Inhaltsverzeichnis

Development of
Algorithmic Constructions

12:06:02
Deutsch
29.Mar 2024

Polynom = x^2-44x+17

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) = 17 = 17
f(1) = 13 = 13
f(2) = 67 = 67
f(3) = 53 = 53
f(4) = 143 = 11*13
f(5) = 89 = 89
f(6) = 211 = 211
f(7) = 121 = 11*11
f(8) = 271 = 271
f(9) = 149 = 149
f(10) = 323 = 17*19
f(11) = 173 = 173
f(12) = 367 = 367
f(13) = 193 = 193
f(14) = 403 = 13*31
f(15) = 209 = 11*19
f(16) = 431 = 431
f(17) = 221 = 13*17
f(18) = 451 = 11*41
f(19) = 229 = 229
f(20) = 463 = 463
f(21) = 233 = 233
f(22) = 467 = 467
f(23) = 233 = 233
f(24) = 463 = 463
f(25) = 229 = 229
f(26) = 451 = 11*41
f(27) = 221 = 13*17
f(28) = 431 = 431
f(29) = 209 = 11*19
f(30) = 403 = 13*31
f(31) = 193 = 193
f(32) = 367 = 367
f(33) = 173 = 173
f(34) = 323 = 17*19
f(35) = 149 = 149
f(36) = 271 = 271
f(37) = 121 = 11*11
f(38) = 211 = 211
f(39) = 89 = 89
f(40) = 143 = 11*13
f(41) = 53 = 53
f(42) = 67 = 67
f(43) = 13 = 13
f(44) = 17 = 17
f(45) = 31 = 31
f(46) = 109 = 109
f(47) = 79 = 79
f(48) = 209 = 11*19
f(49) = 131 = 131
f(50) = 317 = 317
f(51) = 187 = 11*17
f(52) = 433 = 433
f(53) = 247 = 13*19
f(54) = 557 = 557
f(55) = 311 = 311
f(56) = 689 = 13*53
f(57) = 379 = 379
f(58) = 829 = 829
f(59) = 451 = 11*41
f(60) = 977 = 977
f(61) = 527 = 17*31
f(62) = 1133 = 11*103
f(63) = 607 = 607
f(64) = 1297 = 1297
f(65) = 691 = 691
f(66) = 1469 = 13*113
f(67) = 779 = 19*41
f(68) = 1649 = 17*97
f(69) = 871 = 13*67
f(70) = 1837 = 11*167
f(71) = 967 = 967
f(72) = 2033 = 19*107
f(73) = 1067 = 11*97
f(74) = 2237 = 2237
f(75) = 1171 = 1171
f(76) = 2449 = 31*79
f(77) = 1279 = 1279
f(78) = 2669 = 17*157
f(79) = 1391 = 13*107
f(80) = 2897 = 2897
f(81) = 1507 = 11*137
f(82) = 3133 = 13*241
f(83) = 1627 = 1627
f(84) = 3377 = 11*307
f(85) = 1751 = 17*103
f(86) = 3629 = 19*191
f(87) = 1879 = 1879
f(88) = 3889 = 3889
f(89) = 2011 = 2011
f(90) = 4157 = 4157
f(91) = 2147 = 19*113
f(92) = 4433 = 11*13*31
f(93) = 2287 = 2287
f(94) = 4717 = 53*89
f(95) = 2431 = 11*13*17
f(96) = 5009 = 5009
f(97) = 2579 = 2579
f(98) = 5309 = 5309
f(99) = 2731 = 2731
f(100) = 5617 = 41*137

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^2-44x+17

f(0)=17
f(1)=13
f(2)=67
f(3)=53
f(4)=11
f(5)=89
f(6)=211
f(7)=1
f(8)=271
f(9)=149
f(10)=19
f(11)=173
f(12)=367
f(13)=193
f(14)=31
f(15)=1
f(16)=431
f(17)=1
f(18)=41
f(19)=229
f(20)=463
f(21)=233
f(22)=467
f(23)=1
f(24)=1
f(25)=1
f(26)=1
f(27)=1
f(28)=1
f(29)=1
f(30)=1
f(31)=1
f(32)=1
f(33)=1
f(34)=1
f(35)=1
f(36)=1
f(37)=1
f(38)=1
f(39)=1
f(40)=1
f(41)=1
f(42)=1
f(43)=1
f(44)=1
f(45)=1
f(46)=109
f(47)=79
f(48)=1
f(49)=131
f(50)=317
f(51)=1
f(52)=433
f(53)=1
f(54)=557
f(55)=311
f(56)=1
f(57)=379
f(58)=829
f(59)=1
f(60)=977
f(61)=1
f(62)=103
f(63)=607
f(64)=1297
f(65)=691
f(66)=113
f(67)=1
f(68)=97
f(69)=1
f(70)=167
f(71)=967
f(72)=107
f(73)=1
f(74)=2237
f(75)=1171
f(76)=1
f(77)=1279
f(78)=157
f(79)=1
f(80)=2897
f(81)=137
f(82)=241
f(83)=1627
f(84)=307
f(85)=1
f(86)=191
f(87)=1879
f(88)=3889
f(89)=2011
f(90)=4157
f(91)=1
f(92)=1
f(93)=2287
f(94)=1
f(95)=1
f(96)=5009
f(97)=2579
f(98)=5309
f(99)=2731

b) Substitution of the polynom
The polynom f(x)=x^2-44x+17 could be written as f(y)= y^2-467 with x=y+22

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-22
f'(x)>2x-45 with x > 22

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

17, 13, 67, 53, 11, 89, 211, 1, 271, 149, 19, 173, 367, 193, 31, 1, 431, 1, 41, 229, 463, 233, 467, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 109, 79, 1, 131, 317, 1, 433, 1, 557, 311, 1, 379, 829, 1, 977, 1, 103, 607, 1297, 691, 113, 1, 97, 1, 167, 967, 107, 1, 2237, 1171, 1, 1279, 157, 1, 2897, 137, 241, 1627, 307, 1, 191, 1879, 3889, 2011, 4157, 1, 1, 2287, 1, 1, 5009, 2579, 5309, 2731, 1, 2887, 349, 277, 6257, 1, 599, 1, 1, 1, 383, 3727, 449, 3907, 727, 4091, 8369, 389, 673, 263, 9137, 359, 9533, 1, 523, 461, 1, 5279, 1, 1, 11197, 439, 11633, 5927, 929, 6151, 1, 6379, 419, 601, 13457, 1, 13933, 373, 1109, 7331, 877, 1, 811, 1, 1447, 8087, 16433, 491, 547, 1, 17489, 683, 1, 9151, 1429, 857, 1, 571, 19697, 1, 20269, 541, 20849, 1, 1, 10867, 2003, 859, 22637, 11471, 347, 11779, 23869, 1, 1, 653, 613, 1, 1, 421, 1, 787, 1, 13711, 677, 1277, 28433, 14387, 2647, 14731, 2293, 887, 30509, 1187, 1, 15787, 2903, 1, 1, 1, 1, 16879, 34129, 1327, 34877, 17627, 2741, 1637, 2141, 1, 1, 1, 1, 1009, 38737, 1151, 3041, 487, 1, 1567, 41149, 1889, 41969, 21191, 42797, 1, 43633, 22027, 563, 1, 45329, 1, 1, 23311, 47057, 23747, 47933, 1, 48817, 24631, 4519, 809, 1, 1, 51517, 1999, 52433, 499, 1, 1583, 1, 1, 55229, 27851, 5107, 2179, 1, 28807, 1, 1723, 59069, 1, 1, 30271, 911, 2797, 1, 31267, 1, 31771, 3371, 1, 2099, 1, 1, 1753, 1, 33827, 863, 34351, 647, 2683, 70289, 2083, 1, 1, 641, 1, 73517, 1949, 74609, 37579, 75709, 2243, 1, 3517, 77933, 3019, 7187, 971, 1, 1, 1, 1321, 1231, 1, 7603, 1, 4463, 3881, 1, 1, 87149, 2309, 88337, 839, 89533, 1, 2927, 1, 643, 46279, 93169, 3607, 94397, 47507, 95633, 1, 8807, 48751, 98129, 1, 5231, 3847, 1, 50647, 7841, 51287, 103217, 4721, 2549, 52579, 9619, 53231, 6301, 53887, 1, 54547, 2677, 1, 10099, 1, 112429, 1, 113777, 1, 115133, 733, 1201, 4507, 1, 1, 9173, 59971, 997, 1, 122033, 1, 1, 62071, 6571, 937, 883, 1, 1, 1, 1, 64927, 7681, 65651, 4259, 1619, 1019, 6101, 7103, 1, 1, 1, 1, 1, 139409, 70079, 140909, 1, 1, 4211, 4643, 6577, 1, 1, 146989, 5683, 8737, 3929, 150077, 6857, 2861, 76207, 1, 76991, 154769, 1, 1, 78571, 12149, 79367, 14503, 80167, 161137, 1, 162749, 1543, 1, 82591, 1, 83407, 167633, 1, 1553, 5003, 1, 1, 172589, 86711, 174257, 87547, 1, 1, 1, 89231, 1061, 1, 5839, 90931, 2053, 1, 1, 92647, 186157, 8501, 1, 94379, 17239, 1, 907, 991, 193133, 97007, 6287, 1847, 941, 98779, 1039, 1, 200237, 1, 15541, 101467, 1979, 1, 12097, 1, 207469, 3361, 1, 105107, 1, 1, 2393, 1, 12637, 1, 19699, 1, 11503, 1, 1609, 1, 222317, 1, 224209, 1093, 17393, 10321, 2131, 114487, 20903, 6791, 12203, 116411, 1, 1097, 18133, 6229, 1, 1, 239633, 10937, 241597, 121291, 1, 122279, 1, 1, 14561, 1, 249533, 1, 1759, 126271, 8179, 7487, 13451, 128291, 1489, 1, 23603, 2459, 20129, 11941, 263729, 1, 3967, 1499, 267857, 4337, 14207, 1, 16001, 136531, 24919, 1, 276209, 1, 1, 3407, 1621, 1451, 1511, 141811, 284689, 1, 2957, 3511, 22229, 1, 1, 11239, 1217, 13381, 15551, 148279, 27059, 149371, 1, 1, 9743, 1, 1, 152671, 2143, 1493, 1, 14081, 16363, 156007, 313133, 157127, 315377, 8329, 1, 14489, 1, 12347, 29287, 1483, 1, 162787, 2081, 9643, 329009, 165079, 30119, 1, 1, 15217, 25841, 5437, 1, 1, 1, 1597, 342929, 15641, 1789, 1, 1, 174407, 3271, 1, 352369, 176779, 18671, 5741, 32467, 179167, 21149, 1, 6829, 13967, 364349, 182779, 1, 1, 369197, 1, 371633, 4547, 1, 187651, 376529, 9941, 29153, 1, 1, 1, 1, 192587, 1, 1, 22877, 1, 2857, 196331, 1, 15199, 3637, 18077, 30689, 2063, 1, 1, 404029, 1, 406577, 10733, 4597, 12071, 2879, 6661, 414269, 1453, 416849, 209071, 1, 1, 6299, 12451, 10357, 1, 427249, 1, 39079, 2729, 1, 6997, 10613, 218227, 1381, 1, 40039, 220879, 443089, 20201, 2017, 223547, 8461, 17299, 23743, 226231, 14639, 1, 1487, 12049, 1, 1, 461933, 1, 3547, 1, 1, 1, 1, 235751, 1, 21557, 1, 12553, 1, 5851, 37013, 2711, 25471, 1697, 2089, 2281, 1, 5987, 28961, 1427, 495149, 8009, 497969, 19207, 1, 14771, 2039, 2087, 1, 253951, 509329, 13441, 3067, 3833, 1, 1, 39841, 8377, 1, 1, 1, 2707, 30977, 264031, 529517, 1, 1, 266947, 3593, 1877, 1, 269879, 1, 271351, 544177, 2503, 547133, 2267, 13417, 16223, 1, 277279, 42773, 2467, 29423, 21559, 2029, 281767, 1657, 1, 568049, 25889, 571069, 4273, 3853, 1, 1997, 1, 3433, 1, 583229, 3701, 53299, 1, 589357, 1, 34849, 296987, 2411, 1, 54419, 1, 11353, 1613, 1733, 303187, 607933, 1579, 611057, 1, 1, 2153, 32491, 1, 4339, 311027, 623633, 16453, 626797, 1, 37057, 315779, 57559, 317371, 1, 1, 639533, 24659, 642737, 322171, 37997, 17041, 6067, 29581, 1, 327007, 3137, 1, 1, 2521, 1, 4201, 16229, 1, 1, 1, 671933, 1801, 6961, 338431, 1, 340079, 681809, 1, 1, 1, 688433, 345047, 5717, 346711, 695089, 3083, 1, 26927, 41281, 351727, 4931, 2251, 708497, 1699, 9011, 1, 715249, 358471, 1, 360167, 55541, 1, 725437, 27967, 1, 21487, 1, 8951, 735697, 1, 1, 370427, 1, 28627, 7243, 1, 57653, 1, 1747, 377347, 756433, 379087, 44701, 1, 763409, 382579, 1, 384331, 770417, 1, 773933, 1, 14669, 2333, 70999, 391379, 1, 1, 788077, 1787, 6043, 1, 61169, 398491, 10111, 36389, 1, 402071, 1, 403867, 1, 405667, 62549, 407471, 3389, 1, 1, 1, 823997, 37537, 827633, 1, 3001, 1, 1, 32183, 838589, 38201, 1, 1, 1, 423887, 27407, 1, 1, 427579, 857009, 429431, 1, 431287, 50849, 1, 868157, 1867, 1, 8243, 875629, 1, 1, 1, 51949, 34039, 80627, 1, 1, 446279, 894449, 1, 7949, 450067, 82003, 451967, 1, 1, 1, 1, 4733, 1, 13691, 459607, 921133, 41957, 1, 463451, 4967, 465379, 2999, 1, 936557, 15137, 72341, 471187, 85847, 1, 1, 43189, 50111, 477031,

6. Sequence of the polynom (only primes)

17, 13, 67, 53, 11, 89, 211, 271, 149, 19, 173, 367, 193, 31, 431, 41, 229, 463, 233, 467, 109, 79, 131, 317, 433, 557, 311, 379, 829, 977, 103, 607, 1297, 691, 113, 97, 167, 967, 107, 2237, 1171, 1279, 157, 2897, 137, 241, 1627, 307, 191, 1879, 3889, 2011, 4157, 2287, 5009, 2579, 5309, 2731, 2887, 349, 277, 6257, 599, 383, 3727, 449, 3907, 727, 4091, 8369, 389, 673, 263, 9137, 359, 9533, 523, 461, 5279, 11197, 439, 11633, 5927, 929, 6151, 6379, 419, 601, 13457, 13933, 373, 1109, 7331, 877, 811, 1447, 8087, 16433, 491, 547, 17489, 683, 9151, 1429, 857, 571, 19697, 20269, 541, 20849, 10867, 2003, 859, 22637, 11471, 347, 11779, 23869, 653, 613, 421, 787, 13711, 677, 1277, 28433, 14387, 2647, 14731, 2293, 887, 30509, 1187, 15787, 2903, 16879, 34129, 1327, 34877, 17627, 2741, 1637, 2141, 1009, 38737, 1151, 3041, 487, 1567, 41149, 1889, 41969, 21191, 42797, 43633, 22027, 563, 45329, 23311, 47057, 23747, 47933, 48817, 24631, 4519, 809, 51517, 1999, 52433, 499, 1583, 55229, 27851, 5107, 2179, 28807, 1723, 59069, 30271, 911, 2797, 31267, 31771, 3371, 2099, 1753, 33827, 863, 34351, 647, 2683, 70289, 2083, 641, 73517, 1949, 74609, 37579, 75709, 2243, 3517, 77933, 3019, 7187, 971, 1321, 1231, 7603, 4463, 3881, 87149, 2309, 88337, 839, 89533, 2927, 643, 46279, 93169, 3607, 94397, 47507, 95633, 8807, 48751, 98129, 5231, 3847, 50647, 7841, 51287, 103217, 4721, 2549, 52579, 9619, 53231, 6301, 53887, 54547, 2677, 10099, 112429, 113777, 115133, 733, 1201, 4507, 9173, 59971, 997, 122033, 62071, 6571, 937, 883, 64927, 7681, 65651, 4259, 1619, 1019, 6101, 7103, 139409, 70079, 140909, 4211, 4643, 6577, 146989, 5683, 8737, 3929, 150077, 6857, 2861, 76207, 76991, 154769, 78571, 12149, 79367, 14503, 80167, 161137, 162749, 1543, 82591, 83407, 167633, 1553, 5003, 172589, 86711, 174257, 87547, 89231, 1061, 5839, 90931, 2053, 92647, 186157, 8501, 94379, 17239, 907, 991, 193133, 97007, 6287, 1847, 941, 98779, 1039, 200237, 15541, 101467, 1979, 12097, 207469, 3361, 105107, 2393, 12637, 19699, 11503, 1609, 222317, 224209, 1093, 17393, 10321, 2131, 114487, 20903, 6791, 12203, 116411, 1097, 18133, 6229, 239633, 10937, 241597, 121291, 122279, 14561, 249533, 1759, 126271, 8179, 7487, 13451, 128291, 1489, 23603, 2459, 20129, 11941, 263729, 3967, 1499, 267857, 4337, 14207, 16001, 136531, 24919, 276209, 3407, 1621, 1451, 1511, 141811, 284689, 2957, 3511, 22229, 11239, 1217, 13381, 15551, 148279, 27059, 149371, 9743, 152671, 2143, 1493, 14081, 16363, 156007, 313133, 157127, 315377, 8329, 14489, 12347, 29287, 1483, 162787, 2081, 9643, 329009, 165079, 30119, 15217, 25841, 5437, 1597, 342929, 15641, 1789, 174407, 3271, 352369, 176779, 18671, 5741, 32467, 179167, 21149, 6829, 13967, 364349, 182779, 369197, 371633, 4547, 187651, 376529, 9941, 29153, 192587, 22877, 2857, 196331, 15199, 3637, 18077, 30689, 2063, 404029, 406577, 10733, 4597, 12071, 2879, 6661, 414269, 1453, 416849, 209071, 6299, 12451, 10357, 427249, 39079, 2729, 6997, 10613, 218227, 1381, 40039, 220879, 443089, 20201, 2017, 223547, 8461, 17299, 23743, 226231, 14639, 1487, 12049, 461933, 3547, 235751, 21557, 12553, 5851, 37013, 2711, 25471, 1697, 2089, 2281, 5987, 28961, 1427, 495149, 8009, 497969, 19207, 14771, 2039, 2087, 253951, 509329, 13441, 3067, 3833, 39841, 8377, 2707, 30977, 264031, 529517, 266947, 3593, 1877, 269879, 271351, 544177, 2503, 547133, 2267, 13417, 16223, 277279, 42773, 2467, 29423, 21559, 2029, 281767, 1657, 568049, 25889, 571069, 4273, 3853, 1997, 3433, 583229, 3701, 53299, 589357, 34849, 296987, 2411, 54419, 11353, 1613, 1733, 303187, 607933, 1579, 611057, 2153, 32491, 4339, 311027, 623633, 16453, 626797, 37057, 315779, 57559, 317371, 639533, 24659, 642737, 322171, 37997, 17041, 6067, 29581, 327007, 3137, 2521, 4201, 16229, 671933, 1801, 6961, 338431, 340079, 681809, 688433, 345047, 5717, 346711, 695089, 3083, 26927, 41281, 351727, 4931, 2251, 708497, 1699, 9011, 715249, 358471, 360167, 55541, 725437, 27967, 21487, 8951, 735697, 370427, 28627, 7243, 57653, 1747, 377347, 756433, 379087, 44701, 763409, 382579, 384331, 770417, 773933, 14669, 2333, 70999, 391379, 788077, 1787, 6043, 61169, 398491, 10111, 36389, 402071, 403867, 405667, 62549, 407471, 3389, 823997, 37537, 827633, 3001, 32183, 838589, 38201, 423887, 27407, 427579, 857009, 429431, 431287, 50849, 868157, 1867, 8243, 875629, 51949, 34039, 80627, 446279, 894449, 7949, 450067, 82003, 451967, 4733, 13691, 459607, 921133, 41957, 463451, 4967, 465379, 2999, 936557, 15137, 72341, 471187, 85847, 43189, 50111, 477031,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-44x+17 and
the reducible primes which appear as divisor for the first time
p | x^2-44x+17 and p < x^2-44x+17

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)
1109810.9000000.8000000.1000000.0000000.0000000.000000
2100514380.5100000.4300000.0800005.6666675.3750008.000000
31.0006643053590.6640000.3050000.35900013.0196087.09302344.875000
410.0007.0142.1234.8910.7014000.2123000.48910010.5632536.96065613.623956
5100.00070.27316.55453.7190.7027300.1655400.53719010.0189627.79745610.983234
61.000.000701.766134.600567.1660.7017660.1346000.5671669.9862828.13096510.558015
710.000.0007.002.3891.134.7625.867.6270.7002390.1134760.5867639.9782398.43062410.345520
8100.000.00069.920.4859.827.91660.092.5690.6992050.0982790.6009269.9852338.66077310.241375
91.000.000.000698.445.81986.604.312611.841.5070.6984460.0866040.6118429.9891448.81207310.181651
1010.000.000.0006.978.385.869774.274.9936.204.110.8760.6978390.0774270.6204119.9913068.94037410.140061


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
245411.2500001.0000000.2500001.6666671.333333inf
388711.0000000.8750000.1250001.6000001.7500001.000000
416141220.8750000.7500000.1250001.7500001.7142862.000000
532191630.5937500.5000000.0937501.3571431.3333331.500000
664302730.4687500.4218750.0468751.5789471.6875001.000000
71287252200.5625000.4062500.1562502.4000001.9259266.666667
825616298640.6328120.3828120.2500002.2500001.8846153.200000
95123331751580.6503910.3417970.3085942.0555561.7857142.468750
101.0246773083690.6611330.3007810.3603522.0330331.7600002.335443
112.0481.4055588470.6860350.2724610.4135742.0753321.8116882.295393
124.0962.8489851.8630.6953120.2404790.4548342.0270461.7652332.199528
138.1925.7291.7763.9530.6993410.2167970.4825442.0115871.8030462.121846
1416.38411.5143.2628.2520.7027590.1990970.5036622.0097751.8367122.087528
1532.76823.0046.11916.8850.7020260.1867370.5152891.9979161.8758432.046171
1665.53646.05811.37034.6880.7027890.1734920.5292972.0021731.8581472.054368
17131.07292.14521.12671.0190.7030110.1611790.5418322.0006301.8580472.047365
18262.144184.14439.533144.6110.7024540.1508060.5516471.9984161.8712962.036230
19524.288368.25174.404293.8470.7023830.1419140.5604691.9997991.8820732.031982
201.048.576735.987140.490595.4970.7018920.1339820.5679101.9986021.8882052.026555
212.097.1521.470.803266.1031.204.7000.7013340.1268880.5744461.9984091.8941062.023016
224.194.3042.939.202505.7512.433.4510.7007600.1205800.5801801.9983661.9005842.019964
238.388.6085.874.878963.2054.911.6730.7003400.1148230.5855171.9988001.9045042.018398
2416.777.21611.743.6321.840.1729.903.4600.6999750.1096830.5902921.9989581.9104682.016311
2533.554.43223.476.7443.523.14919.953.5950.6996610.1049980.5946641.9991041.9145762.014811
2667.108.86446.932.4196.754.25140.178.1680.6993480.1006460.5987011.9991031.9171062.013580
27134.217.72893.831.72312.967.31580.864.4080.6991010.0966140.6024871.9992941.9198752.012645
28268.435.456187.600.76224.938.283162.662.4790.6988670.0929020.6059651.9993321.9231652.011546
29536.870.912375.074.72248.035.060327.039.6620.6986310.0894720.6091591.9993241.9261572.010541
301.073.741.824749.929.47592.648.285657.281.1900.6984260.0862850.6121411.9994131.9287642.009791
312.147.483.6481.499.437.198178.935.4481.320.501.7500.6982300.0833230.6149071.9994381.9313412.009036
324.294.967.2962.998.082.719345.986.4452.652.096.2740.6980460.0805560.6174891.9994721.9335822.008400
338.589.934.5925.994.702.819669.767.5435.324.935.2760.6978750.0779710.6199041.9995121.9358202.007821
3417.179.869.18411.986.687.2101.297.900.80510.688.786.4050.6977170.0755480.6221691.9995471.9378382.007308


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
244221120
387432221
41612663243
53216884354
6642714137596
712852272513131313
825698494928232423
9512175849144414446
101.02430815115777757284
112.048558277281130140137151
124.096985496489234255242254
138.1921.776876900432453437454
1416.3843.2621.6241.638805849786822
1532.7686.1193.0483.0711.5131.5541.4821.570
1665.53611.3705.6675.7032.8022.9192.7912.858
17131.07221.12610.56010.5665.1875.4015.1995.339
18262.14439.53319.77819.7559.7289.9979.75410.054
19524.28874.40437.33337.07118.31418.73718.36618.987
201.048.576140.49070.48770.00334.62935.38534.76535.711
212.097.152266.103133.656132.44765.64667.22165.71067.526
224.194.304505.751254.023251.728124.776127.559124.938128.478
238.388.608963.205483.574479.631238.153243.240237.990243.822
2416.777.2161.840.172923.877916.295454.786464.948454.998465.440
2533.554.4323.523.1491.767.8841.755.265871.208890.687871.537889.717
2667.108.8646.754.2513.387.4743.366.7771.672.0261.705.0151.672.3941.704.816
27134.217.72812.967.3156.504.3366.462.9793.209.8763.273.3343.210.7893.273.316
28268.435.45624.938.28312.508.96812.429.3156.174.2476.293.5896.174.6976.295.750
29536.870.91248.035.06024.089.42123.945.63911.894.94912.119.00311.900.44812.120.660
301.073.741.82492.648.28546.460.63246.187.65322.953.48623.369.03422.956.62723.369.138
312.147.483.648178.935.44889.728.85889.206.59044.348.71245.117.01644.344.74945.124.971
324.294.967.296345.986.445173.469.981172.516.46485.775.89187.218.21585.771.69987.220.640
338.589.934.592669.767.543335.786.640333.980.903166.089.951168.794.627166.089.429168.793.536
3417.179.869.1841.297.900.805650.637.854647.262.951321.934.092327.009.867321.944.483327.012.363


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
241010100
381010100
4162110101
5323121101
6643121101
7128209114358
825664313311141920
9512158797932424143
101.02436917819189889696
112.048847412435203205219220
124.0961.863898965457464467475
138.1923.9531.9362.0179779881.000988
1416.3848.2524.0554.1972.0282.1272.0902.007
1532.76816.8858.3998.4864.1994.2924.2284.166
1665.53634.68817.43317.2558.6338.6768.7068.673
17131.07271.01935.47435.54517.62517.81217.78617.796
18262.144144.61172.15572.45636.06536.22036.25536.071
19524.288293.847147.134146.71373.27573.53073.63273.410
201.048.576595.497297.957297.540148.243148.911149.368148.975
212.097.1521.204.700602.065602.635300.717300.794301.647301.542
224.194.3042.433.4511.216.5831.216.868608.155607.723609.092608.481
238.388.6084.911.6732.455.1522.456.5211.227.8261.227.7621.228.5281.227.557
2416.777.2169.903.4604.951.5964.951.8642.475.3642.475.8072.477.6922.474.597
2533.554.43219.953.5959.976.5709.977.0254.989.3234.988.0594.988.6734.987.540
2667.108.86440.178.16820.086.67020.091.49810.046.56310.044.06610.045.04010.042.499
27134.217.72880.864.40840.430.00740.434.40120.214.23420.213.29820.222.29820.214.578
28268.435.456162.662.47981.325.98581.336.49440.661.70940.664.05240.668.66640.668.052
29536.870.912327.039.662163.514.117163.525.54581.751.18181.760.59281.774.85581.753.034
301.073.741.824657.281.190328.627.005328.654.185164.318.715164.318.430164.336.827164.307.218
312.147.483.6481.320.501.750660.223.171660.278.579330.133.478330.112.185330.156.777330.099.310
324.294.967.2962.652.096.2741.326.024.5011.326.071.773663.061.047662.985.937663.077.193662.972.097
338.589.934.5925.324.935.2762.662.424.8512.662.510.4251.331.321.2031.331.187.3041.331.262.7851.331.163.984
3417.179.869.18410.688.786.4055.344.316.2205.344.470.1852.672.337.1932.672.066.1152.672.270.2302.672.112.867


8. Check for existing Integer Sequences by OEIS

Found in Database : 17, 13, 67, 53, 11, 89, 211, 1, 271, 149, 19, 173, 367, 193, 31, 1, 431, 1, 41, 229,
Found in Database : 17, 13, 67, 53, 11, 89, 211, 271, 149, 19, 173, 367, 193, 31, 431, 41, 229, 463, 233, 467,
Found in Database : 11, 13, 17, 19, 31, 41, 53, 67, 79, 89, 97, 103, 107, 109, 113, 131, 137, 149,