0000000.000000
21008358250.8300000.5800000.2500007.5454556.44444512.500000
31.0007733803930.7730000.3800000.3930009.3132536.55172415.720000
410.0007.4422.7424.7000.7442000.2742000.4700009.6274257.21578911.959288
5100.00073.35821.21852.1400.7335800.2121800.5214009.8572967.73814711.093617
61.000.000726.443173.568552.8750.7264430.1735680.5528759.9027108.18022410.603663
710.000.0007.212.9661.466.5165.746.4500.7212970.1466520.5746459.9291568.44923010.393760
8100.000.00071.766.02612.677.32959.088.6970.7176600.1267730.5908879.9495868.64452210.282643
91.000.000.000714.796.348111.716.000603.080.3480.7147960.1117160.6030809.9600948.81226610.206357
1010.000.000.0007.125.062.360998.792.4276.126.269.9330.7125060.0998790.6126279.9679618.94046010.158298


ABCDEFGHIJK
exponent =log2 (x)<=xnumber of all primesnumber of primes p = f(x) number of primes p | f(x) C/xD/xE/xC(n) / C(n-1)D(n) / D(n-1)E(n) / E(n-1)
123301.5000001.5000000.0000000.0000000.0000000.000000
245501.2500001.2500000.0000001.6666671.666667-nan
389811.1250001.0000000.1250001.8000001.600000inf
416161421.0000000.8750000.1250001.7777781.7500002.000000
532292450.9062500.7500000.1562501.8125001.7142862.500000
6645540150.8593750.6250000.2343751.8965521.6666673.000000
712810669370.8281250.5390620.2890621.9272731.7250002.466667
8256207128790.8085940.5000000.3085941.9528301.8550722.135135
95124062251810.7929690.4394530.3535161.9613531.7578122.291139
101.0247933864070.7744140.3769530.3974611.9532021.7155562.248619
112.0481.5607028580.7617190.3427730.4189451.9672131.8186532.108108
124.0963.0681.2701.7980.7490230.3100590.4389651.9666671.8091172.095571
138.1926.1122.2933.8190.7460940.2799070.4661871.9921771.8055122.124027
1416.38412.1654.2357.9300.7424930.2584840.4840091.9903471.8469252.076460
1532.76824.2107.84716.3630.7388310.2394710.4993591.9901361.8528932.063430
1665.53648.22014.49133.7290.7357790.2211150.5146641.9917391.8466932.061297
17131.07296.02727.13968.8880.7326280.2070540.5255741.9914351.8728182.042397
18262.144191.43551.001140.4340.7302670.1945530.5357131.9935541.8792512.038584
19524.288381.57295.995285.5770.7277910.1830960.5446951.9932201.8822182.033532
201.048.576761.546181.378580.1680.7262670.1729760.5532911.9958121.8894532.031564
212.097.1521.519.469344.0261.175.4430.7245390.1640440.5604951.9952431.8967352.026039
224.194.3043.032.527653.6782.378.8490.7230110.1558490.5671621.9957811.9000832.023789
238.388.6086.053.4541.245.2454.808.2090.7216280.1484450.5731831.9961751.9049822.021233
2416.777.21612.086.8432.375.7669.711.0770.7204320.1416070.5788251.9966851.9078702.019687
2533.554.43224.134.5594.544.42619.590.1330.7192660.1354340.5838311.9967631.9128262.017298
2667.108.86448.199.5368.713.56039.485.9760.7182290.1298420.5883871.9971171.9174172.015605
27134.217.72896.266.37416.727.73679.538.6380.7172400.1246310.5926091.9972471.9197362.014352
28268.435.456192.294.11332.170.445160.123.6680.7163510.1198440.5965071.9975211.9231802.013156
29536.870.912384.133.35661.969.103322.164.2530.7155040.1154260.6000781.9976341.9262742.011971
301.073.741.824767.419.417119.518.125647.901.2920.7147150.1113100.6034051.9977941.9286732.011090
312.147.483.6481.533.253.868230.825.2231.302.428.6450.7139770.1074860.6064911.9979351.9312992.010227
324.294.967.2963.063.544.445446.322.7532.617.221.6920.7132870.1039180.6093691.9980671.9335962.009493
338.589.934.5926.121.544.031864.003.5305.257.540.5010.7126420.1005830.6120581.9981901.9358272.008825
3417.179.869.18412.232.707.8571.674.225.07010.558.482.7870.7120370.0974530.6145851.9983041.9377532.008255
3534.359.738.36824.445.917.0593.247.458.03221.198.459.0270.7114700.0945130.6169561.9984061.9396782.007718
3668.719.476.73648.855.096.0596.304.819.17242.550.276.8870.7109350.0917470.6191881.9984971.9414632.007234


ABCDEFGHI
exponent =log2 (x) <=xnumber of primes with p=f(x) number of primes with p=f(x) and p%6=1 number of primes with p=f(x) and p%6=5 number of primes with p=f(x) and p%8=1 number of primes with p=f(x) and p%8=3 number of primes with p=f(x) and p%8=5 number of primes with p=f(x) and p%8=7
123121101
245232102
388353212
41614685324
5322413117647
6644023171311610
712869402920181417
8256128695933332933
951222511411160595155
101.0243861971891029587102
112.048702356346181180157184
124.0961.270637633331307305327
138.1922.2931.1641.129580571556586
1416.3844.2352.1322.1031.0411.0631.0371.094
1532.7687.8473.9603.8871.8951.9891.9452.018
1665.53614.4917.2867.2053.5193.6613.5683.743
17131.07227.13913.62913.5106.6536.8746.7256.887
18262.14451.00125.65425.34712.51512.93812.59512.953
19524.28895.99548.24247.75323.73424.28923.74324.229
201.048.576181.37890.99190.38744.80345.89644.77245.907
212.097.152344.026172.639171.38785.09587.14184.75587.035
224.194.304653.678328.213325.465161.694165.122161.276165.586
238.388.6081.245.245625.194620.051307.650314.646307.737315.212
2416.777.2162.375.7661.193.0661.182.700587.200600.617587.364600.585
2533.554.4324.544.4262.280.7642.263.6621.124.4661.148.0801.123.5081.148.372
2667.108.8648.713.5604.371.6234.341.9372.155.9752.200.3862.155.5002.201.699
27134.217.72816.727.7368.392.6918.335.0454.140.8534.222.6634.138.9364.225.284
28268.435.45632.170.44516.138.70616.031.7397.965.3318.120.5287.964.3448.120.242
29536.870.91261.969.10331.082.59630.886.50715.348.54115.635.73315.350.44915.634.380
301.073.741.824119.518.12559.944.10259.574.02329.611.87430.145.94129.615.91130.144.399
312.147.483.648230.825.223115.750.871115.074.35257.207.97058.202.73557.209.70558.204.813
324.294.967.296446.322.753223.783.991222.538.762110.659.074112.502.454110.651.881112.509.344
338.589.934.592864.003.530433.160.892430.842.638214.275.533217.723.213214.266.618217.738.166
3417.179.869.1841.674.225.070839.276.416834.948.654415.311.668421.802.361415.301.565421.809.476
3534.359.738.3683.247.458.0321.627.787.9421.619.670.090805.776.572817.984.878805.727.026817.969.556
3668.719.476.7366.304.819.1723.160.048.1973.144.770.9751.564.696.1571.587.712.5991.564.666.3241.587.744.092


ABCDEFGHI
exponent =log2 (x) <=xnumber of primes with p|f(x) number of primes with p=f(x) and p%6=1 number of primes with p=f(x) and p%6=5 number of primes with p=f(x) and p%8=1 number of primes with p=f(x) and p%8=3 number of primes with p=f(x) and p%8=5 number of primes with p=f(x) and p%8=7
120000000
240000000
381011000
4162111010
5325231112
66415782364
712837162198119
825679394017202220
9512181968541415247
101.0244071972109610511294
112.048858427431200222225211
124.0961.798877921440452457449
138.1923.8191.8851.934934947961977
1416.3847.9303.9643.9661.9671.9662.0101.987
1532.76816.3638.1688.1954.0624.0934.1034.105
1665.53633.72916.81716.9128.4068.4578.4478.419
17131.07268.88834.32234.56617.21017.28417.23317.161
18262.144140.43470.05770.37735.17735.12735.20334.927
19524.288285.577142.492143.08571.34871.35271.57471.303
201.048.576580.168289.663290.505145.037145.089145.230144.812
212.097.1521.175.443587.330588.113294.119293.568293.945293.811
224.194.3042.378.8491.189.2721.189.577594.370594.733595.069594.677
238.388.6084.808.2092.402.9972.405.2121.201.3131.202.4091.202.5491.201.938
2416.777.2169.711.0774.852.9324.858.1452.428.4872.427.2662.428.5162.426.808
2533.554.43219.590.1339.789.3039.800.8304.900.4664.896.7424.897.2174.895.708
2667.108.86439.485.97619.734.57619.751.4009.873.2299.870.3579.872.8779.869.513
27134.217.72879.538.63839.756.37739.782.26119.889.41319.879.71219.888.97819.880.535
28268.435.456160.123.66880.043.20980.080.45940.042.18040.016.67840.042.79740.022.013
29536.870.912322.164.253161.053.532161.110.72180.556.64180.521.97480.558.95880.526.680
301.073.741.824647.901.292323.883.510324.017.782162.010.018161.943.133162.007.028161.941.113
312.147.483.6481.302.428.645651.091.225651.337.420325.677.280325.539.482325.649.432325.562.451
324.294.967.2962.617.221.6921.308.391.9051.308.829.787654.430.477654.201.019654.398.943654.191.253
338.589.934.5925.257.540.5012.628.394.1082.629.146.3931.314.560.6691.314.194.7691.314.597.0711.314.187.992
3417.179.869.18410.558.482.7875.278.559.1525.279.923.6352.640.006.9882.639.253.3852.639.974.0492.639.248.365
3534.359.738.36821.198.459.02710.597.970.93610.600.488.0915.300.353.6475.298.888.0325.300.310.6935.298.906.655
3668.719.476.73642.550.276.88721.272.870.04421.277.406.84310.638.980.59910.636.150.29710.638.868.43510.636.277.556


8. Check for existing Integer Sequences by OEIS

Found in Database : 151, 41, 11, 31, 137, 107, 293, 17, 457, 271, 37, 359, 809, 1, 997, 547, 1193, 647, 127, 751,
Found in Database : 151, 41, 11, 31, 137, 107, 293, 17, 457, 271, 37, 359, 809, 997, 547, 1193, 647, 127, 751, 1609, 859, 59, 971, 1087, 2293, 71, 43, 2789, 1459, 3049, 157, 3593, 1867, 3877, 2011,
Found in Database : 11, 17, 31, 37, 41, 43, 59, 71, 101, 107, 109, 113, 127, 137,