Inhaltsverzeichnis

Development of
Algorithmic Constructions

01:00:29
Deutsch
29.Mar 2024

Polynom = x^2+9x-83

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) = 83 = 83
f(1) = 73 = 73
f(2) = 61 = 61
f(3) = 47 = 47
f(4) = 31 = 31
f(5) = 13 = 13
f(6) = 7 = 7
f(7) = 29 = 29
f(8) = 53 = 53
f(9) = 79 = 79
f(10) = 107 = 107
f(11) = 137 = 137
f(12) = 169 = 13*13
f(13) = 203 = 7*29
f(14) = 239 = 239
f(15) = 277 = 277
f(16) = 317 = 317
f(17) = 359 = 359
f(18) = 403 = 13*31
f(19) = 449 = 449
f(20) = 497 = 7*71
f(21) = 547 = 547
f(22) = 599 = 599
f(23) = 653 = 653
f(24) = 709 = 709
f(25) = 767 = 13*59
f(26) = 827 = 827
f(27) = 889 = 7*127
f(28) = 953 = 953
f(29) = 1019 = 1019
f(30) = 1087 = 1087
f(31) = 1157 = 13*89
f(32) = 1229 = 1229
f(33) = 1303 = 1303
f(34) = 1379 = 7*197
f(35) = 1457 = 31*47
f(36) = 1537 = 29*53
f(37) = 1619 = 1619
f(38) = 1703 = 13*131
f(39) = 1789 = 1789
f(40) = 1877 = 1877
f(41) = 1967 = 7*281
f(42) = 2059 = 29*71
f(43) = 2153 = 2153
f(44) = 2249 = 13*173
f(45) = 2347 = 2347
f(46) = 2447 = 2447
f(47) = 2549 = 2549
f(48) = 2653 = 7*379
f(49) = 2759 = 31*89
f(50) = 2867 = 47*61
f(51) = 2977 = 13*229
f(52) = 3089 = 3089
f(53) = 3203 = 3203
f(54) = 3319 = 3319
f(55) = 3437 = 7*491
f(56) = 3557 = 3557
f(57) = 3679 = 13*283
f(58) = 3803 = 3803
f(59) = 3929 = 3929
f(60) = 4057 = 4057
f(61) = 4187 = 53*79
f(62) = 4319 = 7*617
f(63) = 4453 = 61*73
f(64) = 4589 = 13*353
f(65) = 4727 = 29*163
f(66) = 4867 = 31*157
f(67) = 5009 = 5009
f(68) = 5153 = 5153
f(69) = 5299 = 7*757
f(70) = 5447 = 13*419
f(71) = 5597 = 29*193
f(72) = 5749 = 5749
f(73) = 5903 = 5903
f(74) = 6059 = 73*83
f(75) = 6217 = 6217
f(76) = 6377 = 7*911
f(77) = 6539 = 13*503
f(78) = 6703 = 6703
f(79) = 6869 = 6869
f(80) = 7037 = 31*227
f(81) = 7207 = 7207
f(82) = 7379 = 47*157
f(83) = 7553 = 7*13*83
f(84) = 7729 = 59*131
f(85) = 7907 = 7907
f(86) = 8087 = 8087
f(87) = 8269 = 8269
f(88) = 8453 = 79*107
f(89) = 8639 = 53*163
f(90) = 8827 = 7*13*97
f(91) = 9017 = 71*127
f(92) = 9209 = 9209
f(93) = 9403 = 9403
f(94) = 9599 = 29*331
f(95) = 9797 = 97*101
f(96) = 9997 = 13*769
f(97) = 10199 = 7*31*47
f(98) = 10403 = 101*103
f(99) = 10609 = 103*103
f(100) = 10817 = 29*373

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+9x-83

f(0)=83
f(1)=73
f(2)=61
f(3)=47
f(4)=31
f(5)=13
f(6)=7
f(7)=29
f(8)=53
f(9)=79
f(10)=107
f(11)=137
f(12)=1
f(13)=1
f(14)=239
f(15)=277
f(16)=317
f(17)=359
f(18)=1
f(19)=449
f(20)=71
f(21)=547
f(22)=599
f(23)=653
f(24)=709
f(25)=59
f(26)=827
f(27)=127
f(28)=953
f(29)=1019
f(30)=1087
f(31)=89
f(32)=1229
f(33)=1303
f(34)=197
f(35)=1
f(36)=1
f(37)=1619
f(38)=131
f(39)=1789
f(40)=1877
f(41)=281
f(42)=1
f(43)=2153
f(44)=173
f(45)=2347
f(46)=2447
f(47)=2549
f(48)=379
f(49)=1
f(50)=1
f(51)=229
f(52)=3089
f(53)=3203
f(54)=3319
f(55)=491
f(56)=3557
f(57)=283
f(58)=3803
f(59)=3929
f(60)=4057
f(61)=1
f(62)=617
f(63)=1
f(64)=353
f(65)=163
f(66)=157
f(67)=5009
f(68)=5153
f(69)=757
f(70)=419
f(71)=193
f(72)=5749
f(73)=5903
f(74)=1
f(75)=6217
f(76)=911
f(77)=503
f(78)=6703
f(79)=6869
f(80)=227
f(81)=7207
f(82)=1
f(83)=1
f(84)=1
f(85)=7907
f(86)=8087
f(87)=8269
f(88)=1
f(89)=1
f(90)=97
f(91)=1
f(92)=9209
f(93)=9403
f(94)=331
f(95)=101
f(96)=769
f(97)=1
f(98)=103
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+9x-83 could be written as f(y)= y^2-103.25 with x=y-4.5

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+4.5
f'(x)>2x+8

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

83, 73, 61, 47, 31, 13, 7, 29, 53, 79, 107, 137, 1, 1, 239, 277, 317, 359, 1, 449, 71, 547, 599, 653, 709, 59, 827, 127, 953, 1019, 1087, 89, 1229, 1303, 197, 1, 1, 1619, 131, 1789, 1877, 281, 1, 2153, 173, 2347, 2447, 2549, 379, 1, 1, 229, 3089, 3203, 3319, 491, 3557, 283, 3803, 3929, 4057, 1, 617, 1, 353, 163, 157, 5009, 5153, 757, 419, 193, 5749, 5903, 1, 6217, 911, 503, 6703, 6869, 227, 7207, 1, 1, 1, 7907, 8087, 8269, 1, 1, 97, 1, 9209, 9403, 331, 101, 769, 1, 103, 1, 373, 11027, 11239, 881, 1667, 11887, 12107, 12329, 12553, 983, 13007, 1, 13469, 1, 263, 14177, 1109, 1, 2129, 15149, 1, 15647, 1223, 557, 269, 2381, 16927, 17189, 563, 1, 17987, 18257, 2647, 18803, 19079, 1489, 1, 19919, 1, 2927, 1, 21067, 1, 367, 467, 22247, 3221, 313, 1, 23459, 23767, 24077, 1, 3529, 1, 1949, 25657, 1, 907, 859, 3851, 2099, 389, 27953, 28289, 28627, 349, 1, 2281, 1, 30347, 30697, 509, 1013, 1, 32117, 691, 32839, 33203, 33569, 33937, 1, 34679, 35053, 499, 587, 36187, 1, 5279, 37339, 1217, 811, 397, 38903, 3023, 1, 1, 40499, 40903, 409, 3209, 1, 1, 42953, 1399, 43787, 44207, 3433, 45053, 1, 1583, 46337, 46769, 3631, 47639, 1, 1, 1, 1, 631, 1, 1637, 51199, 1, 487, 52567, 4079, 601, 1, 54419, 7841, 1, 55829, 1, 56779, 1847, 57737, 8317, 1249, 1, 719, 60167, 60659, 61153, 8807, 2143, 1, 63149, 1201, 773, 64667, 9311, 1, 1, 1, 947, 67757, 68279, 9829, 5333, 69857, 1193, 70919, 71453, 1, 797, 2357, 73609, 2557, 74699, 1601, 75797, 839, 1451, 2671, 78017, 78577, 79139, 6131, 11467, 1, 641, 1123, 2663, 857, 1, 12041, 84869, 85453, 887, 86627, 6709, 1, 1, 1459, 89597, 90197, 1, 1, 92009, 1, 1759, 877, 3257, 1, 929, 1, 1, 97553, 98179, 98807, 7649, 100069, 100703, 1, 101977, 1153, 1, 103903, 104549, 1783, 15121, 1, 1291, 8293, 1373, 1, 109789, 1, 111119, 8599, 1051, 1, 1, 114479, 16451, 115837, 8963, 117203, 117889, 1, 119267, 17137, 9281, 121349, 1, 937, 123449, 124153, 17837, 743, 1, 126989, 127703, 2423, 1, 1427, 130579, 131303, 4259, 132757, 4603, 134219, 1483, 2887, 1, 1879, 137909, 2273, 10723, 20021, 140897, 141649, 142403, 143159, 2027, 1, 1, 146203, 1, 941, 1531, 11483, 150053, 1, 151607, 1571, 1721, 153953, 11903, 1, 1, 157109, 1, 158699, 12269, 1021, 2269, 1, 162709, 163517, 1627, 12703, 1, 2111, 1, 2029, 1, 1, 1597, 171707, 3671, 24767, 6007, 175039, 1, 2897, 1, 178403, 883, 180097, 1, 1327, 182653, 183509, 1, 1, 1, 1, 3079, 188687, 1447, 3593, 27329, 14783, 193057, 193937, 194819, 1, 2693, 28211, 15259, 6871, 200153, 201049, 201947, 2857, 2239, 7057, 205559, 206467, 207377, 6719, 209203, 2309, 1, 2683, 1103, 4549, 214729, 1, 30941, 217519, 218453, 219389, 220327, 1279, 17093, 1, 7229, 1, 7793, 1787, 1, 228859, 32831, 2593, 1, 2399, 233669, 18049, 235607, 33797, 1, 238529, 4519, 1423, 241469, 242453, 1, 4007, 245417, 1031, 19031, 2791, 249397, 35771, 4261, 3041, 1, 1, 3499, 2539, 36779, 258469, 5521, 1, 261529, 262553, 1151, 1, 2579, 1, 5051, 8669, 1, 2531, 1, 272903, 1621, 5851, 276047, 277099, 1, 39887, 21559, 281327, 282389, 2069, 9811, 1, 1321, 22133, 288803, 1277, 1, 292037, 1867, 1, 4159, 296377, 297467, 298559, 299653, 300749, 1, 1, 304049, 1549, 306259, 307367, 1, 1, 1979, 1913, 312937, 5323, 315179, 1, 1, 318557, 319687, 1, 1861, 1, 3643, 1, 6947, 1, 1187, 329947, 25469, 4003, 47629, 4583, 3461, 10867, 26003, 2081, 340369, 1, 4129, 1, 3877, 26633, 4759, 348587, 1723, 4943, 1, 27179, 6689, 1567, 1, 51157, 359297, 360497, 27823, 362903, 1, 3617, 52361, 367739, 1, 370169, 371387, 372607, 1, 1, 1, 1, 1, 6229, 3701, 1, 1, 1, 386159, 387403, 7333, 389897, 1, 1933, 1, 12739, 2011, 397427, 1741, 399953, 4409, 402487, 403757, 405029, 1, 407579, 408857, 4507, 1, 13313, 2393, 1579, 5273, 32143, 59879, 7933, 14543, 9001, 5813, 425653, 32843, 1, 14813, 4027, 432203, 4871, 1, 436157, 62497, 6011, 6199, 7237, 9421, 1, 445453, 1, 3271, 1709, 450809, 34781, 14629, 454847, 65171, 1, 7523, 1, 35509, 1, 4787, 66529, 1, 1, 1, 471179, 1, 473929, 67901, 15377, 478069, 2837, 480839, 482227, 1, 1, 1, 1, 489197, 3581, 9283, 493403, 70687, 16007, 1, 499039, 17257, 4969, 503287, 72101, 38933, 10799, 17551, 3251, 1, 513269, 73529, 39703, 517577, 5039, 1, 521903, 523349, 1, 8627, 527699, 529153, 530609, 10039, 4201, 5879, 536453, 537919, 1, 1, 18701, 1, 1, 1523, 3169, 549719, 1, 4219, 1, 1, 557159, 6277, 560149, 2851, 43319, 6803, 2609, 567659, 10739, 570677, 8059, 44131, 575219, 1753, 1747, 579779, 581303, 1, 584357, 1, 2707, 588953, 590489, 592027, 45659, 20521, 596653, 1, 1699, 601297, 1, 1931, 6247, 607517, 87011, 610639, 11551, 1, 615337, 1, 10139, 1, 621629, 47939, 624787, 13327, 627953, 4957, 3109, 632717, 1, 1, 637499, 10477, 22093, 91757, 49531, 8171, 647117, 9137, 1, 12301, 1, 1, 1733, 2377, 660053, 661679, 21397, 7307, 3853, 668203, 669839, 671477, 673117, 674759, 7433, 1, 6599, 8209, 3467, 9643, 1, 3389, 1, 691289, 1, 694619, 8389, 1, 1, 5119, 14957, 704657, 706337, 1, 709703, 101627, 713077, 23057, 716459, 1, 55373, 721547, 1, 724949, 25057, 15497, 1, 1, 733489, 1, 25411, 23827, 1, 4391, 743803, 745529, 106751, 748987, 750719, 57881, 2161, 1, 1, 1, 761153, 762899, 1, 7901, 1, 769903, 110237, 773417, 59629, 1, 9857, 780469, 4799, 2383, 785779, 2089, 1, 4099, 25577, 794669, 113779, 61403, 2417, 801817, 803609, 805403, 11369, 115571, 1, 812599, 814403, 816209, 818017, 819827, 9029, 1, 1, 827087, 1, 830729, 1987, 1, 14173, 838037, 28961, 10141, 843539, 65029, 1, 1, 1, 852749, 1, 9623, 1, 1, 862009, 863867, 2731, 867589, 1, 3313, 1, 3253, 876929, 3677, 28409, 2341, 884437, 2389, 888203, 1, 2293, 2371, 4547, 1, 1, 901447, 903347, 905249, 2251, 2477, 12479, 130411, 914789, 916703, 70663, 7027, 922457, 11701, 132329, 1, 930157, 71699, 17623, 9649, 32341, 1, 1, 1, 9749, 16061, 1, 951497, 136207, 8929, 73643, 20411, 961277, 963239, 15823, 4457, 1, 1, 1, 975053, 1, 5659, 1, 1, 4339, 1, 1, 990917, 1, 1, 996899, 998897, 1, 1002899, 1004903, 34721,

6. Sequence of the polynom (only primes)

83, 73, 61, 47, 31, 13, 7, 29, 53, 79, 107, 137, 239, 277, 317, 359, 449, 71, 547, 599, 653, 709, 59, 827, 127, 953, 1019, 1087, 89, 1229, 1303, 197, 1619, 131, 1789, 1877, 281, 2153, 173, 2347, 2447, 2549, 379, 229, 3089, 3203, 3319, 491, 3557, 283, 3803, 3929, 4057, 617, 353, 163, 157, 5009, 5153, 757, 419, 193, 5749, 5903, 6217, 911, 503, 6703, 6869, 227, 7207, 7907, 8087, 8269, 97, 9209, 9403, 331, 101, 769, 103, 373, 11027, 11239, 881, 1667, 11887, 12107, 12329, 12553, 983, 13007, 13469, 263, 14177, 1109, 2129, 15149, 15647, 1223, 557, 269, 2381, 16927, 17189, 563, 17987, 18257, 2647, 18803, 19079, 1489, 19919, 2927, 21067, 367, 467, 22247, 3221, 313, 23459, 23767, 24077, 3529, 1949, 25657, 907, 859, 3851, 2099, 389, 27953, 28289, 28627, 349, 2281, 30347, 30697, 509, 1013, 32117, 691, 32839, 33203, 33569, 33937, 34679, 35053, 499, 587, 36187, 5279, 37339, 1217, 811, 397, 38903, 3023, 40499, 40903, 409, 3209, 42953, 1399, 43787, 44207, 3433, 45053, 1583, 46337, 46769, 3631, 47639, 631, 1637, 51199, 487, 52567, 4079, 601, 54419, 7841, 55829, 56779, 1847, 57737, 8317, 1249, 719, 60167, 60659, 61153, 8807, 2143, 63149, 1201, 773, 64667, 9311, 947, 67757, 68279, 9829, 5333, 69857, 1193, 70919, 71453, 797, 2357, 73609, 2557, 74699, 1601, 75797, 839, 1451, 2671, 78017, 78577, 79139, 6131, 11467, 641, 1123, 2663, 857, 12041, 84869, 85453, 887, 86627, 6709, 1459, 89597, 90197, 92009, 1759, 877, 3257, 929, 97553, 98179, 98807, 7649, 100069, 100703, 101977, 1153, 103903, 104549, 1783, 15121, 1291, 8293, 1373, 109789, 111119, 8599, 1051, 114479, 16451, 115837, 8963, 117203, 117889, 119267, 17137, 9281, 121349, 937, 123449, 124153, 17837, 743, 126989, 127703, 2423, 1427, 130579, 131303, 4259, 132757, 4603, 134219, 1483, 2887, 1879, 137909, 2273, 10723, 20021, 140897, 141649, 142403, 143159, 2027, 146203, 941, 1531, 11483, 150053, 151607, 1571, 1721, 153953, 11903, 157109, 158699, 12269, 1021, 2269, 162709, 163517, 1627, 12703, 2111, 2029, 1597, 171707, 3671, 24767, 6007, 175039, 2897, 178403, 883, 180097, 1327, 182653, 183509, 3079, 188687, 1447, 3593, 27329, 14783, 193057, 193937, 194819, 2693, 28211, 15259, 6871, 200153, 201049, 201947, 2857, 2239, 7057, 205559, 206467, 207377, 6719, 209203, 2309, 2683, 1103, 4549, 214729, 30941, 217519, 218453, 219389, 220327, 1279, 17093, 7229, 7793, 1787, 228859, 32831, 2593, 2399, 233669, 18049, 235607, 33797, 238529, 4519, 1423, 241469, 242453, 4007, 245417, 1031, 19031, 2791, 249397, 35771, 4261, 3041, 3499, 2539, 36779, 258469, 5521, 261529, 262553, 1151, 2579, 5051, 8669, 2531, 272903, 1621, 5851, 276047, 277099, 39887, 21559, 281327, 282389, 2069, 9811, 1321, 22133, 288803, 1277, 292037, 1867, 4159, 296377, 297467, 298559, 299653, 300749, 304049, 1549, 306259, 307367, 1979, 1913, 312937, 5323, 315179, 318557, 319687, 1861, 3643, 6947, 1187, 329947, 25469, 4003, 47629, 4583, 3461, 10867, 26003, 2081, 340369, 4129, 3877, 26633, 4759, 348587, 1723, 4943, 27179, 6689, 1567, 51157, 359297, 360497, 27823, 362903, 3617, 52361, 367739, 370169, 371387, 372607, 6229, 3701, 386159, 387403, 7333, 389897, 1933, 12739, 2011, 397427, 1741, 399953, 4409, 402487, 403757, 405029, 407579, 408857, 4507, 13313, 2393, 1579, 5273, 32143, 59879, 7933, 14543, 9001, 5813, 425653, 32843, 14813, 4027, 432203, 4871, 436157, 62497, 6011, 6199, 7237, 9421, 445453, 3271, 1709, 450809, 34781, 14629, 454847, 65171, 7523, 35509, 4787, 66529, 471179, 473929, 67901, 15377, 478069, 2837, 480839, 482227, 489197, 3581, 9283, 493403, 70687, 16007, 499039, 17257, 4969, 503287, 72101, 38933, 10799, 17551, 3251, 513269, 73529, 39703, 517577, 5039, 521903, 523349, 8627, 527699, 529153, 530609, 10039, 4201, 5879, 536453, 537919, 18701, 1523, 3169, 549719, 4219, 557159, 6277, 560149, 2851, 43319, 6803, 2609, 567659, 10739, 570677, 8059, 44131, 575219, 1753, 1747, 579779, 581303, 584357, 2707, 588953, 590489, 592027, 45659, 20521, 596653, 1699, 601297, 1931, 6247, 607517, 87011, 610639, 11551, 615337, 10139, 621629, 47939, 624787, 13327, 627953, 4957, 3109, 632717, 637499, 10477, 22093, 91757, 49531, 8171, 647117, 9137, 12301, 1733, 2377, 660053, 661679, 21397, 7307, 3853, 668203, 669839, 671477, 673117, 674759, 7433, 6599, 8209, 3467, 9643, 3389, 691289, 694619, 8389, 5119, 14957, 704657, 706337, 709703, 101627, 713077, 23057, 716459, 55373, 721547, 724949, 25057, 15497, 733489, 25411, 23827, 4391, 743803, 745529, 106751, 748987, 750719, 57881, 2161, 761153, 762899, 7901, 769903, 110237, 773417, 59629, 9857, 780469, 4799, 2383, 785779, 2089, 4099, 25577, 794669, 113779, 61403, 2417, 801817, 803609, 805403, 11369, 115571, 812599, 814403, 816209, 818017, 819827, 9029, 827087, 830729, 1987, 14173, 838037, 28961, 10141, 843539, 65029, 852749, 9623, 862009, 863867, 2731, 867589, 3313, 3253, 876929, 3677, 28409, 2341, 884437, 2389, 888203, 2293, 2371, 4547, 901447, 903347, 905249, 2251, 2477, 12479, 130411, 914789, 916703, 70663, 7027, 922457, 11701, 132329, 930157, 71699, 17623, 9649, 32341, 9749, 16061, 951497, 136207, 8929, 73643, 20411, 961277, 963239, 15823, 4457, 975053, 5659, 4339, 990917, 996899, 998897, 1002899, 1004903, 34721,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+9x-83 and
the reducible primes which appear as divisor for the first time
p | x^2+9x-83 and p < x^2+9x-83

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)
110111101.1000001.1000001.1000000.0000000.0000000.000000
21007955240.7900000.5500000.7900007.1818185.000000inf
31.0007623504120.7620000.3500000.7620009.6455706.36363617.166666
410.0007.4162.4174.9990.7416000.2417000.7416009.7322846.90571512.133495
5100.00073.44218.77354.6690.7344200.1877300.7344209.9031827.76706610.935987
61.000.000727.502152.825574.6770.7275020.1528250.7275029.9058048.14068010.511935
710.000.0007.225.0871.291.8035.933.2840.7225090.1291800.7225099.9313648.45282610.324554
8100.000.00071.858.74911.203.03860.655.7110.7185880.1120300.7185889.9457288.67240410.222958
91.000.000.000715.594.31898.881.147616.713.1710.7155940.0988810.7155949.9583458.82628010.167438
1010.000.000.0007.132.546.919884.927.0386.247.619.8810.7132550.0884930.7132559.9673058.94940110.130512


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
389901.1250001.1250000.0000001.8000001.800000-nan
416151500.9375000.9375000.0000001.6666671.666667-nan
532302730.9375000.8437500.0937502.0000001.800000inf
6645542130.8593750.6562500.2031251.8333331.5555564.333333
712810368350.8046880.5312500.2734381.8727271.6190482.692308
8256198115830.7734380.4492190.3242191.9223301.6911762.371428
95123982011970.7773440.3925780.3847662.0101011.7478262.373494
101.0247803554250.7617190.3466800.4150391.9597991.7661692.157360
112.0481.5456279180.7543950.3061520.4482421.9807691.7661972.160000
124.0963.0551.1141.9410.7458500.2719730.4738771.9773461.7767152.114379
138.1926.0982.0224.0760.7443850.2468260.4975591.9960721.8150812.099948
1416.38412.1503.7008.4500.7415770.2258300.5157471.9924571.8298712.073111
1532.76824.1906.87017.3200.7382200.2096560.5285641.9909471.8567572.049704
1665.53648.23912.79635.4430.7360690.1952510.5408171.9941711.8625912.046363
17131.07296.16123.89472.2670.7336500.1822970.5513531.9934291.8673022.038964
18262.144191.84244.756147.0860.7318190.1707310.5610891.9950081.8731062.035314
19524.288382.43884.515297.9230.7294430.1612000.5682431.9935051.8883502.025502
201.048.576762.769159.665603.1040.7274330.1522680.5751651.9944911.8891912.024362
212.097.1521.522.255302.3051.219.9500.7258680.1441500.5817171.9956961.8933712.022785
224.194.3043.037.862575.5652.462.2970.7242830.1372250.5870571.9956331.9039212.018359
238.388.6086.063.7751.096.7664.967.0090.7228580.1307450.5921141.9960671.9055472.017226
2416.777.21612.103.9062.096.92310.006.9830.7214490.1249860.5964631.9961011.9119152.014690
2533.554.43224.168.1934.013.10720.155.0860.7202680.1196000.6006681.9967271.9138082.014102
2667.108.86448.262.8357.696.85240.565.9830.7191720.1146920.6044801.9969571.9179282.012692
27134.217.72896.389.77014.786.83281.602.9380.7181600.1101700.6079891.9971841.9211532.011610
28268.435.456192.525.96128.447.878164.078.0830.7172150.1059770.6112381.9973691.9238662.010689
29536.870.912384.576.61054.824.261329.752.3490.7163300.1021180.6142121.9975311.9271832.009728
301.073.741.824768.278.876105.789.158662.489.7180.7155150.0985240.6169921.9977261.9296052.009052
312.147.483.6481.534.941.065204.382.2401.330.558.8250.7147630.0951730.6195901.9978961.9319772.008422
324.294.967.2963.066.850.957395.313.7122.671.537.2450.7140570.0920410.6220161.9980251.9341882.007831
338.589.934.5926.128.003.223765.439.2785.362.563.9450.7133930.0891090.6242851.9981421.9362832.007295
3417.179.869.18412.245.337.5481.483.642.79610.761.694.7520.7127720.0863590.6264131.9982591.9382892.006819


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
123211110
245321112
389541143
41615782265
5322710174599
664421527891312
712868254315131921
8256115407527282634
95122017013151465351
101.02435511723892879383
112.048627209418160158160149
124.0961.114371743279295278262
138.1922.0226781.344500527501494
1416.3843.7001.2452.455921933920926
1532.7686.8702.3024.5681.7261.7381.7031.703
1665.53612.7964.2908.5063.2353.2173.2073.137
17131.07223.8947.96515.9296.0475.9815.9635.903
18262.14444.75614.92429.83211.24111.21811.10511.192
19524.28884.51528.17756.33821.13521.18121.07921.120
201.048.576159.66553.360106.30539.85039.95539.95839.902
212.097.152302.305100.798201.50775.53975.46575.57275.729
224.194.304575.565191.834383.731143.863143.748143.851144.103
238.388.6081.096.766365.876730.890274.002273.865274.337274.562
2416.777.2162.096.923699.4421.397.481523.801523.930524.364524.828
2533.554.4324.013.1071.338.1642.674.9431.003.0841.002.1611.003.5961.004.266
2667.108.8647.696.8522.566.7355.130.1171.923.9771.923.0431.925.0801.924.752
27134.217.72814.786.8324.928.4649.858.3683.697.7163.694.2493.697.7573.697.110
28268.435.45628.447.8789.482.25218.965.6267.111.8707.108.9347.113.9277.113.147
29536.870.91254.824.26118.272.42236.551.83913.703.74313.702.65713.709.04413.708.817
301.073.741.824105.789.15835.263.55070.525.60826.440.77726.446.03926.450.86726.451.475
312.147.483.648204.382.24068.129.527136.252.71351.092.80651.095.43551.097.16051.096.839
324.294.967.296395.313.712131.777.078263.536.63498.829.96198.825.77698.825.57998.832.396
338.589.934.592765.439.278255.142.988510.296.290191.366.210191.355.361191.361.649191.356.058
3417.179.869.1841.483.642.796494.537.839989.104.957370.920.200370.904.912370.904.680370.913.004


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
380000000
4160000000
5323121002
66413494432
7128351124810107
825683354820202023
95121978711045494558
101.02442520222396117112100
112.048918427491222238228230
124.0961.9419281.013482515462482
138.1924.0761.9922.0841.0171.0361.0181.005
1416.3848.4504.0754.3752.1192.1322.1072.092
1532.76817.3208.3898.9314.3504.3684.3384.264
1665.53635.44317.26218.1818.9008.9488.8458.750
17131.07272.26735.23937.02818.07118.12518.11317.958
18262.144147.08671.86875.21836.57536.84036.96636.705
19524.288297.923145.972151.95174.46774.40374.71474.339
201.048.576603.104295.486307.618150.894150.637150.698150.875
212.097.1521.219.950598.158621.792305.033304.949304.652305.316
224.194.3042.462.2971.210.6901.251.607615.762615.496615.184615.855
238.388.6084.967.0092.441.6732.525.3361.242.5211.241.1791.240.9921.242.317
2416.777.21610.006.9834.925.4695.081.5142.503.9632.500.1292.500.0502.502.841
2533.554.43220.155.0869.930.18510.224.9015.040.9305.039.0325.036.4445.038.680
2667.108.86440.565.98319.998.59220.567.39110.143.56110.141.73610.139.53810.141.148
27134.217.72881.602.93840.250.12341.352.81520.401.72620.399.57020.399.30420.402.338
28268.435.456164.078.08380.987.87083.090.21341.024.54641.016.24141.017.83941.019.457
29536.870.912329.752.349162.853.956166.898.39382.440.75082.438.47882.435.50582.437.616
301.073.741.824662.489.718327.338.136335.151.582165.612.752165.622.958165.620.836165.633.172
312.147.483.6481.330.558.825657.740.927672.817.898332.617.110332.647.533332.625.638332.668.544
324.294.967.2962.671.537.2451.321.203.5031.350.333.742667.855.236667.891.856667.897.098667.893.055
338.589.934.5925.362.563.9452.653.084.5692.709.479.3761.340.638.9201.340.647.6191.340.627.3661.340.650.040
3417.179.869.18410.761.694.7525.326.133.1485.435.561.6042.690.432.0912.690.434.2852.690.435.2282.690.393.148


8. Check for existing Integer Sequences by OEIS

Found in Database : 83, 73, 61, 47, 31, 13, 7, 29, 53, 79, 107, 137, 1, 1, 239, 277, 317, 359, 1, 449,
Found in Database : 83, 73, 61, 47, 31, 13, 7, 29, 53, 79, 107, 137, 239, 277, 317, 359, 449, 71, 547, 599, 653, 709, 59, 827, 127, 953, 1019, 1087, 89, 1229, 1303, 197, 1619, 131, 1789,
Found in Database : 7, 13, 29, 31, 47, 53, 59, 61, 71, 73, 79, 83, 89, 97, 101, 103, 107, 127, 131, 137,