Inhaltsverzeichnis

Development of
Algorithmic Constructions

18:59:17
Deutsch
28.Mar 2024

Polynom = x^2+36x-53

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) = 53 = 53
f(1) = 1 = 1
f(2) = 23 = 23
f(3) = 1 = 1
f(4) = 107 = 107
f(5) = 19 = 19
f(6) = 199 = 199
f(7) = 31 = 31
f(8) = 299 = 13*23
f(9) = 11 = 11
f(10) = 407 = 11*37
f(11) = 29 = 29
f(12) = 523 = 523
f(13) = 73 = 73
f(14) = 647 = 647
f(15) = 89 = 89
f(16) = 779 = 19*41
f(17) = 53 = 53
f(18) = 919 = 919
f(19) = 31 = 31
f(20) = 1067 = 11*97
f(21) = 143 = 11*13
f(22) = 1223 = 1223
f(23) = 163 = 163
f(24) = 1387 = 19*73
f(25) = 23 = 23
f(26) = 1559 = 1559
f(27) = 103 = 103
f(28) = 1739 = 37*47
f(29) = 229 = 229
f(30) = 1927 = 41*47
f(31) = 253 = 11*23
f(32) = 2123 = 11*193
f(33) = 139 = 139
f(34) = 2327 = 13*179
f(35) = 19 = 19
f(36) = 2539 = 2539
f(37) = 331 = 331
f(38) = 2759 = 31*89
f(39) = 359 = 359
f(40) = 2987 = 29*103
f(41) = 97 = 97
f(42) = 3223 = 11*293
f(43) = 209 = 11*19
f(44) = 3467 = 3467
f(45) = 449 = 449
f(46) = 3719 = 3719
f(47) = 481 = 13*37
f(48) = 3979 = 23*173
f(49) = 257 = 257
f(50) = 4247 = 31*137
f(51) = 137 = 137
f(52) = 4523 = 4523
f(53) = 583 = 11*53
f(54) = 4807 = 11*19*23
f(55) = 619 = 619
f(56) = 5099 = 5099
f(57) = 41 = 41
f(58) = 5399 = 5399
f(59) = 347 = 347
f(60) = 5707 = 13*439
f(61) = 733 = 733
f(62) = 6023 = 19*317
f(63) = 773 = 773
f(64) = 6347 = 11*577
f(65) = 407 = 11*37
f(66) = 6679 = 6679
f(67) = 107 = 107
f(68) = 7019 = 7019
f(69) = 899 = 29*31
f(70) = 7367 = 53*139
f(71) = 943 = 23*41
f(72) = 7723 = 7723
f(73) = 247 = 13*19
f(74) = 8087 = 8087
f(75) = 517 = 11*47
f(76) = 8459 = 11*769
f(77) = 1081 = 23*47
f(78) = 8839 = 8839
f(79) = 1129 = 1129
f(80) = 9227 = 9227
f(81) = 589 = 19*31
f(82) = 9623 = 9623
f(83) = 307 = 307
f(84) = 10027 = 37*271
f(85) = 1279 = 1279
f(86) = 10439 = 11*13*73
f(87) = 1331 = 11*11*11
f(88) = 10859 = 10859
f(89) = 173 = 173
f(90) = 11287 = 11287
f(91) = 719 = 719
f(92) = 11723 = 19*617
f(93) = 1493 = 1493
f(94) = 12167 = 23*23*23
f(95) = 1549 = 1549
f(96) = 12619 = 12619
f(97) = 803 = 11*73
f(98) = 13079 = 11*29*41
f(99) = 13 = 13
f(100) = 13547 = 19*23*31

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+36x-53

f(0)=53
f(1)=1
f(2)=23
f(3)=1
f(4)=107
f(5)=19
f(6)=199
f(7)=31
f(8)=13
f(9)=11
f(10)=37
f(11)=29
f(12)=523
f(13)=73
f(14)=647
f(15)=89
f(16)=41
f(17)=1
f(18)=919
f(19)=1
f(20)=97
f(21)=1
f(22)=1223
f(23)=163
f(24)=1
f(25)=1
f(26)=1559
f(27)=103
f(28)=47
f(29)=229
f(30)=1
f(31)=1
f(32)=193
f(33)=139
f(34)=179
f(35)=1
f(36)=2539
f(37)=331
f(38)=1
f(39)=359
f(40)=1
f(41)=1
f(42)=293
f(43)=1
f(44)=3467
f(45)=449
f(46)=3719
f(47)=1
f(48)=173
f(49)=257
f(50)=137
f(51)=1
f(52)=4523
f(53)=1
f(54)=1
f(55)=619
f(56)=5099
f(57)=1
f(58)=5399
f(59)=347
f(60)=439
f(61)=733
f(62)=317
f(63)=773
f(64)=577
f(65)=1
f(66)=6679
f(67)=1
f(68)=7019
f(69)=1
f(70)=1
f(71)=1
f(72)=7723
f(73)=1
f(74)=8087
f(75)=1
f(76)=769
f(77)=1
f(78)=8839
f(79)=1129
f(80)=9227
f(81)=1
f(82)=9623
f(83)=307
f(84)=271
f(85)=1279
f(86)=1
f(87)=1
f(88)=10859
f(89)=1
f(90)=11287
f(91)=719
f(92)=617
f(93)=1493
f(94)=1
f(95)=1549
f(96)=12619
f(97)=1
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+36x-53 could be written as f(y)= y^2-377 with x=y-18

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+18
f'(x)>2x+35

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

53, 1, 23, 1, 107, 19, 199, 31, 13, 11, 37, 29, 523, 73, 647, 89, 41, 1, 919, 1, 97, 1, 1223, 163, 1, 1, 1559, 103, 47, 229, 1, 1, 193, 139, 179, 1, 2539, 331, 1, 359, 1, 1, 293, 1, 3467, 449, 3719, 1, 173, 257, 137, 1, 4523, 1, 1, 619, 5099, 1, 5399, 347, 439, 733, 317, 773, 577, 1, 6679, 1, 7019, 1, 1, 1, 7723, 1, 8087, 1, 769, 1, 8839, 1129, 9227, 1, 9623, 307, 271, 1279, 1, 1, 10859, 1, 11287, 719, 617, 1493, 1, 1549, 12619, 1, 1, 1, 1, 1723, 379, 1783, 1, 461, 283, 953, 1409, 1, 16007, 1, 1, 1049, 17047, 541, 17579, 1, 18119, 1, 1697, 1, 409, 1, 421, 1, 20359, 1, 20939, 1327, 1, 1, 22123, 2803, 22727, 2879, 23339, 739, 1, 1, 1069, 1, 2293, 1, 25867, 1637, 1153, 839, 27179, 181, 27847, 1, 2593, 1, 29207, 1847, 1031, 1, 827, 1, 31307, 1979, 1033, 1, 1, 4139, 33479, 4231, 1801, 1, 479, 1, 1, 4513, 1, 419, 1, 1, 38039, 1201, 947, 4903, 39623, 5003, 40427, 1, 1, 1, 1, 5309, 3299, 5413, 1901, 1, 1087, 1, 4129, 521, 46279, 5839, 47147, 1487, 48023, 233, 48907, 1, 2621, 571, 1, 1, 51607, 1627, 991, 1, 1, 1, 1, 857, 1, 1, 56267, 1, 57223, 7213, 1877, 1, 59159, 1, 60139, 1, 5557, 7703, 1, 1, 63127, 1, 2069, 8081, 2833, 8209, 547, 1, 5171, 1, 3593, 8599, 673, 8731, 70379, 277, 1931, 1, 1, 9133, 73607, 1, 74699, 4703, 1, 1193, 76907, 1, 1, 1, 79147, 1, 80279, 1, 6263, 1, 82567, 1, 2887, 1, 7717, 2671, 2099, 10831, 3793, 10979, 88427, 1, 1, 5639, 1, 1039, 2969, 313, 1, 5867, 4973, 743, 95723, 12043, 7459, 1109, 8929, 3089, 3209, 6257, 100747, 1, 102023, 1, 103307, 1, 1, 1, 105899, 701, 2281, 1, 2309, 853, 109847, 6907, 1523, 1, 1, 14149, 1, 7159, 115223, 1811, 1, 1, 117959, 14831, 1, 1, 1, 7589, 1259, 1181, 123527, 1, 1213, 7853, 126359, 1, 11617, 16063, 129223, 1, 3187, 2053, 10163, 1, 1, 1, 12277, 1543, 136523, 373, 138007, 1, 1, 1, 1, 1, 4597, 1, 13093, 9049, 145547, 18289, 7741, 18481, 1, 9337, 6529, 1, 1061, 1733, 153287, 19259, 6733, 1, 2143, 1, 4271, 19853, 159623, 1823, 14657, 1, 162839, 2557, 1, 1, 1, 1, 937, 1, 1, 967, 9001, 21481, 1, 1, 174347, 10949, 176023, 5527, 1, 2029, 1, 22531, 1867, 2843, 5897, 883, 184523, 23173, 186247, 1231, 1, 1, 983, 1489, 191467, 24043, 1, 1277, 1, 6121, 4799, 1123, 18049, 1, 1229, 25153, 1889, 12689, 10733, 1, 1, 1987, 18869, 1, 5107, 1, 11117, 13259, 213067, 1163, 7411, 26981, 216779, 1237, 1, 1, 2141, 1, 6011, 27919, 224299, 7039, 226199, 14197, 1, 1, 1, 2221, 231947, 14557, 233879, 1, 10253, 29599, 237767, 2713, 1, 3761, 241687, 1, 18743, 1, 245639, 30829, 13033, 1, 2063, 1, 8117, 1373, 253639, 1, 255659, 1, 257687, 1, 259723, 2963, 1, 1, 263819, 16553, 5657, 1, 5701, 33623, 20771, 1093, 1, 1, 274199, 17203, 1, 1, 14653, 1, 1721, 17599, 12289, 1, 25889, 35731, 15101, 35999, 289067, 9067, 7103, 18269, 4019, 36809, 1, 3371, 1, 1, 5659, 1, 302123, 1307, 9817, 1, 1, 1, 28069, 1, 310987, 3001, 3041, 39293, 315467, 1, 1, 1, 1531, 1, 322247, 40423, 324523, 10177, 1, 1, 17321, 41281, 331399, 3779, 1319, 20929, 11587, 1, 338347, 1, 1, 1, 343019, 2689, 31397, 1, 3907, 43613, 350087, 2311, 352459, 1, 354839, 5563, 27479, 4073, 32693, 1, 362027, 11351, 19181, 22853, 366859, 1, 9007, 1597, 1, 1, 1, 11731, 12149, 1, 2767, 1, 1, 5981, 1, 1, 1, 2551, 1301, 48781, 1, 24547, 394007, 1, 396523, 2617, 36277, 4549, 1283, 12589, 404119, 1949, 1, 50993, 1787, 1, 21673, 2347, 1, 1, 1423, 2273, 419527, 1, 1709, 1, 424727, 26627, 1, 4871, 4831, 1, 432587, 1, 435223, 1, 437867, 1, 1, 5021, 40289, 1, 445847, 1471, 19501, 1, 451207, 56569, 1, 1, 1, 1, 1, 1, 24317, 57923, 464747, 7283, 467479, 29303, 470219, 1, 1, 4561, 1, 29819, 1, 1, 13007, 60331, 16691, 60679, 44257, 1, 1, 30689, 37879, 61729, 1, 62081, 10597, 1, 500887, 1427, 1, 1, 13691, 63499, 1607, 1, 22273, 1, 1, 2083, 47093, 5903, 2699, 32647, 9883, 1, 1459, 1, 40739, 66383, 532523, 1, 48677, 1, 2351, 67481, 1, 3571, 1, 1483, 547223, 1, 4547, 6269, 1, 1, 4001, 8713, 1499, 1, 18133, 1, 565127, 1, 1, 35603, 1, 2237, 3319, 71963, 10891, 1, 25229, 18181, 2791, 3323, 5693, 5653, 1, 2383, 1, 1, 1, 18661, 3673, 1, 54709, 1, 604907, 1, 46771, 1, 611147, 1, 614279, 3347, 1, 3517, 620567, 9721, 623723, 78163, 626887, 6043, 33161, 19739, 1, 3607, 1, 1, 1, 80153, 1, 40277, 646039, 1, 49943, 81359, 2579, 7433, 655723, 10271, 1, 1, 6827, 82981, 665479, 83389, 668747, 1, 1, 1, 3499, 2287, 678599, 3697, 681899, 1, 1, 42929, 1, 1, 2801, 86689, 1, 1, 1, 21881, 2731, 87943, 37117, 1, 1741, 1, 711959, 1, 715339, 1907, 31249, 90053, 722123, 2381, 65957, 1, 1, 91331, 4493, 1, 56599, 1, 19979, 46309, 23957, 1, 67829, 93481, 14143, 46957, 18367, 1, 1, 1, 760007, 95219, 2239, 1, 766999, 2089, 1, 2609, 774023, 96973, 7549, 1, 60083, 1, 71329, 98299, 788167, 5197, 5779, 1, 16921, 1, 1, 2441, 72949, 1, 806027, 1, 35201, 25357, 813227, 101879, 816839, 3301, 820459, 1, 3943, 1, 63671, 1, 831367, 104149, 835019, 1, 1, 1, 1, 1, 3631, 105983, 6113, 1, 27529, 53453, 857099, 5651, 29683, 9803, 78593, 1, 8951, 27191, 23567, 5749, 1, 1, 2657, 1, 3491, 1, 886987, 111109, 3823, 111581, 1, 1, 898327, 1, 22003, 10273, 7487, 3067, 47881, 1, 2081, 1973, 2243, 114913, 921223, 1, 6469, 1, 2677, 1, 932779, 3769, 936647, 5101, 940523, 1, 1, 1, 86209, 9137, 952199, 2909, 956107, 59879, 13151, 15031, 3557, 120739, 1, 1, 1, 30431, 1, 1, 26479, 1, 2251, 123209, 987659, 5623, 90149, 31051, 32117, 124703, 999623, 9631, 1003627, 1, 1, 63103, 91969, 11519, 1, 5531, 1019723, 3361, 11503, 4007, 79063, 1, 1, 1,

6. Sequence of the polynom (only primes)

53, 23, 107, 19, 199, 31, 13, 11, 37, 29, 523, 73, 647, 89, 41, 919, 97, 1223, 163, 1559, 103, 47, 229, 193, 139, 179, 2539, 331, 359, 293, 3467, 449, 3719, 173, 257, 137, 4523, 619, 5099, 5399, 347, 439, 733, 317, 773, 577, 6679, 7019, 7723, 8087, 769, 8839, 1129, 9227, 9623, 307, 271, 1279, 10859, 11287, 719, 617, 1493, 1549, 12619, 1723, 379, 1783, 461, 283, 953, 1409, 16007, 1049, 17047, 541, 17579, 18119, 1697, 409, 421, 20359, 20939, 1327, 22123, 2803, 22727, 2879, 23339, 739, 1069, 2293, 25867, 1637, 1153, 839, 27179, 181, 27847, 2593, 29207, 1847, 1031, 827, 31307, 1979, 1033, 4139, 33479, 4231, 1801, 479, 4513, 419, 38039, 1201, 947, 4903, 39623, 5003, 40427, 5309, 3299, 5413, 1901, 1087, 4129, 521, 46279, 5839, 47147, 1487, 48023, 233, 48907, 2621, 571, 51607, 1627, 991, 857, 56267, 57223, 7213, 1877, 59159, 60139, 5557, 7703, 63127, 2069, 8081, 2833, 8209, 547, 5171, 3593, 8599, 673, 8731, 70379, 277, 1931, 9133, 73607, 74699, 4703, 1193, 76907, 79147, 80279, 6263, 82567, 2887, 7717, 2671, 2099, 10831, 3793, 10979, 88427, 5639, 1039, 2969, 313, 5867, 4973, 743, 95723, 12043, 7459, 1109, 8929, 3089, 3209, 6257, 100747, 102023, 103307, 105899, 701, 2281, 2309, 853, 109847, 6907, 1523, 14149, 7159, 115223, 1811, 117959, 14831, 7589, 1259, 1181, 123527, 1213, 7853, 126359, 11617, 16063, 129223, 3187, 2053, 10163, 12277, 1543, 136523, 373, 138007, 4597, 13093, 9049, 145547, 18289, 7741, 18481, 9337, 6529, 1061, 1733, 153287, 19259, 6733, 2143, 4271, 19853, 159623, 1823, 14657, 162839, 2557, 937, 967, 9001, 21481, 174347, 10949, 176023, 5527, 2029, 22531, 1867, 2843, 5897, 883, 184523, 23173, 186247, 1231, 983, 1489, 191467, 24043, 1277, 6121, 4799, 1123, 18049, 1229, 25153, 1889, 12689, 10733, 1987, 18869, 5107, 11117, 13259, 213067, 1163, 7411, 26981, 216779, 1237, 2141, 6011, 27919, 224299, 7039, 226199, 14197, 2221, 231947, 14557, 233879, 10253, 29599, 237767, 2713, 3761, 241687, 18743, 245639, 30829, 13033, 2063, 8117, 1373, 253639, 255659, 257687, 259723, 2963, 263819, 16553, 5657, 5701, 33623, 20771, 1093, 274199, 17203, 14653, 1721, 17599, 12289, 25889, 35731, 15101, 35999, 289067, 9067, 7103, 18269, 4019, 36809, 3371, 5659, 302123, 1307, 9817, 28069, 310987, 3001, 3041, 39293, 315467, 1531, 322247, 40423, 324523, 10177, 17321, 41281, 331399, 3779, 1319, 20929, 11587, 338347, 343019, 2689, 31397, 3907, 43613, 350087, 2311, 352459, 354839, 5563, 27479, 4073, 32693, 362027, 11351, 19181, 22853, 366859, 9007, 1597, 11731, 12149, 2767, 5981, 2551, 1301, 48781, 24547, 394007, 396523, 2617, 36277, 4549, 1283, 12589, 404119, 1949, 50993, 1787, 21673, 2347, 1423, 2273, 419527, 1709, 424727, 26627, 4871, 4831, 432587, 435223, 437867, 5021, 40289, 445847, 1471, 19501, 451207, 56569, 24317, 57923, 464747, 7283, 467479, 29303, 470219, 4561, 29819, 13007, 60331, 16691, 60679, 44257, 30689, 37879, 61729, 62081, 10597, 500887, 1427, 13691, 63499, 1607, 22273, 2083, 47093, 5903, 2699, 32647, 9883, 1459, 40739, 66383, 532523, 48677, 2351, 67481, 3571, 1483, 547223, 4547, 6269, 4001, 8713, 1499, 18133, 565127, 35603, 2237, 3319, 71963, 10891, 25229, 18181, 2791, 3323, 5693, 5653, 2383, 18661, 3673, 54709, 604907, 46771, 611147, 614279, 3347, 3517, 620567, 9721, 623723, 78163, 626887, 6043, 33161, 19739, 3607, 80153, 40277, 646039, 49943, 81359, 2579, 7433, 655723, 10271, 6827, 82981, 665479, 83389, 668747, 3499, 2287, 678599, 3697, 681899, 42929, 2801, 86689, 21881, 2731, 87943, 37117, 1741, 711959, 715339, 1907, 31249, 90053, 722123, 2381, 65957, 91331, 4493, 56599, 19979, 46309, 23957, 67829, 93481, 14143, 46957, 18367, 760007, 95219, 2239, 766999, 2089, 2609, 774023, 96973, 7549, 60083, 71329, 98299, 788167, 5197, 5779, 16921, 2441, 72949, 806027, 35201, 25357, 813227, 101879, 816839, 3301, 820459, 3943, 63671, 831367, 104149, 835019, 3631, 105983, 6113, 27529, 53453, 857099, 5651, 29683, 9803, 78593, 8951, 27191, 23567, 5749, 2657, 3491, 886987, 111109, 3823, 111581, 898327, 22003, 10273, 7487, 3067, 47881, 2081, 1973, 2243, 114913, 921223, 6469, 2677, 932779, 3769, 936647, 5101, 940523, 86209, 9137, 952199, 2909, 956107, 59879, 13151, 15031, 3557, 120739, 30431, 26479, 2251, 123209, 987659, 5623, 90149, 31051, 32117, 124703, 999623, 9631, 1003627, 63103, 91969, 11519, 5531, 1019723, 3361, 11503, 4007, 79063,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+36x-53 and
the reducible primes which appear as divisor for the first time
p | x^2+36x-53 and p < x^2+36x-53

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)
1108530.8000000.5000000.8000000.0000000.0000000.000000
21006326370.6300000.2600000.6300007.8750005.20000012.333333
31.0006501684820.6500000.1680000.65000010.3174606.46153813.027027
410.0006.6741.1805.4940.6674000.1180000.66740010.2676937.02380911.398340
5100.00067.1219.06458.0570.6712100.0906400.67121010.0570877.68135610.567347
61.000.000675.61473.259602.3550.6756140.0732590.67561410.0656138.08241410.375235
710.000.0006.782.681620.1106.162.5710.6782680.0620110.67826810.0392858.46462510.230796
8100.000.00068.004.0845.371.63462.632.4500.6800410.0537160.68004110.0261358.66238910.163363
91.000.000.000681.444.99147.411.712634.033.2790.6814450.0474120.68144510.0206498.82631110.123080
1010.000.000.0006.825.917.566424.326.9806.401.590.5860.6825920.0424330.68259210.0168298.94983510.096617


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)
122201.0000001.0000000.0000000.0000000.0000000.000000
243300.7500000.7500000.0000001.5000001.500000-nan
387520.8750000.6250000.2500002.3333331.666667inf
41614770.8750000.4375000.4375002.0000001.4000003.500000
5322210120.6875000.3125000.3750001.5714291.4285711.714286
6644416280.6875000.2500000.4375002.0000001.6000002.333333
71288132490.6328120.2500000.3828121.8409092.0000001.750000
8256161571040.6289060.2226560.4062501.9876541.7812502.122449
9512329992300.6425780.1933590.4492192.0434781.7368422.211539
101.0246661714950.6503910.1669920.4833982.0243161.7272732.152174
112.0481.3472971.0500.6577150.1450200.5126952.0225221.7368422.121212
124.0962.7275392.1880.6657710.1315920.5341802.0244991.8148152.083810
138.1925.4521.0004.4520.6655270.1220700.5434571.9992671.8552882.034735
1416.38410.9181.8309.0880.6663820.1116940.5546882.0025681.8300002.041330
1532.76821.8873.35818.5290.6679380.1024780.5654602.0046711.8349732.038842
1665.53643.9416.21237.7290.6704860.0947880.5756992.0076301.8499112.036213
17131.07288.10811.50876.6000.6722110.0877990.5844122.0051431.8525432.030268
18262.144176.62021.533155.0870.6737520.0821420.5916102.0045851.8711332.024634
19524.288353.76840.562313.2060.6747590.0773660.5973932.0029901.8837132.019550
201.048.576708.53776.535632.0020.6757140.0729890.6027242.0028301.8868652.017848
212.097.1521.419.129145.1951.273.9340.6766930.0692340.6074592.0029001.8971062.015712
224.194.3042.841.158276.2762.564.8820.6773850.0658690.6115162.0020431.9027932.013355
238.388.6085.687.961526.2495.161.7120.6780580.0627340.6153242.0019871.9047942.012456
2416.777.21611.385.8451.005.22610.380.6190.6786490.0599160.6187332.0017451.9101722.011081
2533.554.43222.791.0331.923.76920.867.2640.6792260.0573330.6218932.0016991.9137682.010214
2667.108.86445.618.3253.691.09641.927.2290.6797660.0550020.6247642.0015911.9186792.009235
27134.217.72891.297.8957.090.17084.207.7250.6802220.0528260.6273962.0013431.9208852.008426
28268.435.456182.721.53313.641.462169.080.0710.6806910.0508180.6298722.0013771.9239962.007892
29536.870.912365.660.58626.291.569339.369.0170.6810960.0489720.6321242.0011901.9273282.007150
301.073.741.824731.736.45850.725.523681.010.9350.6814830.0472420.6342412.0011361.9293462.006697
312.147.483.6481.464.266.90498.000.4071.366.266.4970.6818520.0456350.6362172.0010851.9319742.006233
324.294.967.2962.930.017.808189.544.4852.740.473.3230.6821980.0441320.6380662.0010141.9341192.005812
338.589.934.5925.862.826.790367.033.7145.495.793.0760.6825230.0427280.6397952.0009531.9363992.005418
3417.179.869.18411.730.912.890711.406.52411.019.506.3660.6828290.0414090.6414202.0008971.9382592.005080


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
122020011
243030111
385140212
4167250313
53210370316
664164120718
7128321121014117
8256572037026130
9512993366045153
101.02417162109083187
112.0482979819901441152
124.09653918435502681270
138.1921.00034465605021497
1416.3841.8306171.21309281901
1532.7683.3581.1322.22601.68911.668
1665.5366.2122.0664.14603.10613.105
17131.07211.5083.8297.67905.69315.814
18262.14421.5337.11914.414010.689110.843
19524.28840.56213.59926.963020.159120.402
201.048.57676.53525.65050.885038.222138.312
212.097.152145.19548.51596.680072.357172.837
224.194.304276.27692.222184.0540138.0841138.191
238.388.608526.249175.276350.9730262.6521263.596
2416.777.2161.005.226335.434669.7920502.7961502.429
2533.554.4321.923.769641.7981.281.9710961.6141962.154
2667.108.8643.691.0961.231.0602.460.03601.844.69111.846.404
27134.217.7287.090.1702.364.3424.725.82803.544.88913.545.280
28268.435.45613.641.4624.547.6289.093.83406.819.73116.821.730
29536.870.91226.291.5698.764.02217.527.547013.144.280113.147.288
301.073.741.82450.725.52316.906.42333.819.100025.358.667125.366.855
312.147.483.64898.000.40732.664.48465.335.923048.996.700149.003.706
324.294.967.296189.544.48563.180.473126.364.012094.764.882194.779.602
338.589.934.592367.033.714122.341.342244.692.3720183.505.2741183.528.439
3417.179.869.184711.406.524237.138.187474.268.3370355.696.3751355.710.148


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
382200101
4167433121
53212935232
6642815139784
71284928211711138
8256104574731262522
951223012310762536748
101.024495252243127119138111
112.0481.050544506254250284262
124.0962.1881.1321.056532527587542
138.1924.4522.2802.1721.1101.0701.1791.093
1416.3849.0884.6354.4532.3282.1862.3752.199
1532.76818.5299.5149.0154.7874.4834.7834.476
1665.53637.72919.24218.4879.6969.1189.7499.166
17131.07276.60039.05737.54319.61818.64419.78018.558
18262.144155.08778.84176.24639.88137.67039.95337.583
19524.288313.206159.076154.13080.56276.38780.38075.877
201.048.576632.002321.168310.834161.802154.446162.107153.647
212.097.1521.273.934646.834627.100326.137311.162326.466310.169
224.194.3042.564.8821.301.6141.263.268655.884628.241654.906625.851
238.388.6085.161.7122.617.6662.544.0461.319.1091.263.0161.317.7751.261.812
2416.777.21610.380.6195.260.0415.120.5782.649.7552.541.5642.648.1072.541.193
2533.554.43220.867.26410.565.75410.301.5105.319.2925.115.5105.318.0775.114.385
2667.108.86441.927.22921.215.86520.711.36410.675.36110.289.53310.673.49610.288.839
27134.217.72884.207.72542.590.76841.616.95721.422.65320.682.99421.419.40720.682.671
28268.435.456169.080.07185.467.51483.612.55742.986.72241.558.01242.977.94441.557.393
29536.870.912339.369.017171.459.272167.909.74586.218.68383.461.97886.212.30583.476.051
301.073.741.824681.010.935343.936.699337.074.236172.902.872167.612.942172.880.217167.614.904
312.147.483.6481.366.266.497689.781.133676.485.364346.652.115336.489.059346.649.782336.475.541
324.294.967.2962.740.473.3231.383.126.9721.357.346.351694.931.683675.311.273694.944.301675.286.066
338.589.934.5925.495.793.0762.772.825.7792.722.967.2971.392.906.3591.354.979.1091.392.978.9481.354.928.660
3417.179.869.18411.019.506.3665.557.993.5485.461.512.8182.791.596.1062.718.112.7422.791.696.3482.718.101.170


8. Check for existing Integer Sequences by OEIS

Found in Database : 53, 1, 23, 1, 107, 19, 199, 31, 13, 11, 37, 29, 523, 73, 647, 89, 41, 1, 919, 1,
Found in Database : 53, 23, 107, 19, 199, 31, 13, 11, 37, 29, 523, 73, 647, 89, 41, 919, 97, 1223, 163, 1559, 103, 47, 229, 193, 139, 179, 2539, 331, 359,
Found in Database : 11, 13, 19, 23, 29, 31, 37, 41, 47, 53, 73, 89, 97, 103, 107, 137, 139,