Inhaltsverzeichnis

Development of
Algorithmic Constructions

12:28:16
Deutsch
29.Mar 2024

Polynom = x^2+108x+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) = 303 = 3*101
f(3) = 13 = 13
f(4) = 531 = 3*3*59
f(5) = 81 = 3*3*3*3
f(6) = 767 = 13*59
f(7) = 111 = 3*37
f(8) = 1011 = 3*337
f(9) = 71 = 71
f(10) = 1263 = 3*421
f(11) = 87 = 3*29
f(12) = 1523 = 1523
f(13) = 207 = 3*3*23
f(14) = 1791 = 3*3*199
f(15) = 241 = 241
f(16) = 2067 = 3*13*53
f(17) = 69 = 3*23
f(18) = 2351 = 2351
f(19) = 39 = 3*13
f(20) = 2643 = 3*881
f(21) = 349 = 349
f(22) = 2943 = 3*3*3*109
f(23) = 387 = 3*3*43
f(24) = 3251 = 3251
f(25) = 213 = 3*71
f(26) = 3567 = 3*29*41
f(27) = 233 = 233
f(28) = 3891 = 3*1297
f(29) = 507 = 3*13*13
f(30) = 4223 = 41*103
f(31) = 549 = 3*3*61
f(32) = 4563 = 3*3*3*13*13
f(33) = 37 = 37
f(34) = 4911 = 3*1637
f(35) = 159 = 3*53
f(36) = 5267 = 23*229
f(37) = 681 = 3*227
f(38) = 5631 = 3*1877
f(39) = 727 = 727
f(40) = 6003 = 3*3*23*29
f(41) = 387 = 3*3*43
f(42) = 6383 = 13*491
f(43) = 411 = 3*137
f(44) = 6771 = 3*37*61
f(45) = 871 = 13*67
f(46) = 7167 = 3*2389
f(47) = 921 = 3*307
f(48) = 7571 = 67*113
f(49) = 243 = 3*3*3*3*3
f(50) = 7983 = 3*3*887
f(51) = 1 = 1
f(52) = 8403 = 3*2801
f(53) = 1077 = 3*359
f(54) = 8831 = 8831
f(55) = 1131 = 3*13*29
f(56) = 9267 = 3*3089
f(57) = 593 = 593
f(58) = 9711 = 3*3*13*83
f(59) = 621 = 3*3*3*23
f(60) = 10163 = 10163
f(61) = 1299 = 3*433
f(62) = 10623 = 3*3541
f(63) = 1357 = 23*59
f(64) = 11091 = 3*3697
f(65) = 177 = 3*59
f(66) = 11567 = 43*269
f(67) = 369 = 3*3*41
f(68) = 12051 = 3*3*13*103
f(69) = 1537 = 29*53
f(70) = 12543 = 3*37*113
f(71) = 1599 = 3*13*41
f(72) = 13043 = 13043
f(73) = 831 = 3*277
f(74) = 13551 = 3*4517
f(75) = 863 = 863
f(76) = 14067 = 3*3*3*521
f(77) = 1791 = 3*3*199
f(78) = 14591 = 14591
f(79) = 1857 = 3*619
f(80) = 15123 = 3*71*71
f(81) = 481 = 13*37
f(82) = 15663 = 3*23*227
f(83) = 249 = 3*83
f(84) = 16211 = 13*29*43
f(85) = 2061 = 3*3*229
f(86) = 16767 = 3*3*3*3*3*3*23
f(87) = 2131 = 2131
f(88) = 17331 = 3*53*109
f(89) = 1101 = 3*367
f(90) = 17903 = 17903
f(91) = 1137 = 3*379
f(92) = 18483 = 3*61*101
f(93) = 2347 = 2347
f(94) = 19071 = 3*3*13*163
f(95) = 2421 = 3*3*269
f(96) = 19667 = 71*277
f(97) = 39 = 3*13
f(98) = 20271 = 3*29*233
f(99) = 643 = 643
f(100) = 20883 = 3*6961

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+108x+83

f(0)=83
f(1)=3
f(2)=101
f(3)=13
f(4)=59
f(5)=1
f(6)=1
f(7)=37
f(8)=337
f(9)=71
f(10)=421
f(11)=29
f(12)=1523
f(13)=23
f(14)=199
f(15)=241
f(16)=53
f(17)=1
f(18)=2351
f(19)=1
f(20)=881
f(21)=349
f(22)=109
f(23)=43
f(24)=3251
f(25)=1
f(26)=41
f(27)=233
f(28)=1297
f(29)=1
f(30)=103
f(31)=61
f(32)=1
f(33)=1
f(34)=1637
f(35)=1
f(36)=229
f(37)=227
f(38)=1877
f(39)=727
f(40)=1
f(41)=1
f(42)=491
f(43)=137
f(44)=1
f(45)=67
f(46)=2389
f(47)=307
f(48)=113
f(49)=1
f(50)=887
f(51)=1
f(52)=2801
f(53)=359
f(54)=8831
f(55)=1
f(56)=3089
f(57)=593
f(58)=1
f(59)=1
f(60)=10163
f(61)=433
f(62)=3541
f(63)=1
f(64)=3697
f(65)=1
f(66)=269
f(67)=1
f(68)=1
f(69)=1
f(70)=1
f(71)=1
f(72)=13043
f(73)=277
f(74)=4517
f(75)=863
f(76)=521
f(77)=1
f(78)=14591
f(79)=619
f(80)=1
f(81)=1
f(82)=1
f(83)=1
f(84)=1
f(85)=1
f(86)=1
f(87)=2131
f(88)=1
f(89)=367
f(90)=17903
f(91)=379
f(92)=1
f(93)=2347
f(94)=163
f(95)=1
f(96)=1
f(97)=1
f(98)=1
f(99)=643

b) Substitution of the polynom
The polynom f(x)=x^2+108x+83 could be written as f(y)= y^2-2833 with x=y-54

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+54
f'(x)>2x+107

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, 101, 13, 59, 1, 1, 37, 337, 71, 421, 29, 1523, 23, 199, 241, 53, 1, 2351, 1, 881, 349, 109, 43, 3251, 1, 41, 233, 1297, 1, 103, 61, 1, 1, 1637, 1, 229, 227, 1877, 727, 1, 1, 491, 137, 1, 67, 2389, 307, 113, 1, 887, 1, 2801, 359, 8831, 1, 3089, 593, 1, 1, 10163, 433, 3541, 1, 3697, 1, 269, 1, 1, 1, 1, 1, 13043, 277, 4517, 863, 521, 1, 14591, 619, 1, 1, 1, 1, 1, 1, 1, 2131, 1, 367, 17903, 379, 1, 2347, 163, 1, 1, 1, 1, 643, 6961, 883, 21503, 1, 2459, 1, 7589, 1, 571, 1, 617, 3049, 1, 1, 25391, 1, 8689, 3301, 1, 1129, 2111, 193, 1, 1, 1, 1217, 29567, 1, 439, 479, 383, 1, 1381, 1, 10837, 4111, 853, 701, 34031, 239, 1289, 1, 409, 1499, 983, 1, 953, 587, 4219, 1, 38783, 1, 1, 1, 13477, 1, 1423, 1, 4679, 1, 14321, 1, 3371, 461, 14897, 5641, 1, 1, 46451, 977, 1, 1, 1237, 2029, 2137, 1, 5563, 1579, 739, 1, 1, 1, 263, 1, 1993, 1, 54767, 1151, 1429, 7027, 1, 2383, 1559, 1, 1, 1, 19889, 1, 1, 2549, 709, 1, 6967, 1, 631, 2677, 21589, 8161, 21937, 691, 66863, 1, 7547, 1, 1, 2897, 70067, 1471, 641, 4481, 1, 1, 1033, 3079, 1, 293, 25189, 1, 1447, 1, 8647, 1, 26321, 1657, 80111, 1, 1, 787, 1, 1153, 1, 877, 1, 1, 28657, 3607, 1301, 1, 1091, 1, 2297, 1879, 90803, 1, 30677, 313, 1, 1, 3259, 991, 1, 12049, 32341, 1, 1, 1, 1, 6263, 33617, 4229, 4441, 4283, 1, 3253, 1, 1, 1, 4447, 35797, 1039, 36241, 1, 110063, 769, 12379, 14011, 37589, 1, 1871, 1, 38501, 3631, 1, 1, 1171, 4957, 353, 7523, 40357, 1, 2311, 1, 1, 1, 41777, 1, 1163, 1, 42737, 1, 14407, 1811, 5701, 1, 1, 1, 1, 1, 10427, 1, 15227, 2153, 46181, 1451, 3257, 5867, 47189, 17791, 1223, 1, 4987, 1, 1, 18367, 929, 1, 2447, 1, 16759, 1, 1373, 1, 3581, 6449, 3989, 1, 5821, 1097, 158771, 1, 1, 20149, 54001, 1, 1, 1, 6121, 1597, 1, 1, 2857, 3529, 2467, 10691, 1, 2399, 4691, 1, 58417, 5503, 1, 463, 178643, 1, 1, 22651, 60689, 1, 1, 3847, 1, 1013, 20807, 1, 189011, 1, 63589, 1, 2213, 8059, 2341, 2711, 21787, 947, 65957, 1, 15359, 1, 67157, 1, 7529, 1, 205103, 1, 2999, 25981, 1, 8737, 9157, 1, 1, 13337, 1, 8969, 5843, 1, 72689, 1, 24439, 1, 1361, 9283, 5737, 28087, 75217, 4721, 227567, 1, 1, 28807, 77141, 1, 1, 2441, 78437, 1, 1, 1103, 239231, 10009, 1, 15137, 1, 5087, 245171, 1, 1, 1, 6389, 1303, 1, 1, 84401, 1, 1, 3559, 1, 5381, 1, 1, 87121, 1, 263423, 3673, 9833, 8329, 89189, 1399, 269651, 11279, 1, 1, 30427, 1, 21227, 1, 1571, 1, 1583, 11719, 2741, 1, 31607, 8923, 7349, 11987, 288767, 1, 2621, 18251, 1123, 1, 295283, 1, 1, 37321, 1, 1, 3637, 1, 1, 38149, 102101, 12809, 10639, 6451, 1459, 1, 11593, 4363, 24251, 13183, 1997, 1, 1, 3343, 322067, 1, 1, 1, 108881, 6829, 1, 1, 1, 1, 2851, 4649, 1, 3511, 112741, 5303, 1861, 1, 2503, 1, 1, 1, 115877, 1, 350003, 14633, 5107, 1, 39419, 1, 1, 3733, 4133, 3469, 120661, 1, 28031, 2539, 40759, 23003, 1483, 1, 1, 1, 124721, 11731, 1, 1, 1627, 1, 127189, 1, 1, 1, 386543, 2693, 1601, 48787, 130517, 1259, 1, 1, 10169, 12433, 1, 1, 9341, 1, 2011, 25343, 135589, 8501, 1, 1901, 45767, 1, 6007, 1, 7069, 2179, 2371, 1, 46919, 1, 1, 683, 142501, 26801, 1, 17977, 14923, 6029, 1, 1, 146021, 1, 440723, 1, 11369, 2417, 5507, 1, 8467, 9377, 5189, 56599, 151381, 18979, 1, 1, 17021, 1, 154097, 19319, 35771, 19433, 155921, 1, 2273, 1, 12791, 19777, 6899, 719, 12277, 1, 1, 1, 53819, 1481, 162389, 20357, 1, 1, 1, 1, 1, 1, 8171, 1, 12853, 1, 168037, 2633, 506963, 1, 1531, 63907, 1, 10711, 39659, 10771, 4019, 4999, 19309, 1, 524243, 1, 1, 16519, 1621, 22147, 18379, 1, 1, 33587, 1, 11257, 23557, 22637, 1, 68281, 60859, 1907, 550703, 1, 14197, 2393, 185557, 1789, 559667, 1, 1, 1, 1, 23633, 9323, 1, 190577, 1, 63863, 1, 1, 1, 5233, 1693, 3299, 12197, 9949, 1, 65563, 73951, 1, 1, 596243, 1, 1, 1, 1, 8389, 1, 1, 1, 1, 8867, 1, 614963, 8563, 1, 1, 1, 811, 2591, 6521, 209201, 78649, 1, 8783, 48767, 13241, 1, 1, 1889, 1, 643583, 1, 71867, 1, 5039, 1697, 653267, 2099, 4129, 82267, 5639, 1, 2521, 1, 5417, 2879, 1, 27967, 16411, 1, 5779, 1, 1, 1, 682751, 28517, 1, 1, 2837, 4799, 23887, 28933, 3463, 6709, 6301, 1, 54059, 1, 26153, 859, 3331, 29633, 712883, 14887, 8233, 1951, 6151, 1, 723071, 1, 242161, 5689, 243301, 7621, 733331, 1, 1, 92311, 1801, 1, 1, 1, 19157, 93607, 83399, 1, 754067, 7873, 10979, 1, 253681, 1, 2557, 1, 1, 3701, 257189, 16111, 775091, 32369, 1, 97549, 1, 1, 1, 1, 263089, 1, 1, 1, 18521, 1, 29629, 50111, 267857, 1, 19687, 911, 20789, 1, 2207, 1, 1, 34159, 2659, 102931, 1, 17231, 1, 1, 92507, 1, 278741, 34919, 1, 1, 281189, 1, 4093, 3931, 850943, 35533, 284881, 1, 1, 17921, 862067, 1, 96199, 108457, 289841, 1, 873263, 1, 7901, 1, 32621, 1, 10657, 1, 22777, 1, 2887, 1, 8219, 1, 1, 14087, 4937, 9431, 1, 1, 1, 8779, 1, 6367, 918767, 19181, 307537, 1, 1, 1, 930323, 1, 4513, 14627, 1, 1, 2699, 1, 315281, 1, 105527, 2203, 953651, 1, 1, 1, 7817, 1, 965423, 3359, 1, 5279, 324437, 40637, 18439, 1, 1, 1, 12163, 13711, 76091, 41299, 7699, 1, 5449, 1, 1, 13933, 1, 125899, 1, 21067, 34939, 1, 1, 1, 3067, 1, 1, 5351, 343141, 32233, 344497, 3319, 1037567, 4813, 1, 1, 6577, 1, 1049843, 1069, 3109, 132001, 117563, 1, 81707, 1, 355441, 10273, 356821, 1, 1, 7477, 1, 1, 360977, 1, 4789, 3491, 363761, 4271, 3121, 1, 13249, 1, 367957, 138247,

6. Sequence of the polynom (only primes)

83, 3, 101, 13, 59, 37, 337, 71, 421, 29, 1523, 23, 199, 241, 53, 2351, 881, 349, 109, 43, 3251, 41, 233, 1297, 103, 61, 1637, 229, 227, 1877, 727, 491, 137, 67, 2389, 307, 113, 887, 2801, 359, 8831, 3089, 593, 10163, 433, 3541, 3697, 269, 13043, 277, 4517, 863, 521, 14591, 619, 2131, 367, 17903, 379, 2347, 163, 643, 6961, 883, 21503, 2459, 7589, 571, 617, 3049, 25391, 8689, 3301, 1129, 2111, 193, 1217, 29567, 439, 479, 383, 1381, 10837, 4111, 853, 701, 34031, 239, 1289, 409, 1499, 983, 953, 587, 4219, 38783, 13477, 1423, 4679, 14321, 3371, 461, 14897, 5641, 46451, 977, 1237, 2029, 2137, 5563, 1579, 739, 263, 1993, 54767, 1151, 1429, 7027, 2383, 1559, 19889, 2549, 709, 6967, 631, 2677, 21589, 8161, 21937, 691, 66863, 7547, 2897, 70067, 1471, 641, 4481, 1033, 3079, 293, 25189, 1447, 8647, 26321, 1657, 80111, 787, 1153, 877, 28657, 3607, 1301, 1091, 2297, 1879, 90803, 30677, 313, 3259, 991, 12049, 32341, 6263, 33617, 4229, 4441, 4283, 3253, 4447, 35797, 1039, 36241, 110063, 769, 12379, 14011, 37589, 1871, 38501, 3631, 1171, 4957, 353, 7523, 40357, 2311, 41777, 1163, 42737, 14407, 1811, 5701, 10427, 15227, 2153, 46181, 1451, 3257, 5867, 47189, 17791, 1223, 4987, 18367, 929, 2447, 16759, 1373, 3581, 6449, 3989, 5821, 1097, 158771, 20149, 54001, 6121, 1597, 2857, 3529, 2467, 10691, 2399, 4691, 58417, 5503, 463, 178643, 22651, 60689, 3847, 1013, 20807, 189011, 63589, 2213, 8059, 2341, 2711, 21787, 947, 65957, 15359, 67157, 7529, 205103, 2999, 25981, 8737, 9157, 13337, 8969, 5843, 72689, 24439, 1361, 9283, 5737, 28087, 75217, 4721, 227567, 28807, 77141, 2441, 78437, 1103, 239231, 10009, 15137, 5087, 245171, 6389, 1303, 84401, 3559, 5381, 87121, 263423, 3673, 9833, 8329, 89189, 1399, 269651, 11279, 30427, 21227, 1571, 1583, 11719, 2741, 31607, 8923, 7349, 11987, 288767, 2621, 18251, 1123, 295283, 37321, 3637, 38149, 102101, 12809, 10639, 6451, 1459, 11593, 4363, 24251, 13183, 1997, 3343, 322067, 108881, 6829, 2851, 4649, 3511, 112741, 5303, 1861, 2503, 115877, 350003, 14633, 5107, 39419, 3733, 4133, 3469, 120661, 28031, 2539, 40759, 23003, 1483, 124721, 11731, 1627, 127189, 386543, 2693, 1601, 48787, 130517, 1259, 10169, 12433, 9341, 2011, 25343, 135589, 8501, 1901, 45767, 6007, 7069, 2179, 2371, 46919, 683, 142501, 26801, 17977, 14923, 6029, 146021, 440723, 11369, 2417, 5507, 8467, 9377, 5189, 56599, 151381, 18979, 17021, 154097, 19319, 35771, 19433, 155921, 2273, 12791, 19777, 6899, 719, 12277, 53819, 1481, 162389, 20357, 8171, 12853, 168037, 2633, 506963, 1531, 63907, 10711, 39659, 10771, 4019, 4999, 19309, 524243, 16519, 1621, 22147, 18379, 33587, 11257, 23557, 22637, 68281, 60859, 1907, 550703, 14197, 2393, 185557, 1789, 559667, 23633, 9323, 190577, 63863, 5233, 1693, 3299, 12197, 9949, 65563, 73951, 596243, 8389, 8867, 614963, 8563, 811, 2591, 6521, 209201, 78649, 8783, 48767, 13241, 1889, 643583, 71867, 5039, 1697, 653267, 2099, 4129, 82267, 5639, 2521, 5417, 2879, 27967, 16411, 5779, 682751, 28517, 2837, 4799, 23887, 28933, 3463, 6709, 6301, 54059, 26153, 859, 3331, 29633, 712883, 14887, 8233, 1951, 6151, 723071, 242161, 5689, 243301, 7621, 733331, 92311, 1801, 19157, 93607, 83399, 754067, 7873, 10979, 253681, 2557, 3701, 257189, 16111, 775091, 32369, 97549, 263089, 18521, 29629, 50111, 267857, 19687, 911, 20789, 2207, 34159, 2659, 102931, 17231, 92507, 278741, 34919, 281189, 4093, 3931, 850943, 35533, 284881, 17921, 862067, 96199, 108457, 289841, 873263, 7901, 32621, 10657, 22777, 2887, 8219, 14087, 4937, 9431, 8779, 6367, 918767, 19181, 307537, 930323, 4513, 14627, 2699, 315281, 105527, 2203, 953651, 7817, 965423, 3359, 5279, 324437, 40637, 18439, 12163, 13711, 76091, 41299, 7699, 5449, 13933, 125899, 21067, 34939, 3067, 5351, 343141, 32233, 344497, 3319, 1037567, 4813, 6577, 1049843, 1069, 3109, 132001, 117563, 81707, 355441, 10273, 356821, 7477, 360977, 4789, 3491, 363761, 4271, 3121, 13249, 367957, 138247,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+108x+83 and
the reducible primes which appear as divisor for the first time
p | x^2+108x+83 and p < x^2+108x+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)
1109450.9000000.4000000.5000000.0000000.0000000.000000
21005721360.5700000.2100000.3600006.3333335.2500007.200000
31.0005851334520.5850000.1330000.45200010.2631586.33333312.555555
410.0006.1849785.2060.6184000.0978000.52060010.5709407.35338411.517699
5100.00063.5157.63155.8840.6351500.0763100.55884010.2708617.80265910.734537
61.000.000645.84561.333584.5120.6458450.0613330.58451210.1683868.03734810.459380
710.000.0006.529.584511.8796.017.7050.6529580.0511880.60177110.1101418.34589910.295263
8100.000.00065.809.7544.406.36561.403.3890.6580980.0440640.61403410.0787058.60821610.203789
91.000.000.000662.085.22438.671.960623.413.2640.6620850.0386720.62341310.0605958.77638610.152750
1010.000.000.0006.652.547.845344.610.7186.307.937.1270.6652550.0344610.63079410.0478738.91112610.118388


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
245321.2500000.7500000.5000001.6666671.5000002.000000
387340.8750000.3750000.5000001.4000001.0000002.000000
41614680.8750000.3750000.5000002.0000002.0000002.000000
5322210120.6875000.3125000.3750001.5714291.6666671.500000
6644214280.6562500.2187500.4375001.9090911.4000002.333333
71287226460.5625000.2031250.3593751.7142861.8571431.642857
8256147421050.5742190.1640620.4101562.0416671.6153852.282609
9512301772240.5878910.1503910.4375002.0476191.8333332.133333
101.0245971354620.5830080.1318360.4511721.9833891.7532472.062500
112.0481.2192529670.5952150.1230470.4721682.0418761.8666672.093074
124.0962.5074472.0600.6120610.1091310.5029302.0566041.7738102.130300
138.1925.0708304.2400.6188960.1013180.5175782.0223371.8568232.058252
1416.38410.2461.5198.7270.6253660.0927120.5326542.0209071.8301202.058255
1532.76820.6312.79317.8380.6296080.0852360.5443732.0135661.8387102.044001
1665.53641.5045.21936.2850.6333010.0796360.5536652.0117301.8686002.034141
17131.07283.5199.72373.7960.6371990.0741810.5630192.0123121.8630012.033788
18262.144167.81718.136149.6810.6401710.0691830.5709882.0093271.8652682.028308
19524.288337.25134.002303.2490.6432550.0648540.5784022.0096351.8748352.025969
201.048.576677.41364.040613.3730.6460310.0610730.5849582.0086311.8834192.022671
212.097.1521.359.930121.0741.238.8560.6484650.0577330.5907332.0075351.8906002.019743
224.194.3042.728.502229.0792.499.4230.6505260.0546170.5959092.0063551.8920582.017525
238.388.6085.473.483435.0695.038.4140.6524900.0518640.6006262.0060401.8992092.015831
2416.777.21610.976.936829.16410.147.7720.6542760.0494220.6048542.0054761.9058222.014081
2533.554.43222.005.9271.584.82520.421.1020.6558280.0472310.6085962.0047421.9113532.012373
2667.108.86444.111.4223.031.02641.080.3960.6573110.0451660.6121462.0045251.9125302.011664
27134.217.72888.404.6315.810.72882.593.9030.6586660.0432930.6153732.0041211.9170832.010543
28268.435.456177.150.15711.161.915165.988.2420.6599360.0415810.6183542.0038561.9209152.009691
29536.870.912354.929.10921.469.424333.459.6850.6611070.0399900.6211172.0035501.9234532.008936
301.073.741.824711.024.21741.365.869669.658.3480.6621930.0385250.6236682.0032851.9267342.008214
312.147.483.6481.424.243.57279.807.4961.344.436.0760.6632150.0371630.6260522.0030871.9293082.007645
324.294.967.2962.852.588.589154.168.5432.698.420.0460.6641700.0358950.6282752.0028801.9317552.007102
338.589.934.5925.712.855.823298.169.5075.414.686.3160.6650640.0347110.6303522.0026921.9340492.006613
3417.179.869.18411.440.204.286577.299.13610.862.905.1500.6659080.0336030.6323042.0025371.9361442.006193


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
243110210
383110210
4166231311
53210362422
66414493524
71282691649310
8256421427715317
95127730461327730
101.02413552822153853
112.0482528816335922798
124.0964471612856116756163
138.192830299530120306107297
1416.3841.519547971212570206531
1532.7682.7931.0021.7903731.052383985
1665.5365.2191.8803.3386771.9597081.875
17131.0729.7233.4226.3001.3243.5481.2713.580
18262.14418.1366.36211.7732.4366.7062.3826.612
19524.28834.00211.81822.1834.54412.5354.48612.437
201.048.57664.04022.24441.7958.49323.6228.43923.486
212.097.152121.07442.08578.98816.06944.69915.84044.466
224.194.304229.07979.531149.54730.24384.35929.99684.481
238.388.608435.069151.043284.02557.148160.41257.011160.498
2416.777.216829.164287.179541.984109.035305.842108.528305.759
2533.554.4321.584.825547.7371.037.087207.171585.128207.265585.261
2667.108.8643.031.0261.045.3311.985.694394.9371.121.302395.1421.119.645
27134.217.7285.810.7282.001.8493.808.878756.0922.149.442756.7822.148.412
28268.435.45611.161.9153.841.9777.319.9371.451.3714.129.8831.451.2394.129.422
29536.870.91221.469.4247.381.46314.087.9602.785.9437.948.5462.787.0977.947.838
301.073.741.82441.365.86914.202.66627.163.2025.361.87215.322.2625.361.61115.320.124
312.147.483.64879.807.49627.375.98652.431.50910.332.54229.576.13010.330.82029.568.004
324.294.967.296154.168.54352.831.354101.337.18819.935.81857.154.56819.932.93957.145.218
338.589.934.592298.169.507102.098.607196.070.89938.510.700110.571.36738.508.584110.578.856
3417.179.869.184577.299.136197.518.108379.781.02774.481.802214.164.86074.484.580214.167.894


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
242020110
384221120
4168441151
53212753162
66428131594114
7128462323159157
8256105584733183222
951222411111365456549
101.02446223522712810113895
112.048967471496271206279211
124.0962.0601.0151.045561459582458
138.1924.2402.0962.1441.1189741.187961
1416.3848.7274.3564.3712.3392.0292.3552.004
1532.76817.8388.9068.9324.7354.1564.8244.123
1665.53636.28518.11618.1699.6718.4859.7628.367
17131.07273.79636.64437.15219.40417.29619.83317.263
18262.144149.68174.59875.08339.34835.26939.89335.171
19524.288303.249151.063152.18679.59971.68380.30471.663
201.048.576613.373305.868307.505161.020145.542161.596145.215
212.097.1521.238.856618.464620.392324.177294.921324.699295.059
224.194.3042.499.4231.248.3121.251.111652.160596.934653.284597.045
238.388.6085.038.4142.516.9772.521.4371.312.5561.206.3081.314.1951.205.355
2416.777.21610.147.7725.070.9955.076.7772.637.2192.435.8712.639.8952.434.787
2533.554.43220.421.10210.207.03610.214.0665.298.0204.913.1135.301.0194.908.950
2667.108.86441.080.39620.533.33320.547.06310.641.9729.897.63910.644.6669.896.119
27134.217.72882.593.90341.280.75441.313.14921.364.40419.939.79121.362.81619.926.892
28268.435.456165.988.24282.955.07983.033.16342.868.43940.126.96642.875.24040.117.597
29536.870.912333.459.685166.658.023166.801.66286.019.07280.719.58486.011.58380.709.446
301.073.741.824669.658.348334.692.864334.965.484172.538.574162.309.319172.514.264162.296.191
312.147.483.6481.344.436.076671.964.952672.471.124345.995.448326.250.836345.957.072326.232.720
324.294.967.2962.698.420.0461.348.704.9611.349.715.085693.685.612655.542.717693.672.212655.519.505
338.589.934.5925.414.686.3162.706.380.6232.708.305.6931.390.532.8101.316.777.4651.390.592.8101.316.783.231
3417.179.869.18410.862.905.1505.429.573.6565.433.331.4942.787.138.7242.644.229.0072.787.245.8542.644.291.565


8. Check for existing Integer Sequences by OEIS

Found in Database : 83, 3, 101, 13, 59, 1, 1, 37, 337, 71, 421, 29, 1523, 23, 199, 241, 53, 1, 2351, 1,
Found in Database : 83, 3, 101, 13, 59, 37, 337, 71, 421, 29, 1523, 23, 199, 241, 53, 2351, 881, 349, 109, 43, 3251, 41, 233, 1297, 103, 61, 1637, 229, 227, 1877, 727,
Found in Database : 3, 13, 23, 29, 37, 41, 43, 53, 59, 61, 67, 71, 83, 101, 103, 109, 113, 137,