Inhaltsverzeichnis

Development of
Algorithmic Constructions

12:35:08
Deutsch
29.Mar 2024

Polynom = x^2-78x+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) = 3 = 3
f(2) = 69 = 3*23
f(3) = 71 = 71
f(4) = 213 = 3*71
f(5) = 141 = 3*47
f(6) = 349 = 349
f(7) = 207 = 3*3*23
f(8) = 477 = 3*3*53
f(9) = 269 = 269
f(10) = 597 = 3*199
f(11) = 327 = 3*109
f(12) = 709 = 709
f(13) = 381 = 3*127
f(14) = 813 = 3*271
f(15) = 431 = 431
f(16) = 909 = 3*3*101
f(17) = 477 = 3*3*53
f(18) = 997 = 997
f(19) = 519 = 3*173
f(20) = 1077 = 3*359
f(21) = 557 = 557
f(22) = 1149 = 3*383
f(23) = 591 = 3*197
f(24) = 1213 = 1213
f(25) = 621 = 3*3*3*23
f(26) = 1269 = 3*3*3*47
f(27) = 647 = 647
f(28) = 1317 = 3*439
f(29) = 669 = 3*223
f(30) = 1357 = 23*59
f(31) = 687 = 3*229
f(32) = 1389 = 3*463
f(33) = 701 = 701
f(34) = 1413 = 3*3*157
f(35) = 711 = 3*3*79
f(36) = 1429 = 1429
f(37) = 717 = 3*239
f(38) = 1437 = 3*479
f(39) = 719 = 719
f(40) = 1437 = 3*479
f(41) = 717 = 3*239
f(42) = 1429 = 1429
f(43) = 711 = 3*3*79
f(44) = 1413 = 3*3*157
f(45) = 701 = 701
f(46) = 1389 = 3*463
f(47) = 687 = 3*229
f(48) = 1357 = 23*59
f(49) = 669 = 3*223
f(50) = 1317 = 3*439
f(51) = 647 = 647
f(52) = 1269 = 3*3*3*47
f(53) = 621 = 3*3*3*23
f(54) = 1213 = 1213
f(55) = 591 = 3*197
f(56) = 1149 = 3*383
f(57) = 557 = 557
f(58) = 1077 = 3*359
f(59) = 519 = 3*173
f(60) = 997 = 997
f(61) = 477 = 3*3*53
f(62) = 909 = 3*3*101
f(63) = 431 = 431
f(64) = 813 = 3*271
f(65) = 381 = 3*127
f(66) = 709 = 709
f(67) = 327 = 3*109
f(68) = 597 = 3*199
f(69) = 269 = 269
f(70) = 477 = 3*3*53
f(71) = 207 = 3*3*23
f(72) = 349 = 349
f(73) = 141 = 3*47
f(74) = 213 = 3*71
f(75) = 71 = 71
f(76) = 69 = 3*23
f(77) = 3 = 3
f(78) = 83 = 83
f(79) = 81 = 3*3*3*3
f(80) = 243 = 3*3*3*3*3
f(81) = 163 = 163
f(82) = 411 = 3*137
f(83) = 249 = 3*83
f(84) = 587 = 587
f(85) = 339 = 3*113
f(86) = 771 = 3*257
f(87) = 433 = 433
f(88) = 963 = 3*3*107
f(89) = 531 = 3*3*59
f(90) = 1163 = 1163
f(91) = 633 = 3*211
f(92) = 1371 = 3*457
f(93) = 739 = 739
f(94) = 1587 = 3*23*23
f(95) = 849 = 3*283
f(96) = 1811 = 1811
f(97) = 963 = 3*3*107
f(98) = 2043 = 3*3*227
f(99) = 1081 = 23*47
f(100) = 2283 = 3*761

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^2-78x+83

f(0)=83
f(1)=3
f(2)=23
f(3)=71
f(4)=1
f(5)=47
f(6)=349
f(7)=1
f(8)=53
f(9)=269
f(10)=199
f(11)=109
f(12)=709
f(13)=127
f(14)=271
f(15)=431
f(16)=101
f(17)=1
f(18)=997
f(19)=173
f(20)=359
f(21)=557
f(22)=383
f(23)=197
f(24)=1213
f(25)=1
f(26)=1
f(27)=647
f(28)=439
f(29)=223
f(30)=59
f(31)=229
f(32)=463
f(33)=701
f(34)=157
f(35)=79
f(36)=1429
f(37)=239
f(38)=479
f(39)=719
f(40)=1
f(41)=1
f(42)=1
f(43)=1
f(44)=1
f(45)=1
f(46)=1
f(47)=1
f(48)=1
f(49)=1
f(50)=1
f(51)=1
f(52)=1
f(53)=1
f(54)=1
f(55)=1
f(56)=1
f(57)=1
f(58)=1
f(59)=1
f(60)=1
f(61)=1
f(62)=1
f(63)=1
f(64)=1
f(65)=1
f(66)=1
f(67)=1
f(68)=1
f(69)=1
f(70)=1
f(71)=1
f(72)=1
f(73)=1
f(74)=1
f(75)=1
f(76)=1
f(77)=1
f(78)=1
f(79)=1
f(80)=1
f(81)=163
f(82)=137
f(83)=1
f(84)=587
f(85)=113
f(86)=257
f(87)=433
f(88)=107
f(89)=1
f(90)=1163
f(91)=211
f(92)=457
f(93)=739
f(94)=1
f(95)=283
f(96)=1811
f(97)=1
f(98)=227
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-78x+83 could be written as f(y)= y^2-1438 with x=y+39

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-39
f'(x)>2x-79 with x > 38

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, 3, 23, 71, 1, 47, 349, 1, 53, 269, 199, 109, 709, 127, 271, 431, 101, 1, 997, 173, 359, 557, 383, 197, 1213, 1, 1, 647, 439, 223, 59, 229, 463, 701, 157, 79, 1429, 239, 479, 719, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 163, 137, 1, 587, 113, 257, 433, 107, 1, 1163, 211, 457, 739, 1, 283, 1811, 1, 227, 1, 761, 401, 2531, 443, 929, 1459, 1, 1, 3323, 577, 1201, 1873, 1297, 673, 1, 241, 499, 1, 1601, 827, 1, 881, 1, 1, 643, 331, 6131, 1051, 2161, 3331, 2281, 1171, 7211, 1, 281, 3889, 2657, 1361, 8363, 1427, 1, 4483, 1019, 521, 9587, 1, 1, 5113, 1, 1777, 10883, 617, 1259, 5779, 3929, 2003, 12251, 2081, 4241, 6481, 1, 1, 13691, 1, 4729, 7219, 1, 1, 661, 859, 1747, 7993, 5417, 2753, 16787, 2843, 1, 8803, 1987, 1009, 18443, 3121, 6337, 9649, 6529, 3313, 877, 379, 769, 10531, 7121, 1, 1, 1, 7529, 1, 2579, 1307, 1, 4027, 8161, 1, 8377, 4243, 1, 1451, 2939, 1, 9041, 1, 27803, 4691, 9497, 14419, 1, 547, 421, 1, 1, 1, 1, 5281, 32051, 1801, 3643, 1, 11177, 5651, 34283, 1, 11681, 17713, 1, 2011, 36587, 6163, 12457, 821, 12721, 6427, 829, 1, 491, 20089, 1, 6833, 41411, 6971, 14081, 1, 4787, 2417, 1, 7393, 14929, 983, 15217, 7681, 46523, 2609, 1, 509, 16097, 8123, 487, 8273, 1, 1, 1889, 953, 1, 8731, 1, 503, 17929, 9043, 1, 3067, 1, 28081, 1, 9521, 1087, 1, 1, 1, 6619, 1, 853, 10177, 20521, 31033, 20857, 10513, 63587, 1187, 2393, 32563, 21881, 11027, 66683, 1, 1, 34129, 7643, 3851, 3037, 11731, 1, 35731, 24001, 1, 73091, 4091, 8243, 37369, 25097, 12641, 76403, 1, 25841, 39043, 971, 1, 3469, 1, 1, 1, 27361, 599, 83243, 4657, 1, 42499, 607, 1, 86771, 14561, 1, 44281, 9907, 4987, 90371, 1, 30529, 46099, 30937, 677, 1, 1753, 3529, 1, 1399, 16193, 97787, 1, 1, 49843, 1, 1, 101603, 17041, 34297, 51769, 34729, 1, 105491, 5897, 11867, 53731, 1567, 18131, 109451, 18353, 36929, 2423, 4153, 2089, 1, 1, 38281, 1229, 38737, 19483, 1993, 6571, 13219, 59833, 757, 20177, 121763, 20411, 41057, 2693, 1, 6961, 126011, 21121, 1847, 64081, 42961, 21601, 1, 809, 1627, 1, 44417, 1, 1, 1, 1, 68473, 15299, 7691, 139187, 1, 2039, 1, 1, 23827, 1423, 1, 16139, 73009, 1, 24593, 148331, 24851, 1063, 1061, 1, 2819, 2887, 25633, 1, 77689, 52057, 26161, 1997, 1, 17707, 1511, 53657, 1, 7069, 1, 1, 1, 18427, 1, 167483, 28051, 56377, 84979, 56929, 28603, 1, 3209, 6449, 87481, 58601, 1, 7717, 29723, 59729, 90019, 1, 1, 182603, 30577, 61441, 92593, 62017, 31153, 187787, 1, 21059, 95203, 63761, 32027, 193043, 32321, 64937, 97849, 1, 1, 1217, 33211, 66721, 1, 2927, 33811, 203771, 1, 1, 1, 1471, 34721, 2521, 35027, 991, 1493, 1, 1, 214787, 1, 1, 108793, 3167, 1, 220403, 4099, 8233, 1, 74729, 1, 2113, 1, 76001, 1, 1, 1, 4933, 38803, 77929, 1097, 78577, 39451, 237683, 13259, 26627, 5231, 80537, 40433, 243587, 40763, 3559, 1091, 1, 4603, 249563, 41761, 83857, 2687, 1, 42433, 1, 1, 28627, 1, 86561, 43451, 1153, 43793, 3823, 132409, 1, 14827, 2371, 1949, 90001, 2297, 90697, 45523, 274187, 1699, 1, 1373, 92801, 1, 1721, 1, 1327, 1117, 31643, 15881, 3457, 2087, 1, 2459, 97081, 1, 12757, 16361, 32843, 148339, 99257, 49811, 3797, 50177, 2143, 2861, 11273, 5657, 306587, 51283, 1303, 1, 103681, 52027, 1, 17467, 35059, 158329, 105929, 2311, 1181, 1, 107441, 161731, 36067, 18097, 2999, 54673, 1, 1, 2351, 55441, 333803, 6203, 1, 168643, 1, 1, 340787, 56993, 114377, 172153, 1669, 19259, 2539, 58171, 1, 175699, 117529, 58963, 354971, 1, 1, 179281, 119921, 60161, 5101, 1, 2293, 182899, 1, 2273, 2909, 1, 1, 8111, 124777, 1, 376787, 21001, 42139, 190243, 127241, 1, 384203, 1, 1, 4127, 1, 1, 391691, 1, 1301, 8597, 132241, 1, 2543, 7417, 1, 2551, 134777, 67601, 406883, 1, 136481, 1499, 45779, 22961, 8821, 69313, 1, 209233, 139921, 70177, 422363, 23537, 2053, 213139, 142529, 71483, 1531, 1, 1, 217081, 1, 8089, 438131, 73243, 1, 221059, 2789, 74131, 446123, 24859, 1, 1, 150497, 1, 6397, 3301, 152297, 229123, 1, 25609, 20101, 1, 155017, 2309, 1, 78193, 470531, 1, 5843, 237331, 1483, 1693, 478811, 1, 1, 241489, 53819, 26987, 1, 1, 163321, 245683, 1979, 3581, 495587, 27611, 1, 3011, 167081, 83777, 9511, 1, 168977, 254179, 1, 9467, 8689, 1823, 1, 4877, 172801, 3767, 1, 1, 58243, 262819, 7639, 1, 1, 1, 1, 267193, 59539, 29851, 4943, 1, 1, 271603, 3863, 1, 547643, 10169, 20393, 276049, 1, 1, 1, 1, 1, 12197, 62507, 31337, 565571, 94513, 189529, 285049, 190537, 1619, 574643, 32009, 64187, 289603, 193577, 1831, 12421, 97553, 1, 12791, 7283, 1, 593003, 1, 1, 298819, 199729, 1697, 2239, 33547, 1, 1933, 202841, 101681, 1597, 102203, 2029, 1, 68659, 1, 621083, 1, 1, 1, 1, 1, 630587, 1, 23473, 2969, 1, 106427, 640163, 106961, 214457, 1637, 71843, 36011, 6073, 108571, 217681, 327331, 218761, 2333, 659531, 1, 73643, 4679, 1, 1, 29101, 111827, 224201, 1, 25033, 12547, 3413, 2141, 227497, 342073, 228601, 114577, 6823, 38377, 1, 347059, 3931, 116243, 1, 1, 234161, 352081, 78427, 1709, 709211, 1, 1, 357139, 238657, 119611, 3169, 4451, 8923, 362233, 242057, 121313, 729587, 121883, 1, 3251, 1741, 1783, 1, 123601, 1951, 1, 1, 124753, 1, 41777, 1, 7127, 252401, 126491, 9629, 1, 254729, 6491, 28433, 14249, 14551, 2741, 258241, 1, 11279, 130003, 4519, 1, 1, 1, 1, 131777, 792443, 132371, 265337, 6761, 88843, 1, 803171, 134161, 268921, 3709, 270121, 1, 813971, 15107, 1, 1, 1, 1, 824843, 137777, 12007, 415153, 92459, 1, 6581, 1, 279817, 1, 1, 140827, 1, 47147, 1, 1861, 284729, 142673, 18253, 143291, 12487, 431731, 1, 1, 1877, 6311, 4931, 2683, 1, 1, 880283, 49009, 98227, 442963, 295937, 1, 891587, 148913, 1901, 448633, 99907, 1, 902963, 1, 1, 1, 303529, 152083, 1, 1, 1, 460081,

6. Sequence of the polynom (only primes)

83, 3, 23, 71, 47, 349, 53, 269, 199, 109, 709, 127, 271, 431, 101, 997, 173, 359, 557, 383, 197, 1213, 647, 439, 223, 59, 229, 463, 701, 157, 79, 1429, 239, 479, 719, 163, 137, 587, 113, 257, 433, 107, 1163, 211, 457, 739, 283, 1811, 227, 761, 401, 2531, 443, 929, 1459, 3323, 577, 1201, 1873, 1297, 673, 241, 499, 1601, 827, 881, 643, 331, 6131, 1051, 2161, 3331, 2281, 1171, 7211, 281, 3889, 2657, 1361, 8363, 1427, 4483, 1019, 521, 9587, 5113, 1777, 10883, 617, 1259, 5779, 3929, 2003, 12251, 2081, 4241, 6481, 13691, 4729, 7219, 661, 859, 1747, 7993, 5417, 2753, 16787, 2843, 8803, 1987, 1009, 18443, 3121, 6337, 9649, 6529, 3313, 877, 379, 769, 10531, 7121, 7529, 2579, 1307, 4027, 8161, 8377, 4243, 1451, 2939, 9041, 27803, 4691, 9497, 14419, 547, 421, 5281, 32051, 1801, 3643, 11177, 5651, 34283, 11681, 17713, 2011, 36587, 6163, 12457, 821, 12721, 6427, 829, 491, 20089, 6833, 41411, 6971, 14081, 4787, 2417, 7393, 14929, 983, 15217, 7681, 46523, 2609, 509, 16097, 8123, 487, 8273, 1889, 953, 8731, 503, 17929, 9043, 3067, 28081, 9521, 1087, 6619, 853, 10177, 20521, 31033, 20857, 10513, 63587, 1187, 2393, 32563, 21881, 11027, 66683, 34129, 7643, 3851, 3037, 11731, 35731, 24001, 73091, 4091, 8243, 37369, 25097, 12641, 76403, 25841, 39043, 971, 3469, 27361, 599, 83243, 4657, 42499, 607, 86771, 14561, 44281, 9907, 4987, 90371, 30529, 46099, 30937, 677, 1753, 3529, 1399, 16193, 97787, 49843, 101603, 17041, 34297, 51769, 34729, 105491, 5897, 11867, 53731, 1567, 18131, 109451, 18353, 36929, 2423, 4153, 2089, 38281, 1229, 38737, 19483, 1993, 6571, 13219, 59833, 757, 20177, 121763, 20411, 41057, 2693, 6961, 126011, 21121, 1847, 64081, 42961, 21601, 809, 1627, 44417, 68473, 15299, 7691, 139187, 2039, 23827, 1423, 16139, 73009, 24593, 148331, 24851, 1063, 1061, 2819, 2887, 25633, 77689, 52057, 26161, 1997, 17707, 1511, 53657, 7069, 18427, 167483, 28051, 56377, 84979, 56929, 28603, 3209, 6449, 87481, 58601, 7717, 29723, 59729, 90019, 182603, 30577, 61441, 92593, 62017, 31153, 187787, 21059, 95203, 63761, 32027, 193043, 32321, 64937, 97849, 1217, 33211, 66721, 2927, 33811, 203771, 1471, 34721, 2521, 35027, 991, 1493, 214787, 108793, 3167, 220403, 4099, 8233, 74729, 2113, 76001, 4933, 38803, 77929, 1097, 78577, 39451, 237683, 13259, 26627, 5231, 80537, 40433, 243587, 40763, 3559, 1091, 4603, 249563, 41761, 83857, 2687, 42433, 28627, 86561, 43451, 1153, 43793, 3823, 132409, 14827, 2371, 1949, 90001, 2297, 90697, 45523, 274187, 1699, 1373, 92801, 1721, 1327, 1117, 31643, 15881, 3457, 2087, 2459, 97081, 12757, 16361, 32843, 148339, 99257, 49811, 3797, 50177, 2143, 2861, 11273, 5657, 306587, 51283, 1303, 103681, 52027, 17467, 35059, 158329, 105929, 2311, 1181, 107441, 161731, 36067, 18097, 2999, 54673, 2351, 55441, 333803, 6203, 168643, 340787, 56993, 114377, 172153, 1669, 19259, 2539, 58171, 175699, 117529, 58963, 354971, 179281, 119921, 60161, 5101, 2293, 182899, 2273, 2909, 8111, 124777, 376787, 21001, 42139, 190243, 127241, 384203, 4127, 391691, 1301, 8597, 132241, 2543, 7417, 2551, 134777, 67601, 406883, 136481, 1499, 45779, 22961, 8821, 69313, 209233, 139921, 70177, 422363, 23537, 2053, 213139, 142529, 71483, 1531, 217081, 8089, 438131, 73243, 221059, 2789, 74131, 446123, 24859, 150497, 6397, 3301, 152297, 229123, 25609, 20101, 155017, 2309, 78193, 470531, 5843, 237331, 1483, 1693, 478811, 241489, 53819, 26987, 163321, 245683, 1979, 3581, 495587, 27611, 3011, 167081, 83777, 9511, 168977, 254179, 9467, 8689, 1823, 4877, 172801, 3767, 58243, 262819, 7639, 267193, 59539, 29851, 4943, 271603, 3863, 547643, 10169, 20393, 276049, 12197, 62507, 31337, 565571, 94513, 189529, 285049, 190537, 1619, 574643, 32009, 64187, 289603, 193577, 1831, 12421, 97553, 12791, 7283, 593003, 298819, 199729, 1697, 2239, 33547, 1933, 202841, 101681, 1597, 102203, 2029, 68659, 621083, 630587, 23473, 2969, 106427, 640163, 106961, 214457, 1637, 71843, 36011, 6073, 108571, 217681, 327331, 218761, 2333, 659531, 73643, 4679, 29101, 111827, 224201, 25033, 12547, 3413, 2141, 227497, 342073, 228601, 114577, 6823, 38377, 347059, 3931, 116243, 234161, 352081, 78427, 1709, 709211, 357139, 238657, 119611, 3169, 4451, 8923, 362233, 242057, 121313, 729587, 121883, 3251, 1741, 1783, 123601, 1951, 124753, 41777, 7127, 252401, 126491, 9629, 254729, 6491, 28433, 14249, 14551, 2741, 258241, 11279, 130003, 4519, 131777, 792443, 132371, 265337, 6761, 88843, 803171, 134161, 268921, 3709, 270121, 813971, 15107, 824843, 137777, 12007, 415153, 92459, 6581, 279817, 140827, 47147, 1861, 284729, 142673, 18253, 143291, 12487, 431731, 1877, 6311, 4931, 2683, 880283, 49009, 98227, 442963, 295937, 891587, 148913, 1901, 448633, 99907, 902963, 303529, 152083, 460081,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-78x+83 and
the reducible primes which appear as divisor for the first time
p | x^2-78x+83 and p < x^2-78x+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)
1109540.9000000.5000000.9000000.0000000.0000000.000000
21004620260.4600000.2000000.4600005.1111114.0000006.500000
31.0006821725100.6820000.1720000.68200014.8260878.60000019.615385
410.0007.2241.2755.9490.7224000.1275000.72240010.5923767.41279111.664706
5100.00071.8959.79862.0970.7189500.0979800.7189509.9522437.68470610.438225
61.000.000715.04479.296635.7480.7150440.0792960.7150449.9456718.09308110.237983
710.000.0007.116.393668.1386.448.2550.7116390.0668140.7116399.9523858.42587310.142784
8100.000.00070.912.4885.785.00965.127.4790.7091250.0578500.7091259.9646678.65840410.100017
91.000.000.000707.170.97350.992.742656.178.2310.7071710.0509930.7071719.9724458.81463510.075290
1010.000.000.0007.056.350.773455.850.7096.600.500.0640.7056350.0455850.7056359.9782818.93952110.059005


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
244311.0000000.7500000.2500001.3333331.5000001.000000
387430.8750000.5000000.3750001.7500001.3333333.000000
41615780.9375000.4375000.5000002.1428571.7500002.666667
5322711160.8437500.3437500.5000001.8000001.5714292.000000
6643414200.5312500.2187500.3125001.2592591.2727271.250000
71286725420.5234380.1953120.3281251.9705881.7857142.100000
8256162521100.6328120.2031250.4296882.4179102.0800002.619048
95123401002400.6640620.1953120.4687502.0987651.9230772.181818
101.0247021755270.6855470.1708980.5146482.0647061.7500002.195833
112.0481.4583281.1300.7119140.1601560.5517582.0769231.8742862.144212
124.0962.9545842.3700.7211910.1425780.5786132.0260631.7804882.097345
138.1925.9161.0814.8350.7221680.1319580.5902102.0027081.8510272.040084
1416.38411.8471.9429.9050.7230830.1185300.6045532.0025361.7964852.048604
1532.76823.6353.58520.0500.7212830.1094060.6118771.9950201.8460352.024230
1665.53647.1526.68840.4640.7194820.1020510.6174321.9950071.8655512.018155
17131.07294.22212.45781.7650.7188570.0950390.6238171.9982611.8625902.020685
18262.144188.11923.215164.9040.7176170.0885580.6290591.9965511.8636112.016804
19524.288375.54143.754331.7870.7162880.0834540.6328331.9962951.8847302.012001
201.048.576749.76682.903666.8630.7150330.0790620.6359701.9964961.8947532.009913
212.097.1521.496.853156.9991.339.8540.7137550.0748630.6388921.9964271.8937672.009189
224.194.3042.989.708297.9942.691.7140.7128020.0710470.6417551.9973291.8980632.008961
238.388.6085.971.279567.1665.404.1130.7118320.0676110.6442201.9972781.9032802.007685
2416.777.21611.928.9751.082.91810.846.0570.7110220.0645470.6464751.9977251.9093492.007000
2533.554.43223.833.0842.073.11621.759.9680.7102810.0617840.6484981.9979161.9143792.006256
2667.108.86447.614.6893.974.82343.639.8660.7095140.0592290.6502851.9978401.9173182.005512
27134.217.72895.138.6097.632.43287.506.1770.7088380.0568660.6519721.9980941.9201942.005189
28268.435.456190.110.38814.682.491175.427.8970.7082160.0546970.6535201.9982461.9236982.004749
29536.870.912379.916.50328.282.702351.633.8010.7076500.0526810.6549691.9984001.9262882.004435
301.073.741.824759.262.06854.551.380704.710.6880.7071180.0508050.6563131.9984971.9287892.004104
312.147.483.6481.517.461.530105.352.8791.412.108.6510.7066230.0490590.6575641.9986001.9312602.003814
324.294.967.2963.032.945.205203.713.4132.829.231.7920.7061630.0474310.6587321.9986971.9336292.003551
338.589.934.5926.062.137.343394.325.3095.667.812.0340.7057260.0459060.6598201.9987631.9356872.003304
3417.179.869.18412.117.314.062764.110.60411.353.203.4580.7053210.0444770.6608431.9988521.9377672.003102


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
122010200
243020201
384120211
4167240232
53211460263
66414580284
712825101421184
825652242793184
95121004950246484
101.02417589854212184
112.0483281611667624084
124.09658429828514642684
138.1921.08155852227579484
1416.3841.9429879544961.43484
1532.7683.5851.8161.7689252.64884
1665.5366.6883.4103.2771.6964.98084
17131.07212.4576.3376.1193.1489.29784
18262.14423.21511.79111.4235.86017.34384
19524.28843.75422.29121.46211.13032.61284
201.048.57682.90342.14540.75720.96461.92784
212.097.152156.99979.51577.48339.588117.39984
224.194.304297.994150.763147.23075.209222.77384
238.388.608567.166287.029280.136143.306423.84884
2416.777.2161.082.918547.960534.957273.761809.14584
2533.554.4322.073.1161.048.1141.025.001523.3551.549.74984
2667.108.8643.974.8232.008.6241.966.1981.003.9582.970.85384
27134.217.7287.632.4323.855.4443.776.9871.927.5565.704.86484
28268.435.45614.682.4917.413.0147.269.4763.706.83810.975.64184
29536.870.91228.282.70214.272.08714.010.6147.134.68021.148.01084
301.073.741.82454.551.38027.516.60927.034.77013.755.98240.795.38684
312.147.483.648105.352.87953.126.08752.226.79126.560.32678.792.54184
324.294.967.296203.713.413102.702.225101.011.18751.350.691152.362.71084
338.589.934.592394.325.309198.744.874195.580.43499.369.084294.956.21384
3417.179.869.184764.110.604385.019.268379.091.335192.508.128571.602.46484


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
241010001
383030012
4168440035
532168800610
66420101000713
7128422319139713
8256110565450351114
9512240124116118742127
101.0245272592682461605863
112.0481.130565565492315170153
124.0962.3701.1651.205949656391374
138.1924.8352.4122.4231.8641.343823805
1416.3849.9054.9594.9463.6472.7091.7601.789
1532.76820.05010.0749.9767.2115.4683.7063.665
1665.53640.46420.23820.22614.26510.8277.6417.731
17131.07281.76540.85140.91427.99821.91015.90515.952
18262.144164.90482.37782.52755.51843.75032.72832.908
19524.288331.787165.781166.006109.71687.70067.13167.240
201.048.576666.863333.253333.610217.717175.777136.783136.586
212.097.1521.339.854669.821670.033431.764352.555277.840277.695
224.194.3042.691.7141.345.1521.346.562857.408705.321564.282564.703
238.388.6085.404.1132.700.7122.703.4011.701.5181.412.2131.144.9421.145.440
2416.777.21610.846.0575.422.6255.423.4323.379.1962.829.3252.317.9432.319.593
2533.554.43221.759.96810.876.52910.883.4396.715.3505.667.6454.686.4994.690.474
2667.108.86443.639.86621.816.30621.823.56013.356.44311.347.5449.464.2959.471.584
27134.217.72887.506.17743.746.39943.759.77826.577.56922.716.97919.104.62919.107.000
28268.435.456175.427.89787.694.97687.732.92152.897.57245.473.79138.526.95538.529.579
29536.870.912351.633.801175.785.459175.848.342105.306.62891.025.25277.638.18077.663.741
301.073.741.824704.710.688352.290.977352.419.711209.724.998182.190.982156.366.235156.428.473
312.147.483.6481.412.108.651705.935.424706.173.227417.814.881364.615.908314.782.282314.895.580
324.294.967.2962.829.231.7921.414.369.7511.414.862.041832.572.307729.737.779633.351.818633.569.888
338.589.934.5925.667.812.0342.833.484.0462.834.327.9881.659.387.6401.460.373.5041.273.834.3451.274.216.545
3417.179.869.18411.353.203.4585.675.838.3835.677.365.0753.307.955.8342.922.410.0982.561.041.6552.561.795.871


8. Check for existing Integer Sequences by OEIS

Found in Database : 83, 3, 23, 71, 1, 47, 349, 1, 53, 269, 199, 109, 709, 127, 271, 431, 101, 1, 997, 173,
Found in Database : 83, 3, 23, 71, 47, 349, 53, 269, 199, 109, 709, 127, 271, 431, 101, 997, 173, 359, 557, 383, 197, 1213, 647, 439, 223, 59, 229, 463, 701, 157, 79, 1429, 239, 479, 719,
Found in Database : 3, 23, 47, 53, 59, 71, 79, 83, 101, 107, 109, 113, 127, 137,