Inhaltsverzeichnis

Development of
Algorithmic Constructions

14:21:24
Deutsch
29.Mar 2024

Polynom = x^2+12x-59

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) = 59 = 59
f(1) = 23 = 23
f(2) = 31 = 31
f(3) = 7 = 7
f(4) = 5 = 5
f(5) = 13 = 13
f(6) = 49 = 7*7
f(7) = 37 = 37
f(8) = 101 = 101
f(9) = 65 = 5*13
f(10) = 161 = 7*23
f(11) = 97 = 97
f(12) = 229 = 229
f(13) = 133 = 7*19
f(14) = 305 = 5*61
f(15) = 173 = 173
f(16) = 389 = 389
f(17) = 217 = 7*31
f(18) = 481 = 13*37
f(19) = 265 = 5*53
f(20) = 581 = 7*83
f(21) = 317 = 317
f(22) = 689 = 13*53
f(23) = 373 = 373
f(24) = 805 = 5*7*23
f(25) = 433 = 433
f(26) = 929 = 929
f(27) = 497 = 7*71
f(28) = 1061 = 1061
f(29) = 565 = 5*113
f(30) = 1201 = 1201
f(31) = 637 = 7*7*13
f(32) = 1349 = 19*71
f(33) = 713 = 23*31
f(34) = 1505 = 5*7*43
f(35) = 793 = 13*61
f(36) = 1669 = 1669
f(37) = 877 = 877
f(38) = 1841 = 7*263
f(39) = 965 = 5*193
f(40) = 2021 = 43*47
f(41) = 1057 = 7*151
f(42) = 2209 = 47*47
f(43) = 1153 = 1153
f(44) = 2405 = 5*13*37
f(45) = 1253 = 7*179
f(46) = 2609 = 2609
f(47) = 1357 = 23*59
f(48) = 2821 = 7*13*31
f(49) = 1465 = 5*293
f(50) = 3041 = 3041
f(51) = 1577 = 19*83
f(52) = 3269 = 7*467
f(53) = 1693 = 1693
f(54) = 3505 = 5*701
f(55) = 1813 = 7*7*37
f(56) = 3749 = 23*163
f(57) = 1937 = 13*149
f(58) = 4001 = 4001
f(59) = 2065 = 5*7*59
f(60) = 4261 = 4261
f(61) = 2197 = 13*13*13
f(62) = 4529 = 7*647
f(63) = 2333 = 2333
f(64) = 4805 = 5*31*31
f(65) = 2473 = 2473
f(66) = 5089 = 7*727
f(67) = 2617 = 2617
f(68) = 5381 = 5381
f(69) = 2765 = 5*7*79
f(70) = 5681 = 13*19*23
f(71) = 2917 = 2917
f(72) = 5989 = 53*113
f(73) = 3073 = 7*439
f(74) = 6305 = 5*13*97
f(75) = 3233 = 53*61
f(76) = 6629 = 7*947
f(77) = 3397 = 43*79
f(78) = 6961 = 6961
f(79) = 3565 = 5*23*31
f(80) = 7301 = 7*7*149
f(81) = 3737 = 37*101
f(82) = 7649 = 7649
f(83) = 3913 = 7*13*43
f(84) = 8005 = 5*1601
f(85) = 4093 = 4093
f(86) = 8369 = 8369
f(87) = 4277 = 7*13*47
f(88) = 8741 = 8741
f(89) = 4465 = 5*19*47
f(90) = 9121 = 7*1303
f(91) = 4657 = 4657
f(92) = 9509 = 37*257
f(93) = 4853 = 23*211
f(94) = 9905 = 5*7*283
f(95) = 5053 = 31*163
f(96) = 10309 = 13*13*61
f(97) = 5257 = 7*751
f(98) = 10721 = 71*151
f(99) = 5465 = 5*1093
f(100) = 11141 = 13*857

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+12x-59

f(0)=59
f(1)=23
f(2)=31
f(3)=7
f(4)=5
f(5)=13
f(6)=1
f(7)=37
f(8)=101
f(9)=1
f(10)=1
f(11)=97
f(12)=229
f(13)=19
f(14)=61
f(15)=173
f(16)=389
f(17)=1
f(18)=1
f(19)=53
f(20)=83
f(21)=317
f(22)=1
f(23)=373
f(24)=1
f(25)=433
f(26)=929
f(27)=71
f(28)=1061
f(29)=113
f(30)=1201
f(31)=1
f(32)=1
f(33)=1
f(34)=43
f(35)=1
f(36)=1669
f(37)=877
f(38)=263
f(39)=193
f(40)=47
f(41)=151
f(42)=1
f(43)=1153
f(44)=1
f(45)=179
f(46)=2609
f(47)=1
f(48)=1
f(49)=293
f(50)=3041
f(51)=1
f(52)=467
f(53)=1693
f(54)=701
f(55)=1
f(56)=163
f(57)=149
f(58)=4001
f(59)=1
f(60)=4261
f(61)=1
f(62)=647
f(63)=2333
f(64)=1
f(65)=2473
f(66)=727
f(67)=2617
f(68)=5381
f(69)=79
f(70)=1
f(71)=2917
f(72)=1
f(73)=439
f(74)=1
f(75)=1
f(76)=947
f(77)=1
f(78)=6961
f(79)=1
f(80)=1
f(81)=1
f(82)=7649
f(83)=1
f(84)=1601
f(85)=4093
f(86)=8369
f(87)=1
f(88)=8741
f(89)=1
f(90)=1303
f(91)=4657
f(92)=257
f(93)=211
f(94)=283
f(95)=1
f(96)=1
f(97)=751
f(98)=1
f(99)=1093

b) Substitution of the polynom
The polynom f(x)=x^2+12x-59 could be written as f(y)= y^2-95 with x=y-6

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+6
f'(x)>2x+11

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

59, 23, 31, 7, 5, 13, 1, 37, 101, 1, 1, 97, 229, 19, 61, 173, 389, 1, 1, 53, 83, 317, 1, 373, 1, 433, 929, 71, 1061, 113, 1201, 1, 1, 1, 43, 1, 1669, 877, 263, 193, 47, 151, 1, 1153, 1, 179, 2609, 1, 1, 293, 3041, 1, 467, 1693, 701, 1, 163, 149, 4001, 1, 4261, 1, 647, 2333, 1, 2473, 727, 2617, 5381, 79, 1, 2917, 1, 439, 1, 1, 947, 1, 6961, 1, 1, 1, 7649, 1, 1601, 4093, 8369, 1, 8741, 1, 1303, 4657, 257, 211, 283, 1, 1, 751, 1, 1093, 857, 811, 503, 1, 1, 6113, 1, 6337, 1, 1, 431, 971, 13829, 541, 2861, 1039, 643, 7517, 1, 1553, 367, 8017, 1, 8273, 3361, 1, 1, 463, 337, 1, 18401, 9337, 2707, 9613, 1, 761, 1, 10177, 20641, 1, 21221, 347, 1, 1579, 4481, 11353, 1, 11657, 1, 2393, 3463, 12277, 1913, 1, 5101, 349, 331, 1, 26801, 2713, 3923, 1069, 1223, 1, 823, 1, 797, 2131, 30181, 1, 30881, 1, 1019, 15973, 1, 16333, 33029, 1, 1, 3413, 34501, 1, 1, 379, 1, 1, 443, 1429, 1, 3793, 38321, 1489, 1, 19753, 1, 2879, 40709, 1, 41521, 599, 3257, 21377, 881, 1, 677, 1, 1, 22637, 1987, 659, 461, 23497, 1103, 1, 9661, 24373, 7027, 1, 821, 1, 7283, 25717, 2731, 3739, 1, 26633, 4133, 1, 1163, 1, 1, 1, 1, 28513, 1, 1, 991, 4211, 59441, 1, 1, 1, 61409, 2381, 1783, 1, 63409, 31957, 9203, 1, 2111, 673, 5113, 33493, 587, 1, 5273, 34537, 1, 7013, 3719, 35597, 10247, 1571, 14561, 1, 1997, 37217, 773, 1, 76081, 38317, 11027, 38873, 15661, 839, 1621, 1, 6197, 1, 81701, 1327, 6373, 1, 1, 42293, 1, 809, 86341, 8693, 12503, 3389, 2063, 6379, 17981, 1, 2939, 6551, 1, 9293, 1, 1, 977, 1289, 1, 48313, 1231, 6991, 7577, 1, 99761, 1, 3259, 50833, 1, 51473, 1459, 1, 14983, 1, 106181, 1, 2287, 2351, 1, 1117, 110129, 1787, 15923, 11213, 8677, 56737, 709, 57413, 1777, 1, 6151, 1109, 1, 1699, 1, 1399, 1, 1, 24481, 61553, 769, 4789, 125221, 1, 126641, 1721, 1543, 9199, 1, 1, 1439, 65837, 4271, 13313, 1471, 1, 135329, 9719, 27361, 1, 138289, 9931, 1237, 1, 20183, 1511, 142789, 5521, 1, 72533, 145829, 1, 1, 14813, 2441, 10691, 1, 911, 1, 967, 11813, 1087, 1, 1, 2957, 11251, 6883, 1, 31981, 883, 1, 81157, 3329, 1, 1091, 1, 23767, 1, 33601, 1, 169649, 85237, 13177, 2459, 1, 1, 1, 1487, 953, 3851, 1, 89417, 3823, 2579, 5851, 1, 183089, 1877, 1607, 1, 26647, 2179, 188261, 18913, 27143, 5023, 191749, 13759, 1, 1171, 6299, 14011, 1, 19793, 28403, 1637, 200609, 1901, 5783, 101653, 3853, 1, 206021, 20693, 10939, 1, 209669, 1483, 6043, 106213, 213349, 107137, 1, 21613, 1, 1, 1, 2339, 1, 1, 2683, 3607, 32083, 1187, 1, 113717, 1, 8821, 46061, 16519, 1, 8969, 234161, 3359, 1319, 1049, 1097, 119513, 2087, 1193, 2659, 1, 1, 3499, 18917, 123457, 1433, 1, 1, 125453, 1, 126457, 8191, 1, 36563, 128477, 4229, 1423, 1, 130513, 262049, 1, 264101, 26513, 1, 133597, 1, 1, 7723, 1, 1, 19531, 3307, 1, 276581, 2833, 278689, 2293, 1, 1, 14891, 1, 1, 1, 12487, 1, 289349, 145213, 1, 20899, 7937, 147377, 3251, 1291, 298021, 1, 3299, 1907, 1951, 1, 304609, 152857, 306821, 1, 7187, 1, 1, 6791, 62701, 12101, 1, 3371, 318001, 1, 10331, 160697, 1, 1, 1, 162973, 46727, 164117, 1949, 33053, 1, 166417, 333989, 1, 67261, 5443, 14723, 1867, 5779, 34213, 49043, 13249, 1931, 9127, 1, 7591, 3469, 25111, 352741, 35393, 1, 1, 357509, 1, 1, 2543, 362309, 181757, 52103, 1, 5171, 1, 1, 1, 1583, 1, 1, 14449, 2341, 37813, 10253, 190297, 54547, 191533, 1, 27539, 29753, 194017, 2017, 1, 30137, 10343, 1, 197773, 1301, 199033, 57047, 200297, 401861, 1, 5119, 8819, 4903, 2243, 81901, 1, 1, 1, 414641, 41593, 3137, 209257, 1, 4297, 84481, 1, 32693, 1, 427621, 1, 61463, 4591, 432869, 1, 1, 1, 4517, 31391, 8317, 1, 1567, 31771, 446129, 1, 12823, 1, 6359, 1, 1, 45553, 456881, 1, 35353, 6229, 92461, 33119, 3121, 10139, 3517, 1, 470501, 18149, 67607, 237313, 1, 1, 478769, 2377, 1, 6899, 4993, 10559, 9941, 244253, 7537, 245653, 1, 13003, 1, 1, 1, 3163, 3319, 35899, 100801, 252713, 1, 1, 1, 1, 1979, 1, 515429, 36919, 4507, 1, 27431, 5333, 524081, 52553, 5791, 1, 12323, 11551, 1, 267133, 2539, 38371, 12527, 54013, 17471, 38791, 544549, 21001, 15643, 14447, 550469, 1, 79063, 1, 1, 5693, 7879, 1, 112481, 1, 1, 1, 81203, 56993, 1, 286477, 82067, 2969, 6079, 1, 580549, 291037, 15773, 1, 1, 1, 84247, 22741, 2237, 6911, 12161, 1, 598981, 1, 602081, 7019, 3581, 2281, 121661, 304933, 6719, 9887, 614561, 61613, 1, 309637, 620849, 1933, 3373, 24061, 627169, 1, 1, 4861, 1847, 317557, 1, 1, 1, 8669, 4259, 46051, 1, 64793, 649541, 46511, 1, 327193, 18743, 328813, 28663, 5417, 1, 66413, 665761, 1, 669029, 4723, 1, 1, 6689, 6389, 1, 68053, 6037, 9241, 97927, 11083, 10597, 1, 11731, 1, 1, 1, 36779, 350237, 2711, 351913, 141101, 1, 101267, 27329, 1, 1, 715621, 1, 719009, 51479, 144481, 362053, 103687, 363757, 1, 3847, 1, 8539, 32003, 1, 1, 370613, 742949, 1, 20173, 1, 107123, 16339, 753329, 1, 3089, 379273, 760289, 1, 1, 2069, 1, 1, 770789, 6547, 22123, 16871, 59833, 389797, 111623, 1, 1, 1, 788449, 395113, 1, 1, 16927, 20983, 1, 1, 802721, 1, 1, 31081, 3767, 1, 1, 407657, 35527, 11699, 19087, 13267, 9059, 413093, 165601, 414913, 1, 18119, 1, 11959, 5147, 420397, 1, 1, 1, 32621, 3917, 4217, 2693, 6581, 2311, 18679, 861089, 1, 1, 22807, 868529, 62171, 1, 1, 5441, 438937, 1, 1, 25243, 9419, 887269, 1, 11279, 89293, 4999, 1, 2371, 1, 1, 34781, 11471, 1, 130003, 91193, 1, 1, 917669, 19991, 14177, 1, 925349, 1, 10211, 93113, 933061, 467497, 19121, 1, 2267, 67339, 944689, 1, 948581, 1, 952481, 36709, 1, 9041, 1, 481133, 1, 483097, 31231, 13859, 1, 5021, 1, 69859, 15077, 490993, 1, 1, 987941, 98993, 1, 8423, 995909, 5483, 6451, 500953, 1003909, 5527, 1007921, 4391,

6. Sequence of the polynom (only primes)

59, 23, 31, 7, 5, 13, 37, 101, 97, 229, 19, 61, 173, 389, 53, 83, 317, 373, 433, 929, 71, 1061, 113, 1201, 43, 1669, 877, 263, 193, 47, 151, 1153, 179, 2609, 293, 3041, 467, 1693, 701, 163, 149, 4001, 4261, 647, 2333, 2473, 727, 2617, 5381, 79, 2917, 439, 947, 6961, 7649, 1601, 4093, 8369, 8741, 1303, 4657, 257, 211, 283, 751, 1093, 857, 811, 503, 6113, 6337, 431, 971, 13829, 541, 2861, 1039, 643, 7517, 1553, 367, 8017, 8273, 3361, 463, 337, 18401, 9337, 2707, 9613, 761, 10177, 20641, 21221, 347, 1579, 4481, 11353, 11657, 2393, 3463, 12277, 1913, 5101, 349, 331, 26801, 2713, 3923, 1069, 1223, 823, 797, 2131, 30181, 30881, 1019, 15973, 16333, 33029, 3413, 34501, 379, 443, 1429, 3793, 38321, 1489, 19753, 2879, 40709, 41521, 599, 3257, 21377, 881, 677, 22637, 1987, 659, 461, 23497, 1103, 9661, 24373, 7027, 821, 7283, 25717, 2731, 3739, 26633, 4133, 1163, 28513, 991, 4211, 59441, 61409, 2381, 1783, 63409, 31957, 9203, 2111, 673, 5113, 33493, 587, 5273, 34537, 7013, 3719, 35597, 10247, 1571, 14561, 1997, 37217, 773, 76081, 38317, 11027, 38873, 15661, 839, 1621, 6197, 81701, 1327, 6373, 42293, 809, 86341, 8693, 12503, 3389, 2063, 6379, 17981, 2939, 6551, 9293, 977, 1289, 48313, 1231, 6991, 7577, 99761, 3259, 50833, 51473, 1459, 14983, 106181, 2287, 2351, 1117, 110129, 1787, 15923, 11213, 8677, 56737, 709, 57413, 1777, 6151, 1109, 1699, 1399, 24481, 61553, 769, 4789, 125221, 126641, 1721, 1543, 9199, 1439, 65837, 4271, 13313, 1471, 135329, 9719, 27361, 138289, 9931, 1237, 20183, 1511, 142789, 5521, 72533, 145829, 14813, 2441, 10691, 911, 967, 11813, 1087, 2957, 11251, 6883, 31981, 883, 81157, 3329, 1091, 23767, 33601, 169649, 85237, 13177, 2459, 1487, 953, 3851, 89417, 3823, 2579, 5851, 183089, 1877, 1607, 26647, 2179, 188261, 18913, 27143, 5023, 191749, 13759, 1171, 6299, 14011, 19793, 28403, 1637, 200609, 1901, 5783, 101653, 3853, 206021, 20693, 10939, 209669, 1483, 6043, 106213, 213349, 107137, 21613, 2339, 2683, 3607, 32083, 1187, 113717, 8821, 46061, 16519, 8969, 234161, 3359, 1319, 1049, 1097, 119513, 2087, 1193, 2659, 3499, 18917, 123457, 1433, 125453, 126457, 8191, 36563, 128477, 4229, 1423, 130513, 262049, 264101, 26513, 133597, 7723, 19531, 3307, 276581, 2833, 278689, 2293, 14891, 12487, 289349, 145213, 20899, 7937, 147377, 3251, 1291, 298021, 3299, 1907, 1951, 304609, 152857, 306821, 7187, 6791, 62701, 12101, 3371, 318001, 10331, 160697, 162973, 46727, 164117, 1949, 33053, 166417, 333989, 67261, 5443, 14723, 1867, 5779, 34213, 49043, 13249, 1931, 9127, 7591, 3469, 25111, 352741, 35393, 357509, 2543, 362309, 181757, 52103, 5171, 1583, 14449, 2341, 37813, 10253, 190297, 54547, 191533, 27539, 29753, 194017, 2017, 30137, 10343, 197773, 1301, 199033, 57047, 200297, 401861, 5119, 8819, 4903, 2243, 81901, 414641, 41593, 3137, 209257, 4297, 84481, 32693, 427621, 61463, 4591, 432869, 4517, 31391, 8317, 1567, 31771, 446129, 12823, 6359, 45553, 456881, 35353, 6229, 92461, 33119, 3121, 10139, 3517, 470501, 18149, 67607, 237313, 478769, 2377, 6899, 4993, 10559, 9941, 244253, 7537, 245653, 13003, 3163, 3319, 35899, 100801, 252713, 1979, 515429, 36919, 4507, 27431, 5333, 524081, 52553, 5791, 12323, 11551, 267133, 2539, 38371, 12527, 54013, 17471, 38791, 544549, 21001, 15643, 14447, 550469, 79063, 5693, 7879, 112481, 81203, 56993, 286477, 82067, 2969, 6079, 580549, 291037, 15773, 84247, 22741, 2237, 6911, 12161, 598981, 602081, 7019, 3581, 2281, 121661, 304933, 6719, 9887, 614561, 61613, 309637, 620849, 1933, 3373, 24061, 627169, 4861, 1847, 317557, 8669, 4259, 46051, 64793, 649541, 46511, 327193, 18743, 328813, 28663, 5417, 66413, 665761, 669029, 4723, 6689, 6389, 68053, 6037, 9241, 97927, 11083, 10597, 11731, 36779, 350237, 2711, 351913, 141101, 101267, 27329, 715621, 719009, 51479, 144481, 362053, 103687, 363757, 3847, 8539, 32003, 370613, 742949, 20173, 107123, 16339, 753329, 3089, 379273, 760289, 2069, 770789, 6547, 22123, 16871, 59833, 389797, 111623, 788449, 395113, 16927, 20983, 802721, 31081, 3767, 407657, 35527, 11699, 19087, 13267, 9059, 413093, 165601, 414913, 18119, 11959, 5147, 420397, 32621, 3917, 4217, 2693, 6581, 2311, 18679, 861089, 22807, 868529, 62171, 5441, 438937, 25243, 9419, 887269, 11279, 89293, 4999, 2371, 34781, 11471, 130003, 91193, 917669, 19991, 14177, 925349, 10211, 93113, 933061, 467497, 19121, 2267, 67339, 944689, 948581, 952481, 36709, 9041, 481133, 483097, 31231, 13859, 5021, 69859, 15077, 490993, 987941, 98993, 8423, 995909, 5483, 6451, 500953, 1003909, 5527, 1007921, 4391,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+12x-59 and
the reducible primes which appear as divisor for the first time
p | x^2+12x-59 and p < x^2+12x-59

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)
1108440.8000000.4000000.8000000.0000000.0000000.000000
21006419450.6400000.1900000.6400008.0000004.75000011.250000
31.0006801105700.6800000.1100000.68000010.6250005.78947412.666667
410.0006.8877816.1060.6887000.0781000.68870010.1279417.10000010.712280
5100.00068.8595.86862.9910.6885900.0586800.6885909.9984037.51344410.316246
61.000.000689.95648.090641.8660.6899560.0480900.68995610.0198378.19529610.189805
710.000.0006.899.878406.6966.493.1820.6899880.0406700.68998810.0004618.45697710.116102
8100.000.00069.025.6423.528.01065.497.6320.6902560.0352800.69025610.0038938.67480910.087139
91.000.000.000690.465.66931.140.414659.325.2550.6904660.0311400.69046610.0030328.82662310.066399
1010.000.000.0006.906.551.223278.630.7686.627.920.4550.6906550.0278630.69065510.0027448.94756210.052581


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)
123211.5000001.0000000.5000000.0000000.0000000.000000
245321.2500000.7500000.5000001.6666671.5000002.000000
388441.0000000.5000000.5000001.6000001.3333332.000000
41614680.8750000.3750000.5000001.7500001.5000002.000000
532249150.7500000.2812500.4687501.7142861.5000001.875000
6644314290.6718750.2187500.4531251.7916671.5555561.933333
71288320630.6484380.1562500.4921881.9302331.4285712.172414
8256167341330.6523440.1328120.5195312.0120481.7000002.111111
9512350572930.6835940.1113280.5722662.0958081.6764712.203007
101.0246981125860.6816410.1093750.5722661.9942861.9649122.000000
112.0481.4031981.2050.6850590.0966800.5883792.0100291.7678572.056314
124.0962.8063642.4420.6850590.0888670.5961912.0000001.8383842.026556
138.1925.6526495.0030.6899410.0792240.6107182.0142551.7829672.048731
1416.38411.2631.19910.0640.6874390.0731810.6142581.9927461.8474582.011593
1532.76822.5812.17420.4070.6891170.0663450.6227722.0048831.8131782.027723
1665.53645.1154.06541.0500.6884000.0620270.6263731.9979191.8698252.011565
17131.07290.2887.52782.7610.6888430.0574260.6314162.0012861.8516602.016102
18262.144180.62514.058166.5670.6890300.0536270.6354032.0005431.8676762.012627
19524.288361.56526.523335.0420.6896310.0505890.6390422.0017441.8866842.011455
201.048.576723.43950.232673.2070.6899250.0479050.6420202.0008551.8939032.009321
212.097.1521.446.85695.5201.351.3360.6899150.0455470.6443671.9999701.9015772.007311
224.194.3042.893.888181.6092.712.2790.6899570.0432990.6466582.0001221.9012672.007109
238.388.6085.787.835345.3745.442.4610.6899640.0411720.6487922.0000211.9017452.006601
2416.777.21611.577.607659.75810.917.8490.6900790.0393250.6507552.0003351.9102712.006050
2533.554.43223.157.4201.263.40921.894.0110.6901450.0376530.6524922.0001901.9149582.005341
2667.108.86446.319.0692.423.98043.895.0890.6902080.0361200.6540882.0001831.9186032.004890
27134.217.72892.647.6124.656.43187.991.1810.6902790.0346930.6555852.0002051.9209862.004579
28268.435.456185.312.9258.960.633176.352.2920.6903440.0333810.6569632.0001911.9243562.004204
29536.870.912370.661.32117.264.409353.396.9120.6904100.0321570.6582532.0001911.9266952.003926
301.073.741.824741.388.98233.316.884708.072.0980.6904720.0310290.6594442.0001791.9298022.003617
312.147.483.6481.482.905.88864.357.1221.418.548.7660.6905320.0299690.6605632.0001721.9316672.003396
324.294.967.2962.966.062.224124.471.7782.841.590.4460.6905900.0289810.6616092.0001691.9340792.003167
338.589.934.5925.932.580.981241.016.8835.691.564.0980.6906430.0280580.6625852.0001541.9363182.002950
3417.179.869.18411.866.084.823467.149.89511.398.934.9280.6906970.0271920.6635052.0001551.9382462.002777


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
122110101
243120111
384130121
4166240141
5329362151
66414595171
71282061481101
8256341123171151
9512571839281271
101.0241123676511591
112.04819866132981981
124.09636411624818311791
138.19264921143831713301
1416.3841.19939980058616111
1532.7682.1747091.4651.08111.0911
1665.5364.0651.3482.7172.01812.0451
17131.0727.5272.4745.0533.74413.7811
18262.14414.0584.7029.3567.07916.9771
19524.28826.5238.84917.67413.333113.1881
201.048.57650.23216.68333.54925.186125.0441
212.097.15295.52031.74363.77747.869147.6491
224.194.304181.60960.472121.13790.909190.6981
238.388.608345.374114.885230.489172.8581172.5141
2416.777.216659.758219.687440.071330.2061329.5501
2533.554.4321.263.409420.836842.573632.8111630.5961
2667.108.8642.423.980808.3411.615.6391.212.70411.211.2741
27134.217.7284.656.4311.552.0873.104.3442.328.97912.327.4501
28268.435.4568.960.6332.986.2425.974.3914.481.23014.479.4011
29536.870.91217.264.4095.753.08711.511.3228.632.87018.631.5371
301.073.741.82433.316.88411.106.44222.210.44216.660.521116.656.3611
312.147.483.64864.357.12221.454.89542.902.22732.177.824132.179.2961
324.294.967.296124.471.77841.497.27682.974.50262.235.077162.236.6991
338.589.934.592241.016.88380.345.875160.671.008120.507.0291120.509.8521
3417.179.869.184467.149.895155.718.746311.431.149233.569.5831233.580.3101


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
121010001
242110002
384310022
4168621052
53215873183
66429141554146
712863362718102015
8256133736039294124
951229314714678638765
101.024586298288147137165137
112.0481.205628577312265345283
124.0962.4421.2621.180636572687547
138.1925.0032.6112.3921.2971.1881.3761.142
1416.38410.0645.2364.8282.6332.3632.7102.358
1532.76820.40710.6279.7805.3454.8495.4164.797
1665.53641.05021.38119.66910.6909.77710.8729.711
17131.07282.76142.79039.97121.61019.67321.77519.703
18262.144166.56785.62380.94443.38139.93443.46039.792
19524.288335.042171.968163.07487.08280.47187.22080.269
201.048.576673.207344.965328.242175.008161.471175.279161.449
212.097.1521.351.336691.928659.408350.830325.104351.318324.084
224.194.3042.712.2791.386.8681.325.411703.433652.845703.593652.408
238.388.6085.442.4612.781.0062.661.4551.409.0891.311.9241.410.1251.311.323
2416.777.21610.917.8495.572.8685.344.9812.822.0182.636.3652.821.5772.637.889
2533.554.43221.894.01111.162.13610.731.8755.651.8545.294.3395.651.2935.296.525
2667.108.86443.895.08922.363.28021.531.80911.324.01610.627.47711.314.95510.628.641
27134.217.72887.991.18144.790.08943.201.09222.665.64121.332.93522.660.39621.332.209
28268.435.456176.352.29289.701.45586.650.83745.377.96742.805.20545.367.61642.801.504
29536.870.912353.396.912179.637.920173.758.99290.832.26285.865.69090.825.05685.873.904
301.073.741.824708.072.098359.706.342348.365.756181.817.773172.210.754181.801.695172.241.876
312.147.483.6481.418.548.766720.224.003698.324.763363.894.345345.357.763363.895.852345.400.806
324.294.967.2962.841.590.4461.441.977.6801.399.612.766728.315.900692.465.064728.316.938692.492.544
338.589.934.5925.691.564.0982.886.779.2412.804.784.8571.457.667.4591.388.154.8191.457.607.4281.388.134.392
3417.179.869.18411.398.934.9285.778.855.6415.620.079.2872.917.223.6652.782.298.8942.917.134.4522.782.277.917


8. Check for existing Integer Sequences by OEIS

Found in Database : 59, 23, 31, 7, 5, 13, 1, 37, 101, 1, 1, 97, 229, 19, 61, 173, 389, 1, 1, 53,
Found in Database : 59, 23, 31, 7, 5, 13, 37, 101, 97, 229, 19, 61, 173, 389, 53, 83, 317, 373, 433, 929, 71, 1061, 113, 1201, 43, 1669, 877, 263, 193,
Found in Database : 5, 7, 13, 19, 23, 31, 37, 43, 47, 53, 59, 61, 71, 79, 83, 97, 101, 113, 149,