Inhaltsverzeichnis

Development of
Algorithmic Constructions

16:55:28
Deutsch
29.Mar 2024

Polynom = x^2+32x-53

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) = 53 = 53
f(1) = 5 = 5
f(2) = 15 = 3*5
f(3) = 13 = 13
f(4) = 91 = 7*13
f(5) = 33 = 3*11
f(6) = 175 = 5*5*7
f(7) = 55 = 5*11
f(8) = 267 = 3*89
f(9) = 79 = 79
f(10) = 367 = 367
f(11) = 105 = 3*5*7
f(12) = 475 = 5*5*19
f(13) = 133 = 7*19
f(14) = 591 = 3*197
f(15) = 163 = 163
f(16) = 715 = 5*11*13
f(17) = 195 = 3*5*13
f(18) = 847 = 7*11*11
f(19) = 229 = 229
f(20) = 987 = 3*7*47
f(21) = 265 = 5*53
f(22) = 1135 = 5*227
f(23) = 303 = 3*101
f(24) = 1291 = 1291
f(25) = 343 = 7*7*7
f(26) = 1455 = 3*5*97
f(27) = 385 = 5*7*11
f(28) = 1627 = 1627
f(29) = 429 = 3*11*13
f(30) = 1807 = 13*139
f(31) = 475 = 5*5*19
f(32) = 1995 = 3*5*7*19
f(33) = 523 = 523
f(34) = 2191 = 7*313
f(35) = 573 = 3*191
f(36) = 2395 = 5*479
f(37) = 625 = 5*5*5*5
f(38) = 2607 = 3*11*79
f(39) = 679 = 7*97
f(40) = 2827 = 11*257
f(41) = 735 = 3*5*7*7
f(42) = 3055 = 5*13*47
f(43) = 793 = 13*61
f(44) = 3291 = 3*1097
f(45) = 853 = 853
f(46) = 3535 = 5*7*101
f(47) = 915 = 3*5*61
f(48) = 3787 = 7*541
f(49) = 979 = 11*89
f(50) = 4047 = 3*19*71
f(51) = 1045 = 5*11*19
f(52) = 4315 = 5*863
f(53) = 1113 = 3*7*53
f(54) = 4591 = 4591
f(55) = 1183 = 7*13*13
f(56) = 4875 = 3*5*5*5*13
f(57) = 1255 = 5*251
f(58) = 5167 = 5167
f(59) = 1329 = 3*443
f(60) = 5467 = 7*11*71
f(61) = 1405 = 5*281
f(62) = 5775 = 3*5*5*7*11
f(63) = 1483 = 1483
f(64) = 6091 = 6091
f(65) = 1563 = 3*521
f(66) = 6415 = 5*1283
f(67) = 1645 = 5*7*47
f(68) = 6747 = 3*13*173
f(69) = 1729 = 7*13*19
f(70) = 7087 = 19*373
f(71) = 1815 = 3*5*11*11
f(72) = 7435 = 5*1487
f(73) = 1903 = 11*173
f(74) = 7791 = 3*7*7*53
f(75) = 1993 = 1993
f(76) = 8155 = 5*7*233
f(77) = 2085 = 3*5*139
f(78) = 8527 = 8527
f(79) = 2179 = 2179
f(80) = 8907 = 3*2969
f(81) = 2275 = 5*5*7*13
f(82) = 9295 = 5*11*13*13
f(83) = 2373 = 3*7*113
f(84) = 9691 = 11*881
f(85) = 2473 = 2473
f(86) = 10095 = 3*5*673
f(87) = 2575 = 5*5*103
f(88) = 10507 = 7*19*79
f(89) = 2679 = 3*19*47
f(90) = 10927 = 7*7*223
f(91) = 2785 = 5*557
f(92) = 11355 = 3*5*757
f(93) = 2893 = 11*263
f(94) = 11791 = 13*907
f(95) = 3003 = 3*7*11*13
f(96) = 12235 = 5*2447
f(97) = 3115 = 5*7*89
f(98) = 12687 = 3*4229
f(99) = 3229 = 3229
f(100) = 13147 = 13147

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+32x-53

f(0)=53
f(1)=5
f(2)=3
f(3)=13
f(4)=7
f(5)=11
f(6)=1
f(7)=1
f(8)=89
f(9)=79
f(10)=367
f(11)=1
f(12)=19
f(13)=1
f(14)=197
f(15)=163
f(16)=1
f(17)=1
f(18)=1
f(19)=229
f(20)=47
f(21)=1
f(22)=227
f(23)=101
f(24)=1291
f(25)=1
f(26)=97
f(27)=1
f(28)=1627
f(29)=1
f(30)=139
f(31)=1
f(32)=1
f(33)=523
f(34)=313
f(35)=191
f(36)=479
f(37)=1
f(38)=1
f(39)=1
f(40)=257
f(41)=1
f(42)=1
f(43)=61
f(44)=1097
f(45)=853
f(46)=1
f(47)=1
f(48)=541
f(49)=1
f(50)=71
f(51)=1
f(52)=863
f(53)=1
f(54)=4591
f(55)=1
f(56)=1
f(57)=251
f(58)=5167
f(59)=443
f(60)=1
f(61)=281
f(62)=1
f(63)=1483
f(64)=6091
f(65)=521
f(66)=1283
f(67)=1
f(68)=173
f(69)=1
f(70)=373
f(71)=1
f(72)=1487
f(73)=1
f(74)=1
f(75)=1993
f(76)=233
f(77)=1
f(78)=8527
f(79)=2179
f(80)=2969
f(81)=1
f(82)=1
f(83)=113
f(84)=881
f(85)=2473
f(86)=673
f(87)=103
f(88)=1
f(89)=1
f(90)=223
f(91)=557
f(92)=757
f(93)=263
f(94)=907
f(95)=1
f(96)=2447
f(97)=1
f(98)=4229
f(99)=3229

b) Substitution of the polynom
The polynom f(x)=x^2+32x-53 could be written as f(y)= y^2-309 with x=y-16

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+16
f'(x)>2x+31

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

53, 5, 3, 13, 7, 11, 1, 1, 89, 79, 367, 1, 19, 1, 197, 163, 1, 1, 1, 229, 47, 1, 227, 101, 1291, 1, 97, 1, 1627, 1, 139, 1, 1, 523, 313, 191, 479, 1, 1, 1, 257, 1, 1, 61, 1097, 853, 1, 1, 541, 1, 71, 1, 863, 1, 4591, 1, 1, 251, 5167, 443, 1, 281, 1, 1483, 6091, 521, 1283, 1, 173, 1, 373, 1, 1487, 1, 1, 1993, 233, 1, 8527, 2179, 2969, 1, 1, 113, 881, 2473, 673, 103, 1, 1, 223, 557, 757, 263, 907, 1, 2447, 1, 4229, 3229, 13147, 1, 389, 3463, 1, 3583, 1, 1, 1, 547, 5189, 1, 643, 1361, 353, 383, 1, 1, 2521, 1493, 1399, 1, 1249, 1, 1, 1, 1, 1, 619, 5179, 3001, 1, 617, 421, 569, 5623, 1, 1, 1, 1, 8009, 1217, 4931, 2081, 3613, 337, 1, 1, 2417, 2243, 2477, 1, 1861, 1009, 28591, 2411, 1171, 1481, 1427, 1, 1, 1, 1, 7933, 563, 1, 6563, 1, 33547, 1, 1039, 1733, 1, 1, 5113, 9043, 2437, 1847, 1, 449, 719, 1, 2593, 1, 2089, 1, 1, 409, 1, 10429, 1, 709, 1, 1549, 1327, 1579, 1, 751, 3499, 883, 2207, 2339, 1, 1, 48091, 1103, 653, 1, 1061, 599, 50767, 1, 1, 13033, 683, 4421, 1, 2699, 18149, 13729, 571, 1, 593, 2029, 1, 1, 1, 1, 8461, 14929, 1, 607, 12239, 1, 62191, 2239, 1, 1, 1, 5393, 3433, 1, 631, 16693, 9613, 5651, 1, 1, 1, 1, 5419, 1, 14303, 18013, 24197, 1, 1, 1237, 971, 991, 1, 3821, 3079, 1, 6007, 1, 5281, 3989, 80347, 613, 1663, 1, 787, 293, 83791, 1, 1307, 1, 1511, 1, 7937, 1, 1609, 22273, 4271, 22573, 1, 1, 92107, 1783, 2393, 1, 18911, 1, 95791, 24103, 6469, 1, 739, 8243, 14221, 5009, 1, 1951, 9281, 1223, 827, 743, 1, 1, 105967, 1777, 1, 1, 5171, 1, 1, 1, 2099, 1, 37529, 809, 22787, 9551, 1, 1, 1, 5867, 1297, 761, 1231, 1201, 8053, 4339, 1721, 1, 1301, 1, 41669, 1, 18061, 1, 1, 32143, 1, 32503, 2377, 1, 1, 1, 44549, 6719, 1, 11321, 1, 1, 1, 1, 1, 1063, 141067, 1013, 1901, 5119, 1619, 12071, 29123, 1, 1, 36979, 1931, 1, 30047, 1987, 2663, 5449, 30671, 1, 154927, 3539, 4013, 1, 4517, 13241, 3259, 40123, 10753, 1621, 162907, 1949, 14957, 1181, 1, 1, 12907, 14051, 1, 8513, 8147, 42979, 1, 1, 34883, 1, 1, 1, 1, 1, 911, 1, 8627, 1, 1, 1, 1, 1, 12421, 1, 4001, 1, 1123, 9533, 1, 4373, 1, 1471, 5573, 1, 65609, 49429, 10453, 1, 3083, 1, 1, 50773, 3709, 1, 29401, 51679, 9887, 10427, 1, 1, 211291, 1, 1093, 1, 214987, 947, 1, 10889, 2083, 1, 31513, 18461, 1, 11171, 1, 1, 226267, 1, 9127, 57283, 76697, 1229, 1, 1, 1759, 1, 1289, 1, 3659, 2843, 3931, 8599, 1, 1, 22157, 20393, 3191, 2467, 1, 1, 19207, 1607, 1, 1, 4451, 1, 5441, 1, 51551, 64693, 1, 65203, 7481, 1, 1, 1, 8059, 1907, 1, 3203, 1657, 67783, 1, 1, 39181, 22943, 3037, 1, 1, 6353, 280591, 1, 56543, 2027, 94949, 71479, 286987, 4801, 1, 5581, 1, 3847, 3089, 4909, 2927, 10597, 1873, 1, 59999, 2281, 302191, 1, 1, 1, 43801, 25643, 308827, 15497, 1, 11149, 1499, 1, 5737, 1, 105929, 6133, 3517, 1, 9209, 80863, 1, 1, 1, 1, 329167, 1, 1, 16631, 1, 1, 1, 84313, 1, 16979, 30977, 28493, 343087, 2459, 23029, 12379, 1187, 2237, 5387, 1597, 16787, 8039, 7243, 1, 3761, 1, 2551, 12889, 1, 1, 2549, 1, 11119, 1, 1, 30881, 53113, 93253, 1, 18773, 376687, 1, 379147, 1, 1, 95713, 7247, 1, 1, 19391, 1, 97579, 35597, 6547, 1433, 2017, 10169, 1, 79823, 6673, 5657, 1, 1, 1, 11621, 1, 409291, 102643, 27457, 1, 1, 1, 417007, 1, 2543, 1, 5483, 35291, 1, 4261, 142469, 5641, 1741, 1, 86531, 1409, 1, 9923, 87587, 7321, 1, 110479, 21107, 22229, 1, 1, 3137, 16069, 1, 1, 23893, 1997, 1459, 22901, 1, 115183, 1, 3511, 7151, 1, 155849, 16747, 5953, 1, 1, 118603, 158597, 119293, 1, 1, 1, 9283, 12413, 1, 97379, 5813, 489691, 17539, 32833, 1, 495307, 1, 71161, 24977, 1, 9661, 503791, 42101, 5333, 1, 15439, 1, 1, 8563, 103043, 1, 24671, 1, 1, 8707, 523867, 11939, 3313, 1, 21187, 6323, 8731, 7027, 1879, 26849, 1, 3461, 1, 27143, 3299, 136453, 547291, 1, 110051, 1, 184409, 138679, 5507, 1, 1, 12743, 1, 7417, 113039, 1889, 568207, 20347, 190409, 4091, 1, 47981, 1, 11131, 5527, 1, 83341, 1, 2087, 29399, 39301, 1, 31189, 1, 23827, 2297, 1, 1, 1, 1, 3457, 2861, 18427, 1, 11113, 1, 3769, 21997, 1, 2381, 1, 2729, 89113, 1, 1, 2857, 630127, 1, 633307, 1, 42433, 1753, 49207, 53441, 11689, 1, 2797, 1, 92761, 1, 6869, 163543, 4651, 1, 10139, 1, 662287, 1, 221849, 33359, 1, 55871, 96013, 168433, 1801, 33851, 1, 1, 1, 1, 9137, 171733, 688591, 1, 1, 34679, 33107, 1, 1979, 1, 10799, 1933, 1, 3607, 141731, 11839, 9013, 9391, 1, 1, 1867, 60041, 2143, 13921, 1, 1, 1, 8699, 12007, 1, 49057, 16763, 105613, 61751, 21221, 37223, 1, 1, 8423, 1789, 13693, 26959, 1, 189583, 30403, 12697, 15583, 1, 36527, 2957, 2371, 5851, 5569, 1, 1, 1, 41113, 1, 784687, 39323, 7507, 1, 1, 5087, 14461, 39857, 3371, 28597, 2063, 1, 1, 201973, 269897, 18443, 1, 1, 1, 204679, 1, 41117, 164831, 9833, 827791, 29629, 5039, 1, 1, 1, 1, 1, 1, 211063, 3011, 3719, 1, 1, 284489, 2777, 857167, 1, 1, 1, 3167, 1, 24809, 14503, 7207, 218479, 26539, 6269, 3319, 1, 46489, 1, 4549, 1, 127261, 6763, 1, 1, 1, 1, 902191, 1, 181199, 1, 1, 1, 6389, 15259, 2383, 12097, 2309, 1, 185027, 15451, 1, 33247, 310949, 1, 14411, 1, 17747, 2333, 1, 1, 2221, 79193, 20261, 47711, 1, 1, 87281, 1, 14831, 48299, 322649, 242479, 138841, 16231, 1, 1, 326597, 1, 196751, 2347, 75979, 2719, 330569, 1, 1, 1, 12983, 250423, 1, 1, 1, 84143, 5297, 1, 5209, 1, 2129, 7741, 2111, 4663, 48947, 1,

6. Sequence of the polynom (only primes)

53, 5, 3, 13, 7, 11, 89, 79, 367, 19, 197, 163, 229, 47, 227, 101, 1291, 97, 1627, 139, 523, 313, 191, 479, 257, 61, 1097, 853, 541, 71, 863, 4591, 251, 5167, 443, 281, 1483, 6091, 521, 1283, 173, 373, 1487, 1993, 233, 8527, 2179, 2969, 113, 881, 2473, 673, 103, 223, 557, 757, 263, 907, 2447, 4229, 3229, 13147, 389, 3463, 3583, 547, 5189, 643, 1361, 353, 383, 2521, 1493, 1399, 1249, 619, 5179, 3001, 617, 421, 569, 5623, 8009, 1217, 4931, 2081, 3613, 337, 2417, 2243, 2477, 1861, 1009, 28591, 2411, 1171, 1481, 1427, 7933, 563, 6563, 33547, 1039, 1733, 5113, 9043, 2437, 1847, 449, 719, 2593, 2089, 409, 10429, 709, 1549, 1327, 1579, 751, 3499, 883, 2207, 2339, 48091, 1103, 653, 1061, 599, 50767, 13033, 683, 4421, 2699, 18149, 13729, 571, 593, 2029, 8461, 14929, 607, 12239, 62191, 2239, 5393, 3433, 631, 16693, 9613, 5651, 5419, 14303, 18013, 24197, 1237, 971, 991, 3821, 3079, 6007, 5281, 3989, 80347, 613, 1663, 787, 293, 83791, 1307, 1511, 7937, 1609, 22273, 4271, 22573, 92107, 1783, 2393, 18911, 95791, 24103, 6469, 739, 8243, 14221, 5009, 1951, 9281, 1223, 827, 743, 105967, 1777, 5171, 2099, 37529, 809, 22787, 9551, 5867, 1297, 761, 1231, 1201, 8053, 4339, 1721, 1301, 41669, 18061, 32143, 32503, 2377, 44549, 6719, 11321, 1063, 141067, 1013, 1901, 5119, 1619, 12071, 29123, 36979, 1931, 30047, 1987, 2663, 5449, 30671, 154927, 3539, 4013, 4517, 13241, 3259, 40123, 10753, 1621, 162907, 1949, 14957, 1181, 12907, 14051, 8513, 8147, 42979, 34883, 911, 8627, 12421, 4001, 1123, 9533, 4373, 1471, 5573, 65609, 49429, 10453, 3083, 50773, 3709, 29401, 51679, 9887, 10427, 211291, 1093, 214987, 947, 10889, 2083, 31513, 18461, 11171, 226267, 9127, 57283, 76697, 1229, 1759, 1289, 3659, 2843, 3931, 8599, 22157, 20393, 3191, 2467, 19207, 1607, 4451, 5441, 51551, 64693, 65203, 7481, 8059, 1907, 3203, 1657, 67783, 39181, 22943, 3037, 6353, 280591, 56543, 2027, 94949, 71479, 286987, 4801, 5581, 3847, 3089, 4909, 2927, 10597, 1873, 59999, 2281, 302191, 43801, 25643, 308827, 15497, 11149, 1499, 5737, 105929, 6133, 3517, 9209, 80863, 329167, 16631, 84313, 16979, 30977, 28493, 343087, 2459, 23029, 12379, 1187, 2237, 5387, 1597, 16787, 8039, 7243, 3761, 2551, 12889, 2549, 11119, 30881, 53113, 93253, 18773, 376687, 379147, 95713, 7247, 19391, 97579, 35597, 6547, 1433, 2017, 10169, 79823, 6673, 5657, 11621, 409291, 102643, 27457, 417007, 2543, 5483, 35291, 4261, 142469, 5641, 1741, 86531, 1409, 9923, 87587, 7321, 110479, 21107, 22229, 3137, 16069, 23893, 1997, 1459, 22901, 115183, 3511, 7151, 155849, 16747, 5953, 118603, 158597, 119293, 9283, 12413, 97379, 5813, 489691, 17539, 32833, 495307, 71161, 24977, 9661, 503791, 42101, 5333, 15439, 8563, 103043, 24671, 8707, 523867, 11939, 3313, 21187, 6323, 8731, 7027, 1879, 26849, 3461, 27143, 3299, 136453, 547291, 110051, 184409, 138679, 5507, 12743, 7417, 113039, 1889, 568207, 20347, 190409, 4091, 47981, 11131, 5527, 83341, 2087, 29399, 39301, 31189, 23827, 2297, 3457, 2861, 18427, 11113, 3769, 21997, 2381, 2729, 89113, 2857, 630127, 633307, 42433, 1753, 49207, 53441, 11689, 2797, 92761, 6869, 163543, 4651, 10139, 662287, 221849, 33359, 55871, 96013, 168433, 1801, 33851, 9137, 171733, 688591, 34679, 33107, 1979, 10799, 1933, 3607, 141731, 11839, 9013, 9391, 1867, 60041, 2143, 13921, 8699, 12007, 49057, 16763, 105613, 61751, 21221, 37223, 8423, 1789, 13693, 26959, 189583, 30403, 12697, 15583, 36527, 2957, 2371, 5851, 5569, 41113, 784687, 39323, 7507, 5087, 14461, 39857, 3371, 28597, 2063, 201973, 269897, 18443, 204679, 41117, 164831, 9833, 827791, 29629, 5039, 211063, 3011, 3719, 284489, 2777, 857167, 3167, 24809, 14503, 7207, 218479, 26539, 6269, 3319, 46489, 4549, 127261, 6763, 902191, 181199, 6389, 15259, 2383, 12097, 2309, 185027, 15451, 33247, 310949, 14411, 17747, 2333, 2221, 79193, 20261, 47711, 87281, 14831, 48299, 322649, 242479, 138841, 16231, 326597, 196751, 2347, 75979, 2719, 330569, 12983, 250423, 84143, 5297, 5209, 2129, 7741, 2111, 4663, 48947,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+32x-53 and
the reducible primes which appear as divisor for the first time
p | x^2+32x-53 and p < x^2+32x-53

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)
1108350.8000000.3000000.8000000.0000000.0000000.000000
21005710470.5700000.1000000.5700007.1250003.3333339.400000
31.000612505620.6120000.0500000.61200010.7368425.00000011.957447
410.0006.3113805.9310.6311000.0380000.63110010.3120927.60000010.553381
5100.00064.4742.97561.4990.6447400.0297500.64474010.2161307.82894810.369078
61.000.000652.75524.308628.4470.6527550.0243080.65275510.1243138.17075610.218817
710.000.0006.586.887205.7796.381.1080.6586890.0205780.65868910.0909028.46548510.153772
8100.000.00066.298.7321.785.65564.513.0770.6629870.0178570.66298710.0652608.67753810.110012
91.000.000.000666.358.53915.742.161650.616.3780.6663590.0157420.66635910.0508498.81590310.085032
1010.000.000.0006.690.338.555140.922.8286.549.415.7270.6690340.0140920.66903410.0401488.95193710.066479


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
244221.0000000.5000000.5000001.3333331.0000002.000000
386240.7500000.2500000.5000001.5000001.0000002.000000
41610370.6250000.1875000.4375001.6666671.5000001.750000
532185130.5625000.1562500.4062501.8000001.6666671.857143
664348260.5312500.1250000.4062501.8888891.6000002.000000
71287110610.5546880.0781250.4765622.0882351.2500002.346154
8256149151340.5820310.0585940.5234382.0985921.5000002.196721
9512303262770.5917970.0507810.5410162.0335571.7333332.067164
101.024626515750.6113280.0498050.5615232.0660071.9615382.075812
112.0481.262911.1710.6162110.0444340.5717772.0159751.7843142.036522
124.0962.5571722.3850.6242680.0419920.5822752.0261491.8901102.036721
138.1925.1513204.8310.6287840.0390620.5897222.0144701.8604652.025577
1416.38410.3855879.7980.6338500.0358280.5980222.0161131.8343752.028152
1532.76820.9221.08619.8360.6384890.0331420.6053472.0146371.8500852.024495
1665.53642.0932.01440.0790.6422880.0307310.6115572.0119011.8545122.020518
17131.07284.6453.77180.8740.6457900.0287700.6170202.0109041.8723932.017865
18262.144170.0117.063162.9480.6485400.0269430.6215972.0085181.8729782.014838
19524.288341.14413.437327.7070.6506810.0256290.6250512.0066001.9024492.011114
201.048.576684.61025.399659.2110.6528950.0242220.6286732.0068071.8902292.011587
212.097.1521.373.33148.1101.325.2210.6548550.0229410.6319152.0060051.8941692.010314
224.194.3042.754.03091.7702.662.2600.6566120.0218800.6347322.0053651.9075042.008918
238.388.6085.522.072174.9525.347.1200.6582820.0208560.6374262.0050881.9064182.008489
2416.777.21611.068.393333.96510.734.4280.6597280.0199060.6398222.0043911.9088952.007516
2533.554.43222.180.402639.29421.541.1080.6610270.0190520.6419752.0039411.9142552.006731
2667.108.86444.445.3271.226.99643.218.3310.6622870.0182840.6440032.0038111.9192992.006319
27134.217.72889.050.1812.355.70286.694.4790.6634760.0175510.6459242.0035891.9198942.005965
28268.435.456178.386.8364.530.565173.856.2710.6645430.0168780.6476652.0032171.9232342.005390
29536.870.912357.308.8758.730.268348.578.6070.6655400.0162610.6492782.0030001.9269712.004982
301.073.741.824715.596.68616.843.172698.753.5140.6664510.0156860.6507652.0027401.9292852.004580
312.147.483.6481.433.030.73332.549.3731.400.481.3600.6673070.0151570.6521502.0025681.9324972.004257
324.294.967.2962.869.541.05062.952.3662.806.588.6840.6681170.0146570.6534602.0024281.9340582.004017
338.589.934.5925.745.582.045121.895.8045.623.686.2410.6688740.0141910.6546832.0022651.9363182.003744
3417.179.869.18411.503.427.454236.268.38311.267.159.0710.6695880.0137530.6558352.0021341.9382812.003519


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
122010011
242010011
382010011
4163110012
5325310212
6648610314
712810810415
8256151310618
951226241013112
101.02451491023127
112.04891891044146
124.0961721701085186
138.192320318101601159
1416.384587585102971289
1532.7681.0861.084105501535
1665.5362.0142.012101.00511.008
17131.0723.7713.769101.89311.877
18262.1447.0637.061103.54113.521
19524.28813.43713.435106.69916.737
201.048.57625.39925.3971012.702112.696
212.097.15248.11048.1081024.087124.022
224.194.30491.77091.7681045.781145.988
238.388.608174.952174.9501087.425187.526
2416.777.216333.965333.96310166.9131167.051
2533.554.432639.294639.29210319.4331319.860
2667.108.8641.226.9961.226.99410613.6291613.366
27134.217.7282.355.7022.355.700101.177.82111.177.880
28268.435.4564.530.5654.530.563102.264.43112.266.133
29536.870.9128.730.2688.730.266104.363.81214.366.455
301.073.741.82416.843.17216.843.170108.420.58518.422.586
312.147.483.64832.549.37332.549.3711016.273.900116.275.472
324.294.967.29662.952.36662.952.3641031.476.937131.475.428
338.589.934.592121.895.804121.895.8021060.949.804160.945.999
3417.179.869.184236.268.383236.268.38110118.135.7421118.132.640


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
242110020
384131120
4167341231
53213672452
6642611156875
712861293217141614
8256134686638323628
951227713214569737164
101.024575281294148145143139
112.0481.171564607295281298297
124.0962.3851.1561.229587601602595
138.1924.8312.3542.4771.1971.2151.2371.182
1416.3849.7984.8204.9782.4802.4152.5242.379
1532.76819.8369.71710.1195.0264.9265.0994.785
1665.53640.07919.79320.28610.2289.80310.2589.790
17131.07280.87439.93840.93620.64919.80420.69619.725
18262.144162.94880.46582.48341.51939.96141.54639.922
19524.288327.707162.033165.67483.51580.37883.74280.072
201.048.576659.211325.545333.666167.758161.473168.203161.777
212.097.1521.325.221655.554669.667337.357324.999337.580325.285
224.194.3042.662.2601.317.8711.344.389676.573654.096677.327654.264
238.388.6085.347.1202.646.7262.700.3941.356.3951.315.1121.359.4371.316.176
2416.777.21610.734.4285.318.7935.415.6352.723.1222.641.9232.724.6482.644.735
2533.554.43221.541.10810.676.63610.864.4725.463.1005.304.1665.464.4225.309.420
2667.108.86443.218.33121.429.97821.788.35310.953.58010.652.46310.957.28610.655.002
27134.217.72886.694.47943.007.68843.686.79121.964.55321.380.14021.966.19221.383.594
28268.435.456173.856.27186.281.89287.574.37944.025.06342.899.98844.022.71442.908.506
29536.870.912348.578.607173.062.453175.516.15488.224.56886.065.93688.217.11586.070.988
301.073.741.824698.753.514347.017.966351.735.548176.761.018172.604.425176.763.759172.624.312
312.147.483.6481.400.481.360695.683.074704.798.286354.134.007346.106.250354.129.631346.111.472
324.294.967.2962.806.588.6841.394.490.7681.412.097.916709.439.423693.900.886709.388.878693.859.497
338.589.934.5925.623.686.2412.794.873.4522.828.812.7891.420.958.6671.390.923.9081.420.939.9371.390.863.729
3417.179.869.18411.267.159.0715.600.779.6315.666.379.4402.845.926.4952.787.672.1032.845.947.1632.787.613.310


8. Check for existing Integer Sequences by OEIS

Found in Database : 53, 5, 3, 13, 7, 11, 1, 1, 89, 79, 367, 1, 19, 1, 197, 163, 1, 1, 1, 229,
Found in Database : 53, 5, 3, 13, 7, 11, 89, 79, 367, 19, 197, 163, 229, 47, 227, 101, 1291, 97, 1627, 139, 523, 313, 191, 479,
Found in Database : 3, 5, 7, 11, 13, 19, 47, 53, 61, 71, 79, 89, 97, 101, 103, 113, 139,