Inhaltsverzeichnis

Development of
Algorithmic Constructions

22:55:47
Deutsch
28.Mar 2024

Polynom = x^2-41

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) = 41 = 41
f(1) = 5 = 5
f(2) = 37 = 37
f(3) = 1 = 1
f(4) = 25 = 5*5
f(5) = 1 = 1
f(6) = 5 = 5
f(7) = 1 = 1
f(8) = 23 = 23
f(9) = 5 = 5
f(10) = 59 = 59
f(11) = 5 = 5
f(12) = 103 = 103
f(13) = 1 = 1
f(14) = 155 = 5*31
f(15) = 23 = 23
f(16) = 215 = 5*43
f(17) = 31 = 31
f(18) = 283 = 283
f(19) = 5 = 5
f(20) = 359 = 359
f(21) = 25 = 5*5
f(22) = 443 = 443
f(23) = 61 = 61
f(24) = 535 = 5*107
f(25) = 73 = 73
f(26) = 635 = 5*127
f(27) = 43 = 43
f(28) = 743 = 743
f(29) = 25 = 5*5
f(30) = 859 = 859
f(31) = 115 = 5*23
f(32) = 983 = 983
f(33) = 131 = 131
f(34) = 1115 = 5*223
f(35) = 37 = 37
f(36) = 1255 = 5*251
f(37) = 83 = 83
f(38) = 1403 = 23*61
f(39) = 185 = 5*37
f(40) = 1559 = 1559
f(41) = 205 = 5*41
f(42) = 1723 = 1723
f(43) = 113 = 113
f(44) = 1895 = 5*379
f(45) = 31 = 31
f(46) = 2075 = 5*5*83
f(47) = 271 = 271
f(48) = 2263 = 31*73
f(49) = 295 = 5*59
f(50) = 2459 = 2459
f(51) = 5 = 5
f(52) = 2663 = 2663
f(53) = 173 = 173
f(54) = 2875 = 5*5*5*23
f(55) = 373 = 373
f(56) = 3095 = 5*619
f(57) = 401 = 401
f(58) = 3323 = 3323
f(59) = 215 = 5*43
f(60) = 3559 = 3559
f(61) = 115 = 5*23
f(62) = 3803 = 3803
f(63) = 491 = 491
f(64) = 4055 = 5*811
f(65) = 523 = 523
f(66) = 4315 = 5*863
f(67) = 139 = 139
f(68) = 4583 = 4583
f(69) = 295 = 5*59
f(70) = 4859 = 43*113
f(71) = 625 = 5*5*5*5
f(72) = 5143 = 37*139
f(73) = 661 = 661
f(74) = 5435 = 5*1087
f(75) = 349 = 349
f(76) = 5735 = 5*31*37
f(77) = 23 = 23
f(78) = 6043 = 6043
f(79) = 775 = 5*5*31
f(80) = 6359 = 6359
f(81) = 815 = 5*163
f(82) = 6683 = 41*163
f(83) = 107 = 107
f(84) = 7015 = 5*23*61
f(85) = 449 = 449
f(86) = 7355 = 5*1471
f(87) = 941 = 941
f(88) = 7703 = 7703
f(89) = 985 = 5*197
f(90) = 8059 = 8059
f(91) = 515 = 5*103
f(92) = 8423 = 8423
f(93) = 269 = 269
f(94) = 8795 = 5*1759
f(95) = 1123 = 1123
f(96) = 9175 = 5*5*367
f(97) = 1171 = 1171
f(98) = 9563 = 73*131
f(99) = 305 = 5*61
f(100) = 9959 = 23*433

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^2-41

f(0)=41
f(1)=5
f(2)=37
f(3)=1
f(4)=1
f(5)=1
f(6)=1
f(7)=1
f(8)=23
f(9)=1
f(10)=59
f(11)=1
f(12)=103
f(13)=1
f(14)=31
f(15)=1
f(16)=43
f(17)=1
f(18)=283
f(19)=1
f(20)=359
f(21)=1
f(22)=443
f(23)=61
f(24)=107
f(25)=73
f(26)=127
f(27)=1
f(28)=743
f(29)=1
f(30)=859
f(31)=1
f(32)=983
f(33)=131
f(34)=223
f(35)=1
f(36)=251
f(37)=83
f(38)=1
f(39)=1
f(40)=1559
f(41)=1
f(42)=1723
f(43)=113
f(44)=379
f(45)=1
f(46)=1
f(47)=271
f(48)=1
f(49)=1
f(50)=2459
f(51)=1
f(52)=2663
f(53)=173
f(54)=1
f(55)=373
f(56)=619
f(57)=401
f(58)=3323
f(59)=1
f(60)=3559
f(61)=1
f(62)=3803
f(63)=491
f(64)=811
f(65)=523
f(66)=863
f(67)=139
f(68)=4583
f(69)=1
f(70)=1
f(71)=1
f(72)=1
f(73)=661
f(74)=1087
f(75)=349
f(76)=1
f(77)=1
f(78)=6043
f(79)=1
f(80)=6359
f(81)=163
f(82)=1
f(83)=1
f(84)=1
f(85)=449
f(86)=1471
f(87)=941
f(88)=7703
f(89)=197
f(90)=8059
f(91)=1
f(92)=8423
f(93)=269
f(94)=1759
f(95)=1123
f(96)=367
f(97)=1171
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-41 could be written as f(y)= y^2-41 with x=y+0

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+0
f'(x)>2x-1 with x > 6

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

41, 5, 37, 1, 1, 1, 1, 1, 23, 1, 59, 1, 103, 1, 31, 1, 43, 1, 283, 1, 359, 1, 443, 61, 107, 73, 127, 1, 743, 1, 859, 1, 983, 131, 223, 1, 251, 83, 1, 1, 1559, 1, 1723, 113, 379, 1, 1, 271, 1, 1, 2459, 1, 2663, 173, 1, 373, 619, 401, 3323, 1, 3559, 1, 3803, 491, 811, 523, 863, 139, 4583, 1, 1, 1, 1, 661, 1087, 349, 1, 1, 6043, 1, 6359, 163, 1, 1, 1, 449, 1471, 941, 7703, 197, 8059, 1, 8423, 269, 1759, 1123, 367, 1171, 1, 1, 433, 1, 241, 1321, 431, 1373, 2239, 1, 1, 1, 389, 307, 12503, 1, 2591, 1, 2683, 853, 13883, 353, 1, 1, 14843, 1, 3067, 487, 3167, 2011, 277, 1, 733, 1, 17383, 1103, 3583, 2273, 3691, 2341, 613, 1, 19559, 1, 20123, 2551, 4139, 1, 1, 337, 21863, 1, 607, 569, 23063, 1, 947, 1499, 1, 769, 24923, 631, 419, 647, 26203, 829, 1, 1699, 5503, 1, 28183, 1, 28859, 1, 953, 467, 6047, 3823, 1, 3911, 31643, 1, 32359, 409, 33083, 1, 6763, 4273, 6911, 1, 821, 1, 1, 911, 1601, 4651, 1, 1187, 1, 2423, 39163, 1, 1289, 1009, 40763, 1, 1663, 1, 1, 5351, 43223, 1091, 44059, 1, 541, 2833, 9151, 1, 9323, 5881, 461, 599, 1307, 1, 2141, 6211, 1, 6323, 1, 1609, 1, 1, 52859, 1, 53783, 6781, 1, 3449, 11131, 877, 1, 1427, 57559, 1451, 1361, 1, 1, 1, 1, 7621, 61463, 1549, 62459, 787, 63463, 1999, 2579, 8123, 13099, 1, 66523, 1, 67559, 1, 2213, 8641, 13931, 1, 14143, 1, 3121, 1, 72859, 1, 73943, 9311, 1, 1181, 15227, 4793, 77243, 1, 617, 1973, 1303, 5003, 701, 1, 1, 1, 82903, 2087, 84059, 1, 85223, 1, 1, 1, 1, 1, 2399, 1117, 89959, 1, 91163, 11471, 739, 1, 18719, 1, 94823, 1193, 96059, 2417, 97303, 12241, 857, 6199, 19963, 1, 1, 2543, 102359, 1, 103643, 3259, 677, 6599, 21247, 1, 1, 1, 4733, 1, 110183, 1, 22303, 1, 22571, 1, 114203, 1, 1583, 1453, 116923, 1, 1, 1, 4787, 7523, 121063, 761, 881, 3079, 123863, 1, 5011, 1, 25339, 7963, 4133, 3221, 1, 3257, 1, 8233, 1, 2081, 26783, 16831, 3659, 1, 136859, 1, 3739, 8693, 27967, 17573, 1229, 17761, 1721, 1, 1, 907, 1291, 797, 29483, 18523, 1, 4679, 150503, 1, 2083, 3821, 153623, 19301, 31039, 9749, 6271, 1231, 158363, 1, 1553, 4019, 161563, 1, 1, 1, 1433, 1, 1021, 1, 4099, 2111, 1, 1, 1, 21523, 34603, 1, 1, 1097, 5689, 1, 7741, 1, 1, 22573, 1, 11393, 1, 1, 184859, 4643, 186583, 23431, 37663, 1, 38011, 11933, 191803, 4817, 193559, 4861, 5279, 12263, 39419, 1, 1, 24971, 6473, 5039, 3319, 1, 1, 12823, 8243, 25873, 41579, 1, 1, 2633, 211559, 1, 3617, 1, 43051, 1, 1, 3407, 9521, 2749, 220859, 1109, 1, 27961, 44927, 1, 45307, 7109, 228443, 1, 1033, 5783, 1, 1, 1, 14699, 1, 29641, 238103, 1, 240059, 1, 5903, 3797, 48799, 1, 9839, 30871, 10781, 1, 5813, 3137, 1279, 1, 10159, 31873, 51199, 16063, 258023, 1619, 8389, 1, 262103, 1061, 2297, 8287, 1439, 16703, 268283, 6733, 7307, 1, 2411, 17093, 54907, 2153, 1, 1, 278743, 1399, 280859, 1, 6581, 1, 1, 1, 1, 1567, 1153, 3631, 291559, 1, 293723, 1, 1, 37123, 11923, 9349, 1, 3767, 302459, 7589, 304663, 1, 1, 19249, 61819, 1, 311323, 1, 13633, 7867, 315803, 1, 63611, 19949, 64063, 1747, 322583, 8093, 1993, 1, 1, 10259, 1607, 1, 66347, 41611, 334043, 1, 5701, 4219, 338683, 1847, 1, 42773, 1597, 1, 1, 1, 1, 8731, 350423, 43951, 1907, 5531, 14207, 22273, 357563, 8969, 6101, 9029, 362363, 1, 14591, 11437, 1, 46051, 3271, 1, 372059, 2333, 1, 1, 75391, 1, 75883, 47581, 1, 4789, 384359, 1, 2953, 1, 77867, 48823, 78367, 1, 2837, 1, 3853, 1, 5471, 1, 2593, 1, 3517, 1, 1, 1, 1, 10271, 412123, 12919, 82939, 25999, 16691, 52321, 6883, 10529, 422459, 5297, 18481, 3331, 17107, 53623, 2099, 53951, 432923, 1, 435559, 1, 438203, 54941, 2383, 1783, 1, 27803, 1, 2797, 448859, 2251, 10501, 56611, 90847, 1, 91387, 28643, 1, 1, 2347, 11593, 1, 29153, 93563, 7331, 94111, 58991, 3613, 11867, 476059, 1, 478823, 30013, 1579, 60373, 1, 1481, 1, 1, 489959, 1, 492763, 1, 1, 1, 99679, 15619, 501223, 1, 6073, 12637, 506903, 63541, 1, 1, 102523, 1, 8737, 12923, 1, 1, 521243, 8167, 104827, 1, 105407, 2131, 23041, 2657, 17189, 6679, 535783, 1, 1, 67523, 1, 67891, 1, 3413, 547559, 6863, 14879, 69001, 4813, 1, 22259, 1, 4951, 1753, 562459, 1, 565463, 1, 22739, 1, 114299, 1, 1, 14401, 577559, 1, 4177, 36383, 116731, 18287, 1, 1, 589783, 14783, 9719, 1, 595943, 1, 5209, 2029, 120427, 1, 8291, 1, 608359, 1, 611483, 76631, 1, 77023, 123551, 9677, 4889, 1, 1, 15641, 20233, 1, 126079, 39499, 25343, 1, 636763, 15959, 639959, 1, 1697, 20149, 5171, 40499, 129919, 81401, 1, 16361, 2137, 8221, 1, 1, 132511, 83023, 1, 83431, 1, 1, 1, 1, 675643, 2731, 135787, 1, 136447, 42743, 9391, 1, 1877, 1, 6469, 2017, 1, 21787, 1, 43783, 702203, 17597, 705559, 17681, 708923, 1931, 3313, 2789, 28627, 89671, 719063, 1, 722459, 1, 725863, 1, 1, 91373, 146539, 91801, 3301, 1, 2729, 1, 3083, 1, 1, 93523, 6521, 1, 753383, 9439, 1, 3793, 760343, 95261, 152767, 1, 1, 1, 770843, 3863, 7237, 19403, 1, 6091, 1, 1, 4243, 2287, 2027, 1, 21407, 9923, 13043, 24919, 159839, 1, 1, 1, 806363, 5051, 3011, 1, 19843, 101921, 32687, 4451, 1, 51413, 1861, 1, 828059, 20747, 36161, 3361, 167071, 1, 5413, 52553, 4871, 1, 846359, 4241, 850043, 1, 2339, 26737, 1, 2903, 37441, 1, 20113, 5417, 868583, 54403, 2957, 4751, 175211, 109741, 14423, 1, 2503, 2767, 887323, 2711, 5749, 111623, 7159, 1, 898663, 11257, 1, 1, 906263, 1, 1, 56999, 182779, 28619, 39901, 1, 24907, 23087, 21521, 28979, 5023, 58199, 1, 116881, 3733, 23473, 1, 2357, 2707, 1, 1, 3833, 190507, 119311, 30853, 1, 3467, 1, 964283, 1, 4723, 1, 194431, 1, 976103, 6113, 7717, 24551, 1, 2089, 197599, 30937, 39679, 1, 2377, 1,

6. Sequence of the polynom (only primes)

41, 5, 37, 23, 59, 103, 31, 43, 283, 359, 443, 61, 107, 73, 127, 743, 859, 983, 131, 223, 251, 83, 1559, 1723, 113, 379, 271, 2459, 2663, 173, 373, 619, 401, 3323, 3559, 3803, 491, 811, 523, 863, 139, 4583, 661, 1087, 349, 6043, 6359, 163, 449, 1471, 941, 7703, 197, 8059, 8423, 269, 1759, 1123, 367, 1171, 433, 241, 1321, 431, 1373, 2239, 389, 307, 12503, 2591, 2683, 853, 13883, 353, 14843, 3067, 487, 3167, 2011, 277, 733, 17383, 1103, 3583, 2273, 3691, 2341, 613, 19559, 20123, 2551, 4139, 337, 21863, 607, 569, 23063, 947, 1499, 769, 24923, 631, 419, 647, 26203, 829, 1699, 5503, 28183, 28859, 953, 467, 6047, 3823, 3911, 31643, 32359, 409, 33083, 6763, 4273, 6911, 821, 911, 1601, 4651, 1187, 2423, 39163, 1289, 1009, 40763, 1663, 5351, 43223, 1091, 44059, 541, 2833, 9151, 9323, 5881, 461, 599, 1307, 2141, 6211, 6323, 1609, 52859, 53783, 6781, 3449, 11131, 877, 1427, 57559, 1451, 1361, 7621, 61463, 1549, 62459, 787, 63463, 1999, 2579, 8123, 13099, 66523, 67559, 2213, 8641, 13931, 14143, 3121, 72859, 73943, 9311, 1181, 15227, 4793, 77243, 617, 1973, 1303, 5003, 701, 82903, 2087, 84059, 85223, 2399, 1117, 89959, 91163, 11471, 739, 18719, 94823, 1193, 96059, 2417, 97303, 12241, 857, 6199, 19963, 2543, 102359, 103643, 3259, 677, 6599, 21247, 4733, 110183, 22303, 22571, 114203, 1583, 1453, 116923, 4787, 7523, 121063, 761, 881, 3079, 123863, 5011, 25339, 7963, 4133, 3221, 3257, 8233, 2081, 26783, 16831, 3659, 136859, 3739, 8693, 27967, 17573, 1229, 17761, 1721, 907, 1291, 797, 29483, 18523, 4679, 150503, 2083, 3821, 153623, 19301, 31039, 9749, 6271, 1231, 158363, 1553, 4019, 161563, 1433, 1021, 4099, 2111, 21523, 34603, 1097, 5689, 7741, 22573, 11393, 184859, 4643, 186583, 23431, 37663, 38011, 11933, 191803, 4817, 193559, 4861, 5279, 12263, 39419, 24971, 6473, 5039, 3319, 12823, 8243, 25873, 41579, 2633, 211559, 3617, 43051, 3407, 9521, 2749, 220859, 1109, 27961, 44927, 45307, 7109, 228443, 1033, 5783, 14699, 29641, 238103, 240059, 5903, 3797, 48799, 9839, 30871, 10781, 5813, 3137, 1279, 10159, 31873, 51199, 16063, 258023, 1619, 8389, 262103, 1061, 2297, 8287, 1439, 16703, 268283, 6733, 7307, 2411, 17093, 54907, 2153, 278743, 1399, 280859, 6581, 1567, 1153, 3631, 291559, 293723, 37123, 11923, 9349, 3767, 302459, 7589, 304663, 19249, 61819, 311323, 13633, 7867, 315803, 63611, 19949, 64063, 1747, 322583, 8093, 1993, 10259, 1607, 66347, 41611, 334043, 5701, 4219, 338683, 1847, 42773, 1597, 8731, 350423, 43951, 1907, 5531, 14207, 22273, 357563, 8969, 6101, 9029, 362363, 14591, 11437, 46051, 3271, 372059, 2333, 75391, 75883, 47581, 4789, 384359, 2953, 77867, 48823, 78367, 2837, 3853, 5471, 2593, 3517, 10271, 412123, 12919, 82939, 25999, 16691, 52321, 6883, 10529, 422459, 5297, 18481, 3331, 17107, 53623, 2099, 53951, 432923, 435559, 438203, 54941, 2383, 1783, 27803, 2797, 448859, 2251, 10501, 56611, 90847, 91387, 28643, 2347, 11593, 29153, 93563, 7331, 94111, 58991, 3613, 11867, 476059, 478823, 30013, 1579, 60373, 1481, 489959, 492763, 99679, 15619, 501223, 6073, 12637, 506903, 63541, 102523, 8737, 12923, 521243, 8167, 104827, 105407, 2131, 23041, 2657, 17189, 6679, 535783, 67523, 67891, 3413, 547559, 6863, 14879, 69001, 4813, 22259, 4951, 1753, 562459, 565463, 22739, 114299, 14401, 577559, 4177, 36383, 116731, 18287, 589783, 14783, 9719, 595943, 5209, 2029, 120427, 8291, 608359, 611483, 76631, 77023, 123551, 9677, 4889, 15641, 20233, 126079, 39499, 25343, 636763, 15959, 639959, 1697, 20149, 5171, 40499, 129919, 81401, 16361, 2137, 8221, 132511, 83023, 83431, 675643, 2731, 135787, 136447, 42743, 9391, 1877, 6469, 2017, 21787, 43783, 702203, 17597, 705559, 17681, 708923, 1931, 3313, 2789, 28627, 89671, 719063, 722459, 725863, 91373, 146539, 91801, 3301, 2729, 3083, 93523, 6521, 753383, 9439, 3793, 760343, 95261, 152767, 770843, 3863, 7237, 19403, 6091, 4243, 2287, 2027, 21407, 9923, 13043, 24919, 159839, 806363, 5051, 3011, 19843, 101921, 32687, 4451, 51413, 1861, 828059, 20747, 36161, 3361, 167071, 5413, 52553, 4871, 846359, 4241, 850043, 2339, 26737, 2903, 37441, 20113, 5417, 868583, 54403, 2957, 4751, 175211, 109741, 14423, 2503, 2767, 887323, 2711, 5749, 111623, 7159, 898663, 11257, 906263, 56999, 182779, 28619, 39901, 24907, 23087, 21521, 28979, 5023, 58199, 116881, 3733, 23473, 2357, 2707, 3833, 190507, 119311, 30853, 3467, 964283, 4723, 194431, 976103, 6113, 7717, 24551, 2089, 197599, 30937, 39679, 2377,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2x-41 and
the reducible primes which appear as divisor for the first time
p | x^2x-41 and p < x^2x-41

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)
1105500.5000000.5000000.5000000.0000000.0000000.000000
21006144170.6100000.4400000.61000012.2000008.800000inf
31.0006743163580.6740000.3160000.67400011.0491807.18181821.058823
410.0006.7742.3204.4540.6774000.2320000.67740010.0504467.34177212.441340
5100.00068.36217.50150.8610.6836200.1750100.68362010.0918227.54353411.419174
61.000.000684.747141.198543.5490.6847470.1411980.68474710.0164868.06799610.686951
710.000.0006.859.6921.180.0665.679.6260.6859690.1180070.68596910.0178498.35752710.449152
8100.000.00068.668.17610.153.94058.514.2360.6866820.1015390.68668210.0103878.60455210.302481
91.000.000.000687.290.84589.101.802598.189.0430.6872910.0891020.68729110.0088698.77509610.222966
1010.000.000.0006.877.986.088794.057.5766.083.928.5120.6877990.0794060.68779910.0073898.91180210.170578


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
243300.7500000.7500000.0000001.0000001.000000-nan
384400.5000000.5000000.0000001.3333331.333333-nan
4168620.5000000.3750000.1250002.0000001.500000inf
532181440.5625000.4375000.1250002.2500002.3333332.000000
664382990.5937500.4531250.1406252.1111112.0714292.250000
71288052280.6250000.4062500.2187502.1052631.7931033.111111
8256169102670.6601560.3984380.2617192.1125001.9615382.392857
95123381791590.6601560.3496090.3105472.0000001.7549022.373134
101.0246903253650.6738280.3173830.3564452.0414201.8156422.295598
112.0481.3885967920.6777340.2910160.3867192.0115941.8338462.169863
124.0962.7741.0891.6850.6772460.2658690.4113771.9985591.8271812.127525
138.1925.5541.9473.6070.6779790.2376710.4403082.0021631.7878792.140653
1416.38411.1513.5637.5880.6806030.2174680.4631352.0077421.8299952.103687
1532.76822.3496.57315.7760.6820370.2005920.4814452.0042151.8447942.079072
1665.53644.77712.02932.7480.6832430.1835480.4996952.0035351.8300622.075811
17131.07289.61922.31567.3040.6837390.1702500.5134892.0014521.8551002.055209
18262.144179.21141.744137.4670.6836360.1592410.5243951.9996991.8706702.042479
19524.288358.83578.308280.5270.6844230.1493610.5350632.0023051.8759102.040686
201.048.576717.999147.483570.5160.6847370.1406510.5440862.0009171.8833712.033729
212.097.1521.437.068278.2341.158.8340.6852470.1326720.5525752.0014901.8865502.031203
224.194.3042.875.324527.3342.347.9900.6855310.1257260.5598042.0008271.8952902.026166
238.388.6085.753.8781.002.0634.751.8150.6859160.1194550.5664602.0011231.9002442.023780
2416.777.21611.510.7761.910.4859.600.2910.6860960.1138740.5722222.0005251.9065522.020342
2533.554.43223.029.0613.648.32219.380.7390.6863190.1087280.5775912.0006521.9096312.018766
2667.108.86446.074.7096.982.39239.092.3170.6865670.1040460.5825212.0007201.9138642.017070
27134.217.72892.175.09313.389.06878.786.0250.6867580.0997560.5870022.0005571.9175472.015384
28268.435.456184.403.80625.713.394158.690.4120.6869580.0957900.5911682.0005821.9204772.014195
29536.870.912368.903.02849.470.656319.432.3720.6871350.0921460.5949892.0005171.9239262.012928
301.073.741.824737.992.13295.314.186642.677.9460.6873090.0887680.5985402.0005041.9266812.011937
312.147.483.6481.476.328.953183.887.9301.292.441.0230.6874690.0856290.6018402.0004671.9292822.011024
324.294.967.2962.953.319.523355.227.1532.598.092.3700.6876230.0827080.6049152.0004481.9317592.010221
338.589.934.5925.907.892.383687.015.2275.220.877.1560.6877690.0799790.6077902.0004241.9340172.009504
3417.179.869.18411.818.134.1761.330.179.94710.487.954.2290.6879060.0774270.6104792.0003981.9361722.008849
3534.359.738.36823.640.795.2252.578.061.11521.062.734.1100.6880380.0750310.6130062.0003831.9381302.008279
3668.719.476.73647.290.207.8765.001.405.06042.288.802.8160.6881630.0727800.6153832.0003651.9399872.007755


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
123121020
243121020
384131021
4166241122
53214682435
66429101941159
71285222306201115
8256102475514391633
95121797710225642565
101.0243251421834012142122
112.0485962593377422281219
124.0961.089473616147404145393
138.1921.9478661.081262709260716
1416.3843.5631.6101.9534841.2874841.308
1532.7686.5732.9763.5978892.3948882.402
1665.53612.0295.4336.5961.6304.3751.6184.406
17131.07222.3159.96112.3542.9728.1213.0188.204
18262.14441.74418.66223.0825.58615.2395.64115.278
19524.28878.30835.07743.23110.44228.58810.52428.754
201.048.576147.48366.07381.41019.52754.08019.62254.254
212.097.152278.234124.656153.57836.764102.07436.793102.603
224.194.304527.334236.523290.81169.478193.77669.542194.538
238.388.6081.002.063449.258552.805131.563368.834131.559370.107
2416.777.2161.910.485856.1891.054.296250.169704.246250.458705.612
2533.554.4323.648.3221.634.3152.014.007476.9201.346.711477.1921.347.499
2667.108.8646.982.3923.128.2853.854.107911.1932.578.970911.4602.580.769
27134.217.72813.389.0685.998.7567.390.3121.744.9784.950.2031.743.6934.950.194
28268.435.45625.713.39411.520.32214.193.0723.344.6289.512.7753.343.1169.512.875
29536.870.91249.470.65622.159.48227.311.1746.422.91018.317.1936.420.40718.310.146
301.073.741.82495.314.18642.679.74052.634.44612.353.76035.305.71312.352.47335.302.240
312.147.483.648183.887.93082.319.841101.568.08923.804.86268.145.74123.803.48568.133.842
324.294.967.296355.227.153158.985.690196.241.46345.931.526131.695.70445.928.809131.671.114
338.589.934.592687.015.227307.421.757379.593.47088.726.277254.787.52488.730.242254.771.184
3417.179.869.1841.330.179.947595.103.862735.076.085171.615.805493.494.257171.617.165493.452.720
3534.359.738.3682.578.061.1151.153.174.3611.424.886.754332.278.283956.745.316332.304.751956.732.765
3668.719.476.7365.001.405.0602.236.691.1242.764.713.936644.054.6681.856.637.245644.050.8711.856.662.276


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
380000000
4162200101
5324310202
6649720603
712828199310312
825667363110241023
9512159738630493050
101.0243651871787811268107
112.048792406386165228174225
124.0961.685872813354485389457
138.1923.6071.8281.779800997811999
1416.3847.5883.8853.7031.7152.0381.7492.086
1532.76815.7767.9967.7803.6514.2373.6924.196
1665.53632.74816.62316.1257.6198.6377.7038.789
17131.07267.30434.20033.10415.72417.80115.90317.876
18262.144137.46769.72267.74532.43936.21132.48236.335
19524.288280.527142.043138.48466.60773.52966.68073.711
201.048.576570.516288.674281.842135.962149.336136.181149.037
212.097.1521.158.834585.456573.378276.776302.447277.711301.900
224.194.3042.347.9901.184.8121.163.178563.100611.243563.109610.538
238.388.6084.751.8152.396.5902.355.2251.141.7571.234.9171.142.9991.232.142
2416.777.2169.600.2914.839.4374.760.8542.312.9822.487.7482.313.9182.485.643
2533.554.43219.380.7399.767.2199.613.5204.679.8145.010.8274.681.8615.008.237
2667.108.86439.092.31719.695.82919.396.4889.454.37810.091.9649.459.00310.086.972
27134.217.72878.786.02539.689.71539.096.31019.082.81720.305.51119.091.51620.306.181
28268.435.456158.690.41279.917.99278.772.42038.492.22740.846.44738.506.49840.845.240
29536.870.912319.432.372160.826.116158.606.25677.587.28282.108.76677.612.10982.124.215
301.073.741.824642.677.946323.481.195319.196.751156.301.546165.018.035156.320.851165.037.514
312.147.483.6481.292.441.023650.351.176642.089.847314.672.171331.529.258314.695.763331.543.831
324.294.967.2962.598.092.3701.307.043.9651.291.048.405633.230.026665.796.447633.249.712665.816.185
338.589.934.5925.220.877.1562.625.893.1932.594.983.9631.273.657.5841.336.751.7611.273.678.7031.336.789.108
3417.179.869.18410.487.954.2295.273.856.5915.214.097.6382.560.836.0192.683.145.9962.560.850.1082.683.122.106
3534.359.738.36821.062.734.11010.589.307.63410.473.426.4765.147.029.6265.384.359.6325.146.961.5045.384.383.348
3668.719.476.73642.288.802.81621.256.877.98521.031.924.83110.341.694.39010.802.754.86610.341.609.72110.802.743.839


8. Check for existing Integer Sequences by OEIS

Found in Database : 41, 5, 37, 1, 1, 1, 1, 1, 23, 1, 59, 1, 103, 1, 31, 1, 43, 1, 283, 1,
Found in Database : 41, 5, 37, 23, 59, 103, 31, 43, 283, 359, 443, 61, 107, 73, 127, 743, 859, 983, 131, 223, 251, 83,
Found in Database : 5, 23, 31, 37, 41, 43, 59, 61, 73, 83, 103, 107, 113, 127, 131, 139,