Inhaltsverzeichnis

Development of
Algorithmic Constructions

13:40:17
Deutsch
29.Mar 2024

Polynom = x^2+36x-23

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) = 23 = 23
f(1) = 7 = 7
f(2) = 53 = 53
f(3) = 47 = 47
f(4) = 137 = 137
f(5) = 91 = 7*13
f(6) = 229 = 229
f(7) = 139 = 139
f(8) = 329 = 7*47
f(9) = 191 = 191
f(10) = 437 = 19*23
f(11) = 247 = 13*19
f(12) = 553 = 7*79
f(13) = 307 = 307
f(14) = 677 = 677
f(15) = 371 = 7*53
f(16) = 809 = 809
f(17) = 439 = 439
f(18) = 949 = 13*73
f(19) = 511 = 7*73
f(20) = 1097 = 1097
f(21) = 587 = 587
f(22) = 1253 = 7*179
f(23) = 667 = 23*29
f(24) = 1417 = 13*109
f(25) = 751 = 751
f(26) = 1589 = 7*227
f(27) = 839 = 839
f(28) = 1769 = 29*61
f(29) = 931 = 7*7*19
f(30) = 1957 = 19*103
f(31) = 1027 = 13*79
f(32) = 2153 = 2153
f(33) = 1127 = 7*7*23
f(34) = 2357 = 2357
f(35) = 1231 = 1231
f(36) = 2569 = 7*367
f(37) = 1339 = 13*103
f(38) = 2789 = 2789
f(39) = 1451 = 1451
f(40) = 3017 = 7*431
f(41) = 1567 = 1567
f(42) = 3253 = 3253
f(43) = 1687 = 7*241
f(44) = 3497 = 13*269
f(45) = 1811 = 1811
f(46) = 3749 = 23*163
f(47) = 1939 = 7*277
f(48) = 4009 = 19*211
f(49) = 2071 = 19*109
f(50) = 4277 = 7*13*47
f(51) = 2207 = 2207
f(52) = 4553 = 29*157
f(53) = 2347 = 2347
f(54) = 4837 = 7*691
f(55) = 2491 = 47*53
f(56) = 5129 = 23*223
f(57) = 2639 = 7*13*29
f(58) = 5429 = 61*89
f(59) = 2791 = 2791
f(60) = 5737 = 5737
f(61) = 2947 = 7*421
f(62) = 6053 = 6053
f(63) = 3107 = 13*239
f(64) = 6377 = 7*911
f(65) = 3271 = 3271
f(66) = 6709 = 6709
f(67) = 3439 = 19*181
f(68) = 7049 = 7*19*53
f(69) = 3611 = 23*157
f(70) = 7397 = 13*569
f(71) = 3787 = 7*541
f(72) = 7753 = 7753
f(73) = 3967 = 3967
f(74) = 8117 = 8117
f(75) = 4151 = 7*593
f(76) = 8489 = 13*653
f(77) = 4339 = 4339
f(78) = 8869 = 7*7*181
f(79) = 4531 = 23*197
f(80) = 9257 = 9257
f(81) = 4727 = 29*163
f(82) = 9653 = 7*7*197
f(83) = 4927 = 13*379
f(84) = 10057 = 89*113
f(85) = 5131 = 7*733
f(86) = 10469 = 19*19*29
f(87) = 5339 = 19*281
f(88) = 10889 = 10889
f(89) = 5551 = 7*13*61
f(90) = 11317 = 11317
f(91) = 5767 = 73*79
f(92) = 11753 = 7*23*73
f(93) = 5987 = 5987
f(94) = 12197 = 12197
f(95) = 6211 = 6211
f(96) = 12649 = 7*13*139
f(97) = 6439 = 47*137
f(98) = 13109 = 13109
f(99) = 6671 = 7*953
f(100) = 13577 = 13577

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-23

f(0)=23
f(1)=7
f(2)=53
f(3)=47
f(4)=137
f(5)=13
f(6)=229
f(7)=139
f(8)=1
f(9)=191
f(10)=19
f(11)=1
f(12)=79
f(13)=307
f(14)=677
f(15)=1
f(16)=809
f(17)=439
f(18)=73
f(19)=1
f(20)=1097
f(21)=587
f(22)=179
f(23)=29
f(24)=109
f(25)=751
f(26)=227
f(27)=839
f(28)=61
f(29)=1
f(30)=103
f(31)=1
f(32)=2153
f(33)=1
f(34)=2357
f(35)=1231
f(36)=367
f(37)=1
f(38)=2789
f(39)=1451
f(40)=431
f(41)=1567
f(42)=3253
f(43)=241
f(44)=269
f(45)=1811
f(46)=163
f(47)=277
f(48)=211
f(49)=1
f(50)=1
f(51)=2207
f(52)=157
f(53)=2347
f(54)=691
f(55)=1
f(56)=223
f(57)=1
f(58)=89
f(59)=2791
f(60)=5737
f(61)=421
f(62)=6053
f(63)=239
f(64)=911
f(65)=3271
f(66)=6709
f(67)=181
f(68)=1
f(69)=1
f(70)=569
f(71)=541
f(72)=7753
f(73)=3967
f(74)=8117
f(75)=593
f(76)=653
f(77)=4339
f(78)=1
f(79)=197
f(80)=9257
f(81)=1
f(82)=1
f(83)=379
f(84)=113
f(85)=733
f(86)=1
f(87)=281
f(88)=10889
f(89)=1
f(90)=11317
f(91)=1
f(92)=1
f(93)=5987
f(94)=12197
f(95)=6211
f(96)=1
f(97)=1
f(98)=13109
f(99)=953

b) Substitution of the polynom
The polynom f(x)=x^2+36x-23 could be written as f(y)= y^2-347 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

23, 7, 53, 47, 137, 13, 229, 139, 1, 191, 19, 1, 79, 307, 677, 1, 809, 439, 73, 1, 1097, 587, 179, 29, 109, 751, 227, 839, 61, 1, 103, 1, 2153, 1, 2357, 1231, 367, 1, 2789, 1451, 431, 1567, 3253, 241, 269, 1811, 163, 277, 211, 1, 1, 2207, 157, 2347, 691, 1, 223, 1, 89, 2791, 5737, 421, 6053, 239, 911, 3271, 6709, 181, 1, 1, 569, 541, 7753, 3967, 8117, 593, 653, 4339, 1, 197, 9257, 1, 1, 379, 113, 733, 1, 281, 10889, 1, 11317, 1, 1, 5987, 12197, 6211, 1, 1, 13109, 953, 13577, 6907, 1, 1021, 14537, 389, 1, 7639, 293, 607, 1, 8147, 16553, 1201, 17077, 1, 17609, 1277, 18149, 151, 2671, 1, 1481, 9767, 149, 1, 20389, 1, 1613, 10631, 21557, 1, 22153, 1, 3251, 887, 23369, 11839, 1, 419, 1, 1, 25253, 673, 1, 1873, 1, 1, 1, 1, 457, 14107, 4079, 14447, 173, 2113, 1, 15139, 1, 2213, 31337, 1, 1, 853, 449, 1, 4787, 1303, 1181, 2473, 443, 769, 35753, 1, 1, 18451, 761, 18839, 38069, 19231, 1, 1033, 2087, 2861, 1759, 20431, 41269, 1, 42089, 1, 6131, 461, 43753, 1699, 1, 22511, 967, 1, 46309, 23371, 1, 1, 1657, 24247, 6991, 24691, 3833, 1093, 7247, 1, 51637, 1, 52553, 2039, 1009, 3853, 54409, 1193, 7907, 1, 2963, 28387, 8179, 28867, 58217, 599, 1, 563, 60169, 619, 2659, 1063, 683, 31327, 1, 1, 1, 1, 1, 1, 2879, 547, 617, 1, 1, 2647, 9907, 34939, 1, 1, 10211, 36007, 5581, 1, 73637, 1279, 74729, 283, 1, 1, 1, 38747, 877, 1709, 11311, 3067, 80309, 1, 1031, 41011, 1, 1, 1373, 1, 1733, 42751, 1, 2281, 1, 1, 1669, 6361, 89653, 45127, 1, 1, 4003, 311, 13327, 1619, 2011, 3659, 13679, 48187, 96997, 1, 5171, 3803, 659, 1, 1, 50707, 1, 51347, 7949, 51991, 14947, 52639, 105929, 331, 1, 739, 108553, 1, 5783, 2909, 15887, 1, 3881, 56611, 1, 57287, 5011, 1, 1, 1, 117989, 1, 1511, 347, 1327, 60727, 1, 1, 929, 62131, 9613, 1, 1, 1, 127817, 9181, 4457, 4999, 18671, 2857, 1283, 1, 19087, 5167, 135077, 1, 7187, 3613, 1, 1, 10733, 70139, 20147, 70891, 142537, 71647, 1583, 1187, 145577, 10453, 937, 73939, 1, 821, 150197, 1, 1, 1439, 1103, 5927, 1, 77839, 6803, 1, 883, 79427, 12281, 1, 1, 81031, 1, 81839, 12653, 82651, 1249, 1, 1, 12041, 2777, 6547, 171049, 12277, 172709, 86771, 859, 1, 176053, 1, 25391, 3079, 179429, 1, 13933, 4789, 9623, 13121, 184553, 92707, 1, 1, 1, 94439, 27107, 95311, 191497, 1, 1297, 1, 1, 1999, 1, 7603, 1493, 1, 200357, 2141, 28879, 101527, 1, 14633, 1381, 4493, 207589, 1, 1, 105167, 1, 106087, 213097, 947, 30707, 1, 11411, 1, 218677, 109807, 220553, 1217, 222437, 1831, 1, 1543, 1, 113591, 1, 114547, 1, 1, 1013, 116471, 1, 1, 235849, 118411, 1, 1, 239753, 1, 4933, 121351, 1237, 17477, 5227, 1, 1091, 17761, 1, 5449, 35951, 1, 1, 4391, 36527, 128351, 257717, 18481, 1, 5669, 1, 18773, 263849, 132439, 37987, 10267, 1549, 134507, 1, 1523, 14323, 1, 11923, 137639, 2017, 19813, 278437, 139747, 3083, 1, 282677, 141871, 1, 142939, 22073, 2939, 6151, 5003, 291253, 1, 15443, 1, 42227, 148339, 1559, 1, 1, 11579, 5701, 21661, 304357, 1, 2713, 21977, 23753, 6737, 1, 1, 16487, 8273, 3467, 1, 5209, 22777, 320009, 1, 11113, 1777, 1, 162847, 46691, 163991, 1163, 12703, 1, 1, 333737, 1259, 1, 1213, 26029, 1, 340709, 3637, 7001, 1579, 1, 1, 1, 174467, 12073, 1, 352489, 1, 354869, 1, 18803, 9433, 1, 13879, 4583, 1, 52067, 182839, 366889, 26293, 28409, 185267, 4177, 26641, 3433, 187711, 4139, 188939, 1, 10009, 1, 1, 5261, 1, 386537, 193891, 389029, 1, 1, 15107, 1, 197647, 1, 1, 1, 200171, 30893, 4111, 1, 1, 406697, 1, 31481, 205267, 58831, 206551, 414389, 1, 59567, 16087, 419557, 1307, 14557, 211727, 424757, 2341, 427369, 1, 1, 1, 1, 216967, 4783, 218287, 1, 1, 440549, 220939, 1, 1, 445877, 4219, 1, 224947, 1, 1, 3413, 11981, 2551, 32713, 459337, 1, 1, 1, 464777, 233071, 1, 10193, 1, 235811, 1, 2099, 475753, 1, 36809, 1, 1, 1499, 4441, 242731, 1, 1, 489653, 245527, 70351, 246931, 495269, 2729, 498089, 1823, 1, 1, 503753, 1, 1, 1, 509449, 1627, 1, 4211, 1723, 36901, 3727, 2383, 9829, 37313, 523829, 1, 1601, 264091, 529637, 265547, 76079, 1, 28183, 5479, 1, 269939, 541349, 1, 1, 272887, 78179, 1, 18973, 1, 6079, 1, 1, 1, 559157, 280327, 29587, 1, 3793, 1, 3529, 284839, 3191, 22027, 82031, 287867, 7307, 41341, 10949, 1, 1951, 41777, 1, 4027, 84211, 1, 2399, 10243, 85091, 6353, 598729, 1, 5843, 1, 1, 43321, 5381, 304807, 12473, 23567, 2549, 307939, 12601, 13457, 1, 2339, 47981, 1637, 626917, 44893, 1, 10891, 6959, 317431, 2663, 319027, 1, 320627, 5689, 3541, 1, 2971, 1, 2447, 1, 1, 4073, 328687, 658997, 330311, 1, 6263, 51193, 47653, 2999, 335207, 1, 48121, 51949, 14717, 3343, 7237, 1889, 17989, 1847, 1, 6317, 7043, 691877, 15077, 1, 1, 1, 350111, 100271, 1, 9661, 1, 1, 1, 5197, 2683, 1, 358531, 4253, 51461, 722153, 361927, 103651, 12539, 728969, 1, 4549, 1, 25373, 1, 739253, 28499, 742697, 53173, 1, 19681, 1, 7993, 1, 1, 108079, 379147, 1, 54413, 58733, 1, 767029, 1, 770537, 1, 15797, 1, 26813, 20509, 1, 1987, 784649, 1, 4001, 394987, 2579, 56681, 1, 3527, 8779, 400339, 2897, 402131, 115151, 5113, 1, 1, 1861, 1, 816869, 58477, 820489, 31627, 117731, 412967, 3923, 14303, 1, 1, 1, 1, 28921, 420271, 10663, 1, 1, 1, 6389, 425791, 853429, 18593, 9419, 1, 4507, 8803, 14173, 433207, 868277, 1, 16453, 2441, 125107, 3203, 38239, 1, 1, 23293, 887017, 63493, 890789, 15391, 68813, 64033, 1, 450127, 128879, 452027, 1, 453931, 129967, 1, 913589, 65393, 1, 1, 48487, 1, 925097, 463511, 132707, 35803, 1, 467371, 133811, 16183, 940553, 2927, 72649, 473191, 1, 1, 952229, 477091, 1, 1, 960053, 3511, 1, 482971, 42083, 1, 13313, 9187, 975797, 1, 5413, 1, 1, 1, 42943, 1, 141667, 1, 1, 71261, 999653, 2767, 1, 2477, 77513, 1, 1, 22037, 21611, 508867, 145679, 3023, 6521, 1, 3821, 514939, 54311, 1,

6. Sequence of the polynom (only primes)

23, 7, 53, 47, 137, 13, 229, 139, 191, 19, 79, 307, 677, 809, 439, 73, 1097, 587, 179, 29, 109, 751, 227, 839, 61, 103, 2153, 2357, 1231, 367, 2789, 1451, 431, 1567, 3253, 241, 269, 1811, 163, 277, 211, 2207, 157, 2347, 691, 223, 89, 2791, 5737, 421, 6053, 239, 911, 3271, 6709, 181, 569, 541, 7753, 3967, 8117, 593, 653, 4339, 197, 9257, 379, 113, 733, 281, 10889, 11317, 5987, 12197, 6211, 13109, 953, 13577, 6907, 1021, 14537, 389, 7639, 293, 607, 8147, 16553, 1201, 17077, 17609, 1277, 18149, 151, 2671, 1481, 9767, 149, 20389, 1613, 10631, 21557, 22153, 3251, 887, 23369, 11839, 419, 25253, 673, 1873, 457, 14107, 4079, 14447, 173, 2113, 15139, 2213, 31337, 853, 449, 4787, 1303, 1181, 2473, 443, 769, 35753, 18451, 761, 18839, 38069, 19231, 1033, 2087, 2861, 1759, 20431, 41269, 42089, 6131, 461, 43753, 1699, 22511, 967, 46309, 23371, 1657, 24247, 6991, 24691, 3833, 1093, 7247, 51637, 52553, 2039, 1009, 3853, 54409, 1193, 7907, 2963, 28387, 8179, 28867, 58217, 599, 563, 60169, 619, 2659, 1063, 683, 31327, 2879, 547, 617, 2647, 9907, 34939, 10211, 36007, 5581, 73637, 1279, 74729, 283, 38747, 877, 1709, 11311, 3067, 80309, 1031, 41011, 1373, 1733, 42751, 2281, 1669, 6361, 89653, 45127, 4003, 311, 13327, 1619, 2011, 3659, 13679, 48187, 96997, 5171, 3803, 659, 50707, 51347, 7949, 51991, 14947, 52639, 105929, 331, 739, 108553, 5783, 2909, 15887, 3881, 56611, 57287, 5011, 117989, 1511, 347, 1327, 60727, 929, 62131, 9613, 127817, 9181, 4457, 4999, 18671, 2857, 1283, 19087, 5167, 135077, 7187, 3613, 10733, 70139, 20147, 70891, 142537, 71647, 1583, 1187, 145577, 10453, 937, 73939, 821, 150197, 1439, 1103, 5927, 77839, 6803, 883, 79427, 12281, 81031, 81839, 12653, 82651, 1249, 12041, 2777, 6547, 171049, 12277, 172709, 86771, 859, 176053, 25391, 3079, 179429, 13933, 4789, 9623, 13121, 184553, 92707, 94439, 27107, 95311, 191497, 1297, 1999, 7603, 1493, 200357, 2141, 28879, 101527, 14633, 1381, 4493, 207589, 105167, 106087, 213097, 947, 30707, 11411, 218677, 109807, 220553, 1217, 222437, 1831, 1543, 113591, 114547, 1013, 116471, 235849, 118411, 239753, 4933, 121351, 1237, 17477, 5227, 1091, 17761, 5449, 35951, 4391, 36527, 128351, 257717, 18481, 5669, 18773, 263849, 132439, 37987, 10267, 1549, 134507, 1523, 14323, 11923, 137639, 2017, 19813, 278437, 139747, 3083, 282677, 141871, 142939, 22073, 2939, 6151, 5003, 291253, 15443, 42227, 148339, 1559, 11579, 5701, 21661, 304357, 2713, 21977, 23753, 6737, 16487, 8273, 3467, 5209, 22777, 320009, 11113, 1777, 162847, 46691, 163991, 1163, 12703, 333737, 1259, 1213, 26029, 340709, 3637, 7001, 1579, 174467, 12073, 352489, 354869, 18803, 9433, 13879, 4583, 52067, 182839, 366889, 26293, 28409, 185267, 4177, 26641, 3433, 187711, 4139, 188939, 10009, 5261, 386537, 193891, 389029, 15107, 197647, 200171, 30893, 4111, 406697, 31481, 205267, 58831, 206551, 414389, 59567, 16087, 419557, 1307, 14557, 211727, 424757, 2341, 427369, 216967, 4783, 218287, 440549, 220939, 445877, 4219, 224947, 3413, 11981, 2551, 32713, 459337, 464777, 233071, 10193, 235811, 2099, 475753, 36809, 1499, 4441, 242731, 489653, 245527, 70351, 246931, 495269, 2729, 498089, 1823, 503753, 509449, 1627, 4211, 1723, 36901, 3727, 2383, 9829, 37313, 523829, 1601, 264091, 529637, 265547, 76079, 28183, 5479, 269939, 541349, 272887, 78179, 18973, 6079, 559157, 280327, 29587, 3793, 3529, 284839, 3191, 22027, 82031, 287867, 7307, 41341, 10949, 1951, 41777, 4027, 84211, 2399, 10243, 85091, 6353, 598729, 5843, 43321, 5381, 304807, 12473, 23567, 2549, 307939, 12601, 13457, 2339, 47981, 1637, 626917, 44893, 10891, 6959, 317431, 2663, 319027, 320627, 5689, 3541, 2971, 2447, 4073, 328687, 658997, 330311, 6263, 51193, 47653, 2999, 335207, 48121, 51949, 14717, 3343, 7237, 1889, 17989, 1847, 6317, 7043, 691877, 15077, 350111, 100271, 9661, 5197, 2683, 358531, 4253, 51461, 722153, 361927, 103651, 12539, 728969, 4549, 25373, 739253, 28499, 742697, 53173, 19681, 7993, 108079, 379147, 54413, 58733, 767029, 770537, 15797, 26813, 20509, 1987, 784649, 4001, 394987, 2579, 56681, 3527, 8779, 400339, 2897, 402131, 115151, 5113, 1861, 816869, 58477, 820489, 31627, 117731, 412967, 3923, 14303, 28921, 420271, 10663, 6389, 425791, 853429, 18593, 9419, 4507, 8803, 14173, 433207, 868277, 16453, 2441, 125107, 3203, 38239, 23293, 887017, 63493, 890789, 15391, 68813, 64033, 450127, 128879, 452027, 453931, 129967, 913589, 65393, 48487, 925097, 463511, 132707, 35803, 467371, 133811, 16183, 940553, 2927, 72649, 473191, 952229, 477091, 960053, 3511, 482971, 42083, 13313, 9187, 975797, 5413, 42943, 141667, 71261, 999653, 2767, 2477, 77513, 22037, 21611, 508867, 145679, 3023, 6521, 3821, 514939, 54311,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+36x-23 and
the reducible primes which appear as divisor for the first time
p | x^2+36x-23 and p < x^2+36x-23

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
21007443310.7400000.4300000.3100008.2222225.37500031.000000
31.0006872734140.6870000.2730000.4140009.2837846.34883713.354838
410.0006.9781.9445.0340.6978000.1944000.50340010.1572067.12087912.159420
5100.00069.82614.85354.9730.6982600.1485300.54973010.0065927.64043210.920341
61.000.000697.319120.263577.0560.6973190.1202630.5770569.9865248.09688310.497081
710.000.0006.963.7871.016.0465.947.7410.6963790.1016050.5947749.9865158.44853410.307043
8100.000.00069.586.3578.786.12460.800.2330.6958640.0878610.6080029.9926038.64736810.222407
91.000.000.000695.461.27377.438.862618.022.4110.6954610.0774390.6180229.9942188.81376810.164804
1010.000.000.0006.951.540.061692.341.1176.259.198.9440.6951540.0692340.6259209.9955838.94048710.127787


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
388711.0000000.8750000.1250001.6000001.400000inf
416131120.8125000.6875000.1250001.6250001.5714292.000000
532251780.7812500.5312500.2500001.9230771.5454554.000000
6645029210.7812500.4531250.3281252.0000001.7058822.625000
71289353400.7265620.4140620.3125001.8600001.8275861.904762
825618189920.7070310.3476560.3593751.9462371.6792452.300000
95123551571980.6933590.3066410.3867191.9613261.7640452.152174
101.0247032764270.6865230.2695310.4169921.9802821.7579622.156566
112.0481.4195009190.6928710.2441410.4487302.0184921.8115942.152225
124.0962.8498991.9500.6955570.2194820.4760742.0077521.7980002.121872
138.1925.7111.6234.0880.6971440.1981200.4990232.0045631.8053392.096410
1416.38411.3862.9998.3870.6949460.1830440.5119021.9936961.8478132.051615
1532.76822.8705.47017.4000.6979370.1669310.5310062.0086071.8239412.074639
1665.53645.74710.14335.6040.6980440.1547700.5432742.0003061.8542962.046207
17131.07291.47118.99672.4750.6978680.1449280.5529401.9994971.8728192.035586
18262.144182.97435.391147.5830.6979900.1350060.5629842.0003501.8630762.036330
19524.288365.78566.630299.1550.6976800.1270870.5705931.9991091.8826822.027029
201.048.576731.163125.648605.5150.6972910.1198270.5774641.9988871.8857572.024085
212.097.1521.461.851237.8841.223.9670.6970650.1134320.5836331.9993501.8932572.021365
224.194.3042.922.279452.4472.469.8320.6967260.1078720.5888541.9990271.9019652.017891
238.388.6085.842.244862.1644.980.0800.6964500.1027780.5936721.9992081.9055582.016364
2416.777.21611.680.7541.647.24710.033.5070.6962270.0981840.5980441.9993611.9105962.014728
2533.554.43223.357.1273.150.42120.206.7060.6960970.0938900.6022071.9996251.9125372.013922
2667.108.86446.703.7286.036.18240.667.5460.6959400.0899460.6059941.9995491.9159922.012577
27134.217.72893.390.61511.594.60981.796.0060.6958140.0863870.6094281.9996391.9208512.011334
28268.435.456186.744.33322.299.477164.444.8560.6956770.0830720.6126051.9996051.9232622.010427
29536.870.912373.422.47942.954.551330.467.9280.6955540.0800090.6155441.9996451.9262582.009597
301.073.741.824746.734.24182.846.081663.888.1600.6954500.0771560.6182941.9997041.9286922.008934
312.147.483.6481.493.251.293160.003.7381.333.247.5550.6953490.0745080.6208421.9997091.9313382.008241
324.294.967.2962.986.109.279309.382.1912.676.727.0880.6952580.0720340.6232241.9997371.9335932.007674
338.589.934.5925.971.475.455598.894.0875.372.581.3680.6951710.0697200.6254511.9997511.9357742.007146
3417.179.869.18411.941.568.7721.160.584.83210.780.983.9400.6950910.0675550.6275361.9997681.9378802.006667
3534.359.738.36823.880.593.3752.251.190.04721.629.403.3280.6950170.0655180.6294981.9997871.9397032.006255
3668.719.476.73647.756.507.6604.370.568.76043.385.938.9000.6949490.0636000.6313481.9998041.9414482.005878


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
123120012
245141013
387341123
41611472234
532176114337
66429121756711
712853233012111515
825689444523192225
9512157827536373747
101.02427614413261706877
112.048500259241112133122133
124.096899463436216236209238
138.1921.623813810389425393416
1416.3842.9991.5131.486735778725761
1532.7685.4702.7602.7101.3301.4121.3291.399
1665.53610.1435.1145.0292.4702.5992.4722.602
17131.07218.9969.5899.4074.6854.8214.6594.831
18262.14435.39117.80017.5918.7029.0298.6828.978
19524.28866.63033.52533.10516.39616.91816.39016.926
201.048.576125.64863.18262.46630.84431.81831.04331.943
212.097.152237.884119.714118.17058.47860.38858.64160.377
224.194.304452.447227.331225.116111.365114.642111.826114.614
238.388.608862.164432.751429.413212.836218.232212.770218.326
2416.777.2161.647.247826.745820.502407.319416.610406.614416.704
2533.554.4323.150.4211.580.3301.570.091778.908796.023779.264796.226
2667.108.8646.036.1823.028.5283.007.6541.492.5591.524.1591.493.8051.525.659
27134.217.72811.594.6095.817.4835.777.1262.869.2912.927.7402.869.6062.927.972
28268.435.45622.299.47711.185.23111.114.2465.520.4415.629.2735.521.9425.627.821
29536.870.91242.954.55121.541.86121.412.69010.639.11410.838.55910.640.98510.835.893
301.073.741.82482.846.08141.547.27341.298.80820.529.41120.895.74720.528.14920.892.774
312.147.483.648160.003.73880.228.67879.775.06039.663.59140.341.01739.658.59840.340.532
324.294.967.296309.382.191155.116.104154.266.08776.705.97377.980.67576.708.12477.987.419
338.589.934.592598.894.087300.248.469298.645.618148.519.971150.922.342148.530.147150.921.627
3417.179.869.1841.160.584.832581.797.132578.787.700287.884.145292.390.061287.909.278292.401.348
3534.359.738.3682.251.190.0471.128.430.2561.122.759.791558.560.824567.025.090558.569.943567.034.190
3668.719.476.7364.370.568.7602.190.618.6702.179.950.0901.084.646.5071.100.635.6391.084.675.3491.100.611.265


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
381100010
4162200011
5328621232
664211562577
712840231786179
825692494322222523
9512198999941545449
101.02442720821996115112104
112.048919445474224224237234
124.0961.950983967494480481495
138.1924.0882.0692.0191.0279861.0391.036
1416.3848.3874.2054.1822.1532.0592.0922.083
1532.76817.4008.6508.7504.3974.3584.3484.297
1665.53635.60417.69517.9099.0158.8448.8978.848
17131.07272.47536.08636.38918.31517.97018.26717.923
18262.144147.58373.80373.78036.98336.87036.93036.800
19524.288299.155149.690149.46574.93474.68074.82274.719
201.048.576605.515303.201302.314151.434151.474151.285151.322
212.097.1521.223.967612.173611.794305.855305.845306.387305.880
224.194.3042.469.8321.235.2871.234.545617.388617.702618.227616.515
238.388.6084.980.0802.491.0902.488.9901.245.3511.245.1391.246.6371.242.953
2416.777.21610.033.5075.018.7375.014.7702.509.6062.507.1532.510.5972.506.151
2533.554.43220.206.70610.109.60610.097.1005.054.0985.047.7155.055.2525.049.641
2667.108.86440.667.54620.340.82020.326.72610.173.03810.161.88910.169.62010.162.999
27134.217.72881.796.00640.904.78440.891.22220.455.32620.439.99320.455.94020.444.747
28268.435.456164.444.85682.242.53982.202.31741.126.01041.094.73941.126.47441.097.633
29536.870.912330.467.928165.264.168165.203.76082.639.06282.585.06382.642.24682.601.557
301.073.741.824663.888.160332.022.105331.866.055166.025.176165.923.610166.013.608165.925.766
312.147.483.6481.333.247.555666.757.078666.490.477333.409.081333.205.183333.414.120333.219.171
324.294.967.2962.676.727.0881.338.648.5241.338.078.564669.365.842669.012.845669.363.816668.984.585
338.589.934.5925.372.581.3682.686.819.2892.685.762.0791.343.472.8131.342.799.2791.343.515.1291.342.794.147
3417.179.869.18410.780.983.9405.391.397.7875.389.586.1532.695.874.5552.694.630.5082.695.901.8762.694.577.001
3534.359.738.36821.629.403.32810.816.317.80710.813.085.5215.408.473.1545.406.255.5135.408.517.3765.406.157.285
3668.719.476.73643.385.938.90021.695.998.05321.689.940.84710.848.534.94410.844.396.60510.848.699.96010.844.307.391


8. Check for existing Integer Sequences by OEIS

Found in Database : 23, 7, 53, 47, 137, 13, 229, 139, 1, 191, 19, 1, 79, 307, 677, 1, 809, 439, 73, 1,
Found in Database : 23, 7, 53, 47, 137, 13, 229, 139, 191, 19, 79, 307, 677, 809, 439, 73, 1097, 587, 179, 29, 109, 751, 227, 839, 61, 103, 2153, 2357, 1231, 367, 2789, 1451,
Found in Database : 7, 13, 19, 23, 29, 47, 53, 61, 73, 79, 89, 103, 109, 113, 137, 139, 149,