Inhaltsverzeichnis

Development of
Algorithmic Constructions

16:13:50
Deutsch
29.Mar 2024

Polynom = x^2+11

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) = 11 = 11
f(1) = 3 = 3
f(2) = 15 = 3*5
f(3) = 5 = 5
f(4) = 27 = 3*3*3
f(5) = 9 = 3*3
f(6) = 47 = 47
f(7) = 15 = 3*5
f(8) = 75 = 3*5*5
f(9) = 23 = 23
f(10) = 111 = 3*37
f(11) = 33 = 3*11
f(12) = 155 = 5*31
f(13) = 45 = 3*3*5
f(14) = 207 = 3*3*23
f(15) = 59 = 59
f(16) = 267 = 3*89
f(17) = 75 = 3*5*5
f(18) = 335 = 5*67
f(19) = 93 = 3*31
f(20) = 411 = 3*137
f(21) = 113 = 113
f(22) = 495 = 3*3*5*11
f(23) = 135 = 3*3*3*5
f(24) = 587 = 587
f(25) = 159 = 3*53
f(26) = 687 = 3*229
f(27) = 185 = 5*37
f(28) = 795 = 3*5*53
f(29) = 213 = 3*71
f(30) = 911 = 911
f(31) = 243 = 3*3*3*3*3
f(32) = 1035 = 3*3*5*23
f(33) = 275 = 5*5*11
f(34) = 1167 = 3*389
f(35) = 309 = 3*103
f(36) = 1307 = 1307
f(37) = 345 = 3*5*23
f(38) = 1455 = 3*5*97
f(39) = 383 = 383
f(40) = 1611 = 3*3*179
f(41) = 423 = 3*3*47
f(42) = 1775 = 5*5*71
f(43) = 465 = 3*5*31
f(44) = 1947 = 3*11*59
f(45) = 509 = 509
f(46) = 2127 = 3*709
f(47) = 555 = 3*5*37
f(48) = 2315 = 5*463
f(49) = 603 = 3*3*67
f(50) = 2511 = 3*3*3*3*31
f(51) = 653 = 653
f(52) = 2715 = 3*5*181
f(53) = 705 = 3*5*47
f(54) = 2927 = 2927
f(55) = 759 = 3*11*23
f(56) = 3147 = 3*1049
f(57) = 815 = 5*163
f(58) = 3375 = 3*3*3*5*5*5
f(59) = 873 = 3*3*97
f(60) = 3611 = 23*157
f(61) = 933 = 3*311
f(62) = 3855 = 3*5*257
f(63) = 995 = 5*199
f(64) = 4107 = 3*37*37
f(65) = 1059 = 3*353
f(66) = 4367 = 11*397
f(67) = 1125 = 3*3*5*5*5
f(68) = 4635 = 3*3*5*103
f(69) = 1193 = 1193
f(70) = 4911 = 3*1637
f(71) = 1263 = 3*421
f(72) = 5195 = 5*1039
f(73) = 1335 = 3*5*89
f(74) = 5487 = 3*31*59
f(75) = 1409 = 1409
f(76) = 5787 = 3*3*643
f(77) = 1485 = 3*3*3*5*11
f(78) = 6095 = 5*23*53
f(79) = 1563 = 3*521
f(80) = 6411 = 3*2137
f(81) = 1643 = 31*53
f(82) = 6735 = 3*5*449
f(83) = 1725 = 3*5*5*23
f(84) = 7067 = 37*191
f(85) = 1809 = 3*3*3*67
f(86) = 7407 = 3*3*823
f(87) = 1895 = 5*379
f(88) = 7755 = 3*5*11*47
f(89) = 1983 = 3*661
f(90) = 8111 = 8111
f(91) = 2073 = 3*691
f(92) = 8475 = 3*5*5*113
f(93) = 2165 = 5*433
f(94) = 8847 = 3*3*983
f(95) = 2259 = 3*3*251
f(96) = 9227 = 9227
f(97) = 2355 = 3*5*157
f(98) = 9615 = 3*5*641
f(99) = 2453 = 11*223
f(100) = 10011 = 3*47*71

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+11

f(0)=11
f(1)=3
f(2)=5
f(3)=1
f(4)=1
f(5)=1
f(6)=47
f(7)=1
f(8)=1
f(9)=23
f(10)=37
f(11)=1
f(12)=31
f(13)=1
f(14)=1
f(15)=59
f(16)=89
f(17)=1
f(18)=67
f(19)=1
f(20)=137
f(21)=113
f(22)=1
f(23)=1
f(24)=587
f(25)=53
f(26)=229
f(27)=1
f(28)=1
f(29)=71
f(30)=911
f(31)=1
f(32)=1
f(33)=1
f(34)=389
f(35)=103
f(36)=1307
f(37)=1
f(38)=97
f(39)=383
f(40)=179
f(41)=1
f(42)=1
f(43)=1
f(44)=1
f(45)=509
f(46)=709
f(47)=1
f(48)=463
f(49)=1
f(50)=1
f(51)=653
f(52)=181
f(53)=1
f(54)=2927
f(55)=1
f(56)=1049
f(57)=163
f(58)=1
f(59)=1
f(60)=157
f(61)=311
f(62)=257
f(63)=199
f(64)=1
f(65)=353
f(66)=397
f(67)=1
f(68)=1
f(69)=1193
f(70)=1637
f(71)=421
f(72)=1039
f(73)=1
f(74)=1
f(75)=1409
f(76)=643
f(77)=1
f(78)=1
f(79)=521
f(80)=2137
f(81)=1
f(82)=449
f(83)=1
f(84)=191
f(85)=1
f(86)=823
f(87)=379
f(88)=1
f(89)=661
f(90)=8111
f(91)=691
f(92)=1
f(93)=433
f(94)=983
f(95)=251
f(96)=9227
f(97)=1
f(98)=641
f(99)=223

b) Substitution of the polynom
The polynom f(x)=x^2+11 could be written as f(y)= y^2+11 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 > 3

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

11, 3, 5, 1, 1, 1, 47, 1, 1, 23, 37, 1, 31, 1, 1, 59, 89, 1, 67, 1, 137, 113, 1, 1, 587, 53, 229, 1, 1, 71, 911, 1, 1, 1, 389, 103, 1307, 1, 97, 383, 179, 1, 1, 1, 1, 509, 709, 1, 463, 1, 1, 653, 181, 1, 2927, 1, 1049, 163, 1, 1, 157, 311, 257, 199, 1, 353, 397, 1, 1, 1193, 1637, 421, 1039, 1, 1, 1409, 643, 1, 1, 521, 2137, 1, 449, 1, 191, 1, 823, 379, 1, 661, 8111, 691, 1, 433, 983, 251, 9227, 1, 641, 223, 1, 1, 2083, 1, 401, 1, 1, 1, 467, 991, 367, 3083, 1, 1, 13007, 1103, 1, 1, 929, 1181, 14411, 1, 331, 757, 1, 1303, 15887, 269, 1093, 1, 1879, 1, 317, 1, 1, 1, 1, 313, 1, 1, 2179, 4973, 1, 1, 20747, 1753, 7109, 1, 487, 617, 22511, 1901, 1, 1171, 719, 2003, 1, 1, 1, 6323, 8537, 2161, 1, 443, 8969, 619, 1021, 1, 5647, 2381, 419, 1, 1973, 499, 977, 1, 1, 1567, 2113, 2671, 32411, 2731, 1, 1, 1, 1, 34607, 1, 2357, 8933, 12037, 3041, 1, 1, 1, 1, 12809, 647, 1, 3301, 13337, 10103, 907, 1, 41627, 1, 14149, 2143, 577, 1, 44111, 1237, 1, 2269, 15269, 3853, 2029, 1, 3169, 1, 1, 1, 9859, 829, 16729, 12659, 17029, 859, 10399, 1, 5879, 1213, 1, 1, 54767, 4603, 599, 1, 1259, 1, 1087, 1, 1, 2953, 863, 5003, 60527, 1, 1367, 1, 1, 1, 12703, 1, 1, 1, 7283, 1, 2663, 5591, 727, 17033, 1, 1153, 6337, 1951, 2621, 1, 4789, 1, 72911, 6121, 4933, 3727, 1, 1, 1621, 1279, 5153, 19463, 1, 6581, 15907, 1, 8963, 883, 1, 1373, 1, 6961, 1, 683, 1, 1, 631, 7253, 29209, 1, 1, 7451, 90011, 839, 2027, 4591, 30809, 7753, 2531, 1571, 1, 23873, 1, 2687, 19471, 1, 32869, 24809, 33289, 1, 1, 1, 3793, 25763, 1, 1, 104987, 8803, 1, 5347, 797, 1, 9901, 1, 7349, 1109, 37189, 1, 4909, 1, 2539, 1, 1, 881, 4679, 1, 1, 29759, 1, 1, 24223, 10151, 1, 30803, 751, 1, 5449, 1, 14083, 6373, 1709, 1, 1, 10861, 8737, 1, 14723, 3701, 133967, 1, 9029, 1, 971, 11471, 1, 773, 1, 35159, 47129, 1, 28579, 11971, 1301, 36293, 1, 1, 1, 1123, 49669, 7489, 10037, 12611, 152111, 1, 1, 7723, 51749, 13003, 1, 1, 1, 1, 1, 1489, 32323, 2707, 54409, 1783, 54949, 1, 6659, 1549, 18679, 1, 11317, 2843, 947, 1, 57689, 1, 1, 4877, 2633, 14771, 1, 1, 59929, 15053, 1871, 1013, 1, 1, 61637, 1, 1, 1, 1697, 47309, 2347, 1061, 38371, 16061, 5867, 48623, 1, 3271, 197147, 5501, 1, 1, 13381, 1, 1, 1, 1, 1, 1, 1, 207947, 1, 2797, 52673, 70537, 1, 3881, 1, 1, 54059, 1, 1, 1, 1, 73637, 1499, 4951, 1, 9769, 18803, 1607, 1, 15233, 19121, 2237, 6427, 1721, 2333, 1, 19603, 236207, 1, 15877, 1, 8893, 1, 1, 4051, 81349, 5569, 82009, 1, 49603, 6917, 27779, 62753, 1, 4217, 254027, 1, 7759, 12853, 1, 2399, 260111, 1, 17477, 13159, 88069, 1, 4513, 1, 1, 67343, 3919, 22621, 54499, 1, 91529, 68909, 1, 1543, 1, 23321, 93637, 1, 18869, 1, 1433, 7951, 3547, 14419, 1, 1, 1, 24391, 3917, 1, 1, 1, 1, 4987, 20021, 75353, 1, 25301, 60943, 1699, 1, 1453, 103049, 5171, 1, 26041, 104537, 1, 7019, 1, 318107, 1, 4643, 1, 1, 26981, 1, 3019, 1, 16417, 109829, 1, 1291, 1, 22273, 83813, 37379, 9377, 1831, 1, 4943, 1277, 1, 5743, 69151, 1, 12893, 87323, 4673, 5861, 32077, 1, 118409, 1, 1, 9967, 1, 1, 1, 18181, 121609, 1, 1, 1, 1, 92723, 1747, 1, 3257, 6263, 125669, 94559, 3833, 1, 76387, 1, 1321, 1439, 25793, 6469, 1, 3617, 43543, 1787, 26293, 32971, 17257, 33181, 1, 4007, 1, 1, 404507, 6763, 2467, 1, 136537, 1, 16487, 2297, 15361, 104009, 139109, 6977, 83983, 3191, 140837, 105953, 1, 1, 427727, 35753, 3877, 1, 1, 36191, 1, 1, 9739, 1, 1, 1, 443567, 1483, 1, 111893, 1609, 1, 1, 7549, 151429, 113909, 1, 7639, 91939, 1423, 1, 1, 2819, 1, 467867, 39103, 1523, 23599, 1, 13187, 476111, 39791, 1, 1, 5179, 40253, 5443, 2699, 1, 1, 163337, 1, 98563, 8237, 1, 1, 18461, 2777, 20051, 1, 168037, 2689, 33797, 1, 9619, 1, 56963, 1, 34369, 1, 518411, 43321, 1511, 1, 58243, 1, 47917, 1, 1, 132863, 4801, 44531, 3457, 1, 59863, 135059, 180569, 1, 1, 1, 182537, 137273, 2447, 3067, 1, 2011, 3947, 27901, 3391, 46751, 1, 15667, 1, 28351, 189509, 1, 1, 9551, 1, 13093, 2377, 16087, 116131, 1, 194569, 146309, 1, 1, 1, 16427, 1, 1, 39733, 1, 599087, 50053, 1, 30187, 13451, 1873, 608411, 4621, 1, 6131, 1, 1, 9221, 1, 13799, 1, 208037, 2267, 2281, 1, 6779, 158009, 3061, 3529, 127363, 53201, 213337, 160403, 1, 1, 17471, 1, 24061, 32563, 1741, 54541, 9241, 1, 1, 33049, 1, 18451, 1, 1, 1, 1, 4229, 56171, 2017, 1, 2039, 1, 1, 11399, 137119, 57271, 229637, 172643, 15383, 1, 695567, 1, 21179, 1, 46817, 58661, 15013, 1, 1, 35533, 1, 1, 715727, 1, 1, 5813, 80279, 20117, 3089, 1, 243109, 1, 7879, 12241, 2677, 1, 1, 5009, 49537, 12413, 746507, 2711, 249989, 7517, 5581, 1907, 12829, 1, 1, 1, 2861, 63803, 14479, 4273, 1, 193163, 1, 1, 1, 1, 260489, 195809, 87223, 1, 6857, 1, 1, 18043, 1, 13291, 11257, 7417, 89203, 40231, 1, 1, 13729, 67651, 4931, 40771, 90803, 22751, 1, 13711, 10993, 1, 276037, 1, 166351, 1, 30941, 5657, 279689, 2803, 5437, 70381, 4211, 1, 2099, 4733, 77617, 1, 5393, 42967, 57413, 1, 864911, 24077, 1, 1, 1, 1, 876107, 14633, 58657, 1, 98179, 1, 1, 14821, 297049, 223259, 1, 14947, 179743, 1, 100279, 226103, 1, 15137, 910127, 76003, 4547, 1, 4079, 1, 921611, 76961, 1, 1, 309769, 1, 3631, 1, 1, 234743, 313637, 78571, 5107, 1, 1, 10333, 35281, 5303, 191299, 1, 1, 5119, 1, 3221, 968267, 26951, 108023, 1, 1, 2203, 89101, 1, 13121, 1, 1, 1, 8779, 16567, 2887, 249503,

6. Sequence of the polynom (only primes)

11, 3, 5, 47, 23, 37, 31, 59, 89, 67, 137, 113, 587, 53, 229, 71, 911, 389, 103, 1307, 97, 383, 179, 509, 709, 463, 653, 181, 2927, 1049, 163, 157, 311, 257, 199, 353, 397, 1193, 1637, 421, 1039, 1409, 643, 521, 2137, 449, 191, 823, 379, 661, 8111, 691, 433, 983, 251, 9227, 641, 223, 2083, 401, 467, 991, 367, 3083, 13007, 1103, 929, 1181, 14411, 331, 757, 1303, 15887, 269, 1093, 1879, 317, 313, 2179, 4973, 20747, 1753, 7109, 487, 617, 22511, 1901, 1171, 719, 2003, 6323, 8537, 2161, 443, 8969, 619, 1021, 5647, 2381, 419, 1973, 499, 977, 1567, 2113, 2671, 32411, 2731, 34607, 2357, 8933, 12037, 3041, 12809, 647, 3301, 13337, 10103, 907, 41627, 14149, 2143, 577, 44111, 1237, 2269, 15269, 3853, 2029, 3169, 9859, 829, 16729, 12659, 17029, 859, 10399, 5879, 1213, 54767, 4603, 599, 1259, 1087, 2953, 863, 5003, 60527, 1367, 12703, 7283, 2663, 5591, 727, 17033, 1153, 6337, 1951, 2621, 4789, 72911, 6121, 4933, 3727, 1621, 1279, 5153, 19463, 6581, 15907, 8963, 883, 1373, 6961, 683, 631, 7253, 29209, 7451, 90011, 839, 2027, 4591, 30809, 7753, 2531, 1571, 23873, 2687, 19471, 32869, 24809, 33289, 3793, 25763, 104987, 8803, 5347, 797, 9901, 7349, 1109, 37189, 4909, 2539, 881, 4679, 29759, 24223, 10151, 30803, 751, 5449, 14083, 6373, 1709, 10861, 8737, 14723, 3701, 133967, 9029, 971, 11471, 773, 35159, 47129, 28579, 11971, 1301, 36293, 1123, 49669, 7489, 10037, 12611, 152111, 7723, 51749, 13003, 1489, 32323, 2707, 54409, 1783, 54949, 6659, 1549, 18679, 11317, 2843, 947, 57689, 4877, 2633, 14771, 59929, 15053, 1871, 1013, 61637, 1697, 47309, 2347, 1061, 38371, 16061, 5867, 48623, 3271, 197147, 5501, 13381, 207947, 2797, 52673, 70537, 3881, 54059, 73637, 1499, 4951, 9769, 18803, 1607, 15233, 19121, 2237, 6427, 1721, 2333, 19603, 236207, 15877, 8893, 4051, 81349, 5569, 82009, 49603, 6917, 27779, 62753, 4217, 254027, 7759, 12853, 2399, 260111, 17477, 13159, 88069, 4513, 67343, 3919, 22621, 54499, 91529, 68909, 1543, 23321, 93637, 18869, 1433, 7951, 3547, 14419, 24391, 3917, 4987, 20021, 75353, 25301, 60943, 1699, 1453, 103049, 5171, 26041, 104537, 7019, 318107, 4643, 26981, 3019, 16417, 109829, 1291, 22273, 83813, 37379, 9377, 1831, 4943, 1277, 5743, 69151, 12893, 87323, 4673, 5861, 32077, 118409, 9967, 18181, 121609, 92723, 1747, 3257, 6263, 125669, 94559, 3833, 76387, 1321, 1439, 25793, 6469, 3617, 43543, 1787, 26293, 32971, 17257, 33181, 4007, 404507, 6763, 2467, 136537, 16487, 2297, 15361, 104009, 139109, 6977, 83983, 3191, 140837, 105953, 427727, 35753, 3877, 36191, 9739, 443567, 1483, 111893, 1609, 7549, 151429, 113909, 7639, 91939, 1423, 2819, 467867, 39103, 1523, 23599, 13187, 476111, 39791, 5179, 40253, 5443, 2699, 163337, 98563, 8237, 18461, 2777, 20051, 168037, 2689, 33797, 9619, 56963, 34369, 518411, 43321, 1511, 58243, 47917, 132863, 4801, 44531, 3457, 59863, 135059, 180569, 182537, 137273, 2447, 3067, 2011, 3947, 27901, 3391, 46751, 15667, 28351, 189509, 9551, 13093, 2377, 16087, 116131, 194569, 146309, 16427, 39733, 599087, 50053, 30187, 13451, 1873, 608411, 4621, 6131, 9221, 13799, 208037, 2267, 2281, 6779, 158009, 3061, 3529, 127363, 53201, 213337, 160403, 17471, 24061, 32563, 1741, 54541, 9241, 33049, 18451, 4229, 56171, 2017, 2039, 11399, 137119, 57271, 229637, 172643, 15383, 695567, 21179, 46817, 58661, 15013, 35533, 715727, 5813, 80279, 20117, 3089, 243109, 7879, 12241, 2677, 5009, 49537, 12413, 746507, 2711, 249989, 7517, 5581, 1907, 12829, 2861, 63803, 14479, 4273, 193163, 260489, 195809, 87223, 6857, 18043, 13291, 11257, 7417, 89203, 40231, 13729, 67651, 4931, 40771, 90803, 22751, 13711, 10993, 276037, 166351, 30941, 5657, 279689, 2803, 5437, 70381, 4211, 2099, 4733, 77617, 5393, 42967, 57413, 864911, 24077, 876107, 14633, 58657, 98179, 14821, 297049, 223259, 14947, 179743, 100279, 226103, 15137, 910127, 76003, 4547, 4079, 921611, 76961, 309769, 3631, 234743, 313637, 78571, 5107, 10333, 35281, 5303, 191299, 5119, 3221, 968267, 26951, 108023, 2203, 89101, 13121, 8779, 16567, 2887, 249503,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2x+11 and
the reducible primes which appear as divisor for the first time
p | x^2x+11 and p < x^2x+11

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)
1106420.6000000.4000000.6000000.0000000.0000000.000000
21005817410.5800000.1700000.5800009.6666674.25000020.500000
31.000617995180.6170000.0990000.61700010.6379315.82352912.634147
410.0006.4256705.7550.6425000.0670000.64250010.4132906.76767711.110039
5100.00065.3525.04960.3030.6535200.0504900.65352010.1715177.53582110.478367
61.000.000660.50841.197619.3110.6605080.0411970.66050810.1069298.15943710.269986
710.000.0006.650.735348.0456.302.6900.6650740.0348050.66507410.0691218.44830910.176939
8100.000.00066.855.5213.002.29163.853.2300.6685550.0300230.66855510.0523518.62615810.131108
91.000.000.000671.233.86526.416.479644.817.3860.6712340.0264160.67123410.0400678.79877410.098431
1010.000.000.0006.733.965.726235.944.2956.498.021.4310.6733970.0235940.67339710.0322208.93170810.077305


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
243210.7500000.5000000.2500001.0000001.0000001.000000
384310.5000000.3750000.1250001.3333331.5000001.000000
4169540.5625000.3125000.2500002.2500001.6666674.000000
53217890.5312500.2500000.2812501.8888891.6000002.250000
6643513220.5468750.2031250.3437502.0588241.6250002.444444
71287521540.5859380.1640620.4218752.1428571.6153852.454545
8256151341170.5898440.1328120.4570312.0133331.6190482.166667
9512310582520.6054690.1132810.4921882.0529801.7058822.153846
101.0246341025320.6191410.0996090.5195312.0451611.7586212.111111
112.0481.2901751.1150.6298830.0854490.5444342.0347001.7156862.095865
124.0962.6133082.3050.6379390.0751950.5627442.0255811.7600002.067265
138.1925.2625684.6940.6423340.0693360.5729982.0137771.8441562.036443
1416.38410.5621.0209.5420.6446530.0622560.5823972.0072221.7957752.032808
1532.76821.2701.89219.3780.6491090.0577390.5913702.0138231.8549022.030811
1665.53642.7553.43539.3200.6523900.0524140.5999762.0101081.8155392.029105
17131.07285.8006.48479.3160.6546020.0494690.6051332.0067831.8876272.017192
18262.144172.23312.179160.0540.6570170.0464590.6105582.0073781.8783162.017928
19524.288345.42022.819322.6010.6588360.0435240.6153132.0055391.8736352.015576
201.048.576692.72843.065649.6630.6606370.0410700.6195672.0054661.8872432.013828
212.097.1521.388.44681.6061.306.8400.6620630.0389130.6231502.0043161.8949492.011566
224.194.3042.782.989154.9792.628.0100.6635160.0369500.6265662.0043911.8991132.010965
238.388.6085.576.429295.4025.281.0270.6647620.0352150.6295472.0037551.9060782.009516
2416.777.21611.173.388563.45510.609.9330.6659860.0335850.6324012.0036821.9074182.009066
2533.554.43222.381.8231.076.82121.305.0020.6670300.0320920.6349392.0031371.9111042.008024
2667.108.86444.831.2492.063.43642.767.8130.6680380.0307480.6372902.0030201.9162292.007407
27134.217.72889.782.5813.960.82185.821.7600.6689320.0295100.6394222.0026791.9195272.006691
28268.435.456179.788.3267.611.968172.176.3580.6697640.0283570.6414072.0024861.9218162.006209
29536.870.912360.008.03714.657.519345.350.5180.6705670.0273020.6432652.0023991.9255892.005795
301.073.741.824720.809.31028.259.307692.550.0030.6713060.0263190.6449872.0022031.9279742.005354
312.147.483.6481.443.119.75654.560.9141.388.558.8420.6720050.0254070.6465982.0020831.9307242.004994
324.294.967.2962.889.031.793105.469.2182.783.562.5750.6726550.0245560.6480992.0019351.9330542.004641
338.589.934.5925.783.325.930204.113.4225.579.212.5080.6732680.0237620.6495062.0018221.9352892.004342
3417.179.869.18411.576.511.240395.407.26111.181.103.9790.6738420.0230160.6508262.0017051.9371942.004065
3534.359.738.36823.171.747.480766.771.73022.404.975.7500.6743870.0223160.6520712.0016171.9391952.003825
3668.719.476.73646.378.831.9301.488.285.12044.890.546.8100.6749010.0216570.6532432.0015251.9409762.003597


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
242010200
383020201
4165040302
5328071403
664130121525
7128210203828
825634033313414
951258057821623
101.024102010114381238
112.048175017425712257
124.09630803074512339101
138.19256805677921573201
1416.3841.02001.019125381129385
1532.7681.89201.891242699235716
1665.5363.43503.4344371.2884371.273
17131.0726.48406.4838182.4298452.392
18262.14412.179012.1781.5494.5611.5714.498
19524.28822.819022.8182.9048.5672.9518.397
201.048.57643.065043.0645.51216.1395.54915.865
212.097.15281.606081.60510.49230.47910.51230.123
224.194.304154.9790154.97819.80557.87019.85757.447
238.388.608295.4020295.40137.739110.11837.968109.577
2416.777.216563.4550563.45472.051209.65172.219209.534
2533.554.4321.076.82101.076.820137.892400.789137.616400.524
2667.108.8642.063.43602.063.435263.607767.711263.278768.840
27134.217.7283.960.82103.960.820505.6021.474.176504.6821.476.361
28268.435.4567.611.96807.611.967971.2052.834.775969.1502.836.838
29536.870.91214.657.519014.657.5181.866.2555.460.6891.864.3905.466.185
301.073.741.82428.259.307028.259.3063.596.49010.530.8893.594.05110.537.877
312.147.483.64854.560.914054.560.9136.940.30020.339.4076.936.49320.344.714
324.294.967.296105.469.2180105.469.21713.407.68139.329.89713.400.92939.330.711


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
121010010
241010010
381010010
4164221021
5329452142
6642212105386
712854302413101615
8256117675028253430
951225213711558617954
101.024532290242128133156115
112.0481.115602513283265314253
124.0962.3051.2371.068608531620546
138.1924.6942.5202.1741.2351.1061.2291.124
1416.3849.5425.0844.4582.5162.2472.4982.281
1532.76819.37810.2789.1005.0574.5975.1074.617
1665.53639.32020.69118.62910.2999.39510.2239.403
17131.07279.31641.57037.74620.72818.96320.62718.998
18262.144160.05483.55076.50441.70038.42341.49238.439
19524.288322.601167.757154.84483.62477.80983.41777.751
201.048.576649.663337.220312.443167.963156.962167.871156.867
212.097.1521.306.840677.042629.798337.250316.440336.617316.533
224.194.3042.628.0101.358.3451.269.665676.397637.403676.297637.913
238.388.6085.281.0272.724.3612.556.6661.358.1171.283.4361.357.2371.282.237
2416.777.21610.609.9335.465.1945.144.7392.723.3502.582.4762.723.2092.580.898
2533.554.43221.305.00210.961.61210.343.3905.461.2925.192.5165.462.9975.188.197
2667.108.86442.767.81321.975.18220.792.63110.950.77510.435.78610.952.96210.428.290
27134.217.72885.821.76044.036.56041.785.20021.952.45120.964.54621.955.18820.949.575
28268.435.456172.176.35888.251.51283.924.84644.003.85242.090.82843.997.71542.083.963
29536.870.912345.350.518176.847.438168.503.08088.174.13184.503.41688.182.80284.490.169
301.073.741.824692.550.003354.293.678338.256.325176.690.787169.594.980176.689.000169.575.236
312.147.483.6481.388.558.842709.746.701678.812.141353.982.707340.286.801354.022.988340.266.346
324.294.967.2962.783.562.5751.421.664.0181.361.898.557709.140.486682.634.341709.182.729682.605.019
338.589.934.5925.579.212.5082.847.370.7292.731.841.7791.420.438.1471.369.129.3721.420.526.7971.369.118.192
3417.179.869.18411.181.103.9795.702.292.5655.478.811.4142.845.022.5082.745.549.3492.845.065.1982.745.466.924
3534.359.738.36822.404.975.75011.418.930.92210.986.044.8285.697.760.9335.504.861.8235.697.697.8045.504.655.190
3668.719.476.73644.890.546.81022.864.979.00222.025.567.80811.409.966.00311.035.396.72611.409.917.69711.035.266.384


8. Check for existing Integer Sequences by OEIS

Found in Database : 11, 3, 5, 1, 1, 1, 47, 1, 1, 23, 37, 1, 31, 1, 1, 59, 89, 1, 67, 1,
Found in Database : 11, 3, 5, 47, 23, 37, 31, 59, 89, 67, 137, 113, 587, 53, 229, 71, 911, 389, 103, 1307, 97, 383,
Found in Database : 3, 5, 11, 23, 31, 37, 47, 53, 59, 67, 71, 89, 97, 103, 113, 137,