Inhaltsverzeichnis

Development of
Algorithmic Constructions

06:46:39
Deutsch
29.Mar 2024

Polynom = x^2+120x-137

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) = 137 = 137
f(1) = 1 = 1
f(2) = 107 = 107
f(3) = 29 = 29
f(4) = 359 = 359
f(5) = 61 = 61
f(6) = 619 = 619
f(7) = 47 = 47
f(8) = 887 = 887
f(9) = 1 = 1
f(10) = 1163 = 1163
f(11) = 163 = 163
f(12) = 1447 = 1447
f(13) = 199 = 199
f(14) = 1739 = 37*47
f(15) = 59 = 59
f(16) = 2039 = 2039
f(17) = 137 = 137
f(18) = 2347 = 2347
f(19) = 313 = 313
f(20) = 2663 = 2663
f(21) = 353 = 353
f(22) = 2987 = 29*103
f(23) = 197 = 197
f(24) = 3319 = 3319
f(25) = 109 = 109
f(26) = 3659 = 3659
f(27) = 479 = 479
f(28) = 4007 = 4007
f(29) = 523 = 523
f(30) = 4363 = 4363
f(31) = 71 = 71
f(32) = 4727 = 29*163
f(33) = 307 = 307
f(34) = 5099 = 5099
f(35) = 661 = 661
f(36) = 5479 = 5479
f(37) = 709 = 709
f(38) = 5867 = 5867
f(39) = 379 = 379
f(40) = 6263 = 6263
f(41) = 101 = 101
f(42) = 6667 = 59*113
f(43) = 859 = 859
f(44) = 7079 = 7079
f(45) = 911 = 911
f(46) = 7499 = 7499
f(47) = 241 = 241
f(48) = 7927 = 7927
f(49) = 509 = 509
f(50) = 8363 = 8363
f(51) = 1073 = 29*37
f(52) = 8807 = 8807
f(53) = 1129 = 1129
f(54) = 9259 = 47*197
f(55) = 593 = 593
f(56) = 9719 = 9719
f(57) = 311 = 311
f(58) = 10187 = 61*167
f(59) = 1303 = 1303
f(60) = 10663 = 10663
f(61) = 1363 = 29*47
f(62) = 11147 = 71*157
f(63) = 89 = 89
f(64) = 11639 = 103*113
f(65) = 743 = 743
f(66) = 12139 = 61*199
f(67) = 1549 = 1549
f(68) = 12647 = 12647
f(69) = 1613 = 1613
f(70) = 13163 = 13163
f(71) = 839 = 839
f(72) = 13687 = 13687
f(73) = 109 = 109
f(74) = 14219 = 59*241
f(75) = 1811 = 1811
f(76) = 14759 = 14759
f(77) = 1879 = 1879
f(78) = 15307 = 15307
f(79) = 487 = 487
f(80) = 15863 = 29*547
f(81) = 1009 = 1009
f(82) = 16427 = 16427
f(83) = 2089 = 2089
f(84) = 16999 = 89*191
f(85) = 2161 = 2161
f(86) = 17579 = 17579
f(87) = 1117 = 1117
f(88) = 18167 = 37*491
f(89) = 577 = 577
f(90) = 18763 = 29*647
f(91) = 2383 = 2383
f(92) = 19367 = 107*181
f(93) = 2459 = 2459
f(94) = 19979 = 19979
f(95) = 317 = 317
f(96) = 20599 = 20599
f(97) = 1307 = 1307
f(98) = 21227 = 21227
f(99) = 2693 = 2693
f(100) = 21863 = 21863

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+120x-137

f(0)=137
f(1)=1
f(2)=107
f(3)=29
f(4)=359
f(5)=61
f(6)=619
f(7)=47
f(8)=887
f(9)=1
f(10)=1163
f(11)=163
f(12)=1447
f(13)=199
f(14)=37
f(15)=59
f(16)=2039
f(17)=1
f(18)=2347
f(19)=313
f(20)=2663
f(21)=353
f(22)=103
f(23)=197
f(24)=3319
f(25)=109
f(26)=3659
f(27)=479
f(28)=4007
f(29)=523
f(30)=4363
f(31)=71
f(32)=1
f(33)=307
f(34)=5099
f(35)=661
f(36)=5479
f(37)=709
f(38)=5867
f(39)=379
f(40)=6263
f(41)=101
f(42)=113
f(43)=859
f(44)=7079
f(45)=911
f(46)=7499
f(47)=241
f(48)=7927
f(49)=509
f(50)=8363
f(51)=1
f(52)=8807
f(53)=1129
f(54)=1
f(55)=593
f(56)=9719
f(57)=311
f(58)=167
f(59)=1303
f(60)=10663
f(61)=1
f(62)=157
f(63)=89
f(64)=1
f(65)=743
f(66)=1
f(67)=1549
f(68)=12647
f(69)=1613
f(70)=13163
f(71)=839
f(72)=13687
f(73)=1
f(74)=1
f(75)=1811
f(76)=14759
f(77)=1879
f(78)=15307
f(79)=487
f(80)=547
f(81)=1009
f(82)=16427
f(83)=2089
f(84)=191
f(85)=2161
f(86)=17579
f(87)=1117
f(88)=491
f(89)=577
f(90)=647
f(91)=2383
f(92)=181
f(93)=2459
f(94)=19979
f(95)=317
f(96)=20599
f(97)=1307
f(98)=21227
f(99)=2693

b) Substitution of the polynom
The polynom f(x)=x^2+120x-137 could be written as f(y)= y^2-3737 with x=y-60

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+60
f'(x)>2x+119

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

137, 1, 107, 29, 359, 61, 619, 47, 887, 1, 1163, 163, 1447, 199, 37, 59, 2039, 1, 2347, 313, 2663, 353, 103, 197, 3319, 109, 3659, 479, 4007, 523, 4363, 71, 1, 307, 5099, 661, 5479, 709, 5867, 379, 6263, 101, 113, 859, 7079, 911, 7499, 241, 7927, 509, 8363, 1, 8807, 1129, 1, 593, 9719, 311, 167, 1303, 10663, 1, 157, 89, 1, 743, 1, 1549, 12647, 1613, 13163, 839, 13687, 1, 1, 1811, 14759, 1879, 15307, 487, 547, 1009, 16427, 2089, 191, 2161, 17579, 1117, 491, 577, 647, 2383, 181, 2459, 19979, 317, 20599, 1307, 21227, 2693, 21863, 1, 1, 1427, 23159, 367, 23819, 3019, 521, 1, 25163, 797, 25847, 1637, 26539, 3361, 27239, 3449, 27947, 1, 28663, 907, 29387, 3719, 30119, 1, 30859, 1, 31607, 1999, 32363, 4093, 211, 1, 1, 2143, 34679, 1, 1223, 4483, 36263, 4583, 1, 1171, 37879, 2393, 38699, 4889, 1, 4993, 223, 2549, 463, 1301, 1, 1, 257, 5419, 43787, 691, 757, 2819, 1231, 5749, 46439, 5861, 47339, 1, 48247, 761, 233, 6203, 50087, 1, 1, 1609, 1, 1, 277, 6673, 883, 6793, 503, 3457, 55799, 1759, 56779, 7159, 947, 7283, 58763, 1, 1013, 3767, 60779, 1, 2131, 7789, 62827, 1, 63863, 1, 1381, 8179, 929, 8311, 2311, 2111, 68087, 4289, 69163, 8713, 1, 8849, 71339, 4493, 677, 2281, 73547, 1, 1, 9403, 75787, 1193, 76919, 1, 78059, 9829, 769, 9973, 80363, 5059, 81527, 1283, 82699, 1, 2267, 10559, 331, 2677, 86263, 1, 1861, 1, 1249, 11161, 89899, 5657, 91127, 1, 92363, 1, 93607, 11779, 3271, 373, 347, 6047, 97387, 12253, 98663, 12413, 1123, 6287, 3491, 1, 102539, 12899, 919, 13063, 1021, 3307, 106487, 1, 787, 1, 109159, 13729, 1873, 6949, 111863, 3517, 113227, 1, 114599, 14411, 115979, 1823, 1, 1, 1051, 1, 120167, 1, 121579, 7643, 2617, 1933, 124427, 15643, 125863, 15823, 2087, 4001, 389, 8093, 1217, 16369, 131687, 16553, 1, 8369, 4643, 4231, 136139, 1, 137639, 17299, 401, 1093, 140663, 8839, 4903, 1, 143719, 18061, 145259, 9127, 146807, 1153, 1667, 1, 1, 18839, 151499, 4759, 3257, 1, 154667, 19433, 1, 1, 157867, 1, 1579, 5009, 2269, 1, 162727, 20443, 164363, 1, 1523, 10427, 431, 21061, 169319, 21269, 1049, 10739, 172663, 2711, 174347, 1, 439, 22111, 177739, 5581, 1, 1, 6247, 1, 182887, 1, 1, 11593, 1187, 5851, 188107, 23623, 6547, 1, 191627, 1, 5227, 12143, 2749, 24509, 457, 24733, 4229, 12479, 1201, 1, 461, 25411, 3461, 25639, 206027, 1, 1907, 13049, 209707, 1, 211559, 26561, 467, 13397, 1, 1, 217163, 1, 1, 1, 220939, 3467, 222839, 1, 1, 1, 226663, 1, 228587, 14347, 3779, 3617, 232459, 29179, 1, 29423, 1, 7417, 3907, 14957, 2333, 30161, 242279, 1, 8423, 15329, 2767, 7727, 248267, 31159, 250279, 1, 1607, 1979, 1291, 15959, 256363, 32173, 258407, 32429, 1439, 1, 262519, 1, 7151, 33203, 266663, 1, 268747, 8431, 1361, 16993, 1429, 1181, 275047, 34513, 277163, 17389, 1, 8761, 281419, 35311, 2753, 1, 285707, 4481, 287863, 18059, 290027, 36389, 6217, 601, 10151, 1, 4177, 4651, 1789, 37483, 300967, 1, 2683, 1, 10531, 19157, 307627, 38593, 1901, 38873, 312107, 19577, 314359, 9859, 1, 39719, 3583, 1, 321163, 1259, 1787, 20287, 5521, 1409, 2903, 41149, 330347, 20719, 3109, 1, 334987, 42019, 7177, 1459, 339659, 10651, 4817, 1, 344363, 1, 9371, 1, 349099, 21893, 351479, 1, 353867, 44383, 356263, 44683, 358667, 5623, 12451, 22643, 1, 45589, 2671, 45893, 368363, 23099, 1, 5813, 1, 46811, 1559, 47119, 1, 1, 1993, 23869, 383147, 48049, 385639, 1, 1, 24337, 390647, 1, 3607, 1, 395687, 1, 398219, 3121, 400759, 25127, 8581, 1, 405863, 50893, 408427, 1, 410999, 3221, 413579, 51859, 416167, 52183, 418763, 13127, 1997, 26417, 1, 53161, 1913, 1, 429227, 1, 431863, 13537, 14983, 1, 437159, 1, 11887, 1, 442487, 27739, 2237, 55813, 15443, 56149, 2287, 1, 453239, 7103, 1, 57163, 458663, 57503, 2069, 14461, 464119, 1, 466859, 1, 1, 1, 6653, 1, 8053, 14891, 477899, 1, 2063, 60259, 483467, 1, 4721, 1, 2099, 1657, 4597, 1, 494699, 1, 1, 1949, 500363, 62723, 503207, 63079, 1, 15859, 508919, 1, 511787, 64153, 17747, 64513, 517547, 1, 8821, 1, 1, 65599, 8627, 65963, 1, 8291, 11321, 33347, 535019, 67061, 8819, 67429, 540907, 1, 14699, 8521, 5309, 68539, 3373, 1, 6211, 17321, 555767, 1, 3559, 1, 561767, 1, 564779, 35393, 1, 17791, 570827, 2467, 573863, 1, 4211, 4519, 1, 36343, 583019, 821, 586087, 73453, 589163, 36919, 12601, 4639, 595339, 74611, 598439, 2027, 20743, 1, 5351, 37889, 1693, 1, 610919, 76561, 3677, 1, 21283, 1, 620363, 77743, 2251, 78139, 5857, 9817, 1987, 1, 633067, 79333, 636263, 1, 5659, 1, 1723, 10067, 1, 2791, 6427, 81343, 11057, 1, 2551, 41077, 17807, 82561, 2129, 2861, 665387, 1, 4259, 20947, 671947, 84199, 675239, 1, 14437, 2657, 2221, 42719, 2843, 85853, 9697, 86269, 1, 1, 23971, 1, 698507, 87523, 701863, 87943, 11953, 22091, 708599, 1, 24551, 89209, 6563, 89633, 11783, 1, 722167, 22621, 725579, 90911, 8191, 1, 12007, 11471, 2351, 46099, 739307, 1, 742759, 3209, 15877, 46747, 1, 1, 2719, 1, 10657, 2017, 1, 1, 4219, 47837, 767147, 1, 770663, 96553, 5651, 48497, 4657, 24359, 4793, 97879, 21211, 98323, 788363, 6173, 791927, 1, 27431, 99661, 7331, 100109, 802667, 1, 806263, 1, 4111, 1, 28051, 1, 7933, 1, 4297, 1, 1, 1, 828007, 2207, 831659, 1, 835319, 26161, 7841, 1, 17929, 105563, 4253, 1, 9551, 1, 853739, 106949, 14533, 1, 861163, 53939, 864887, 1, 4799, 1, 2377, 109279, 876107, 27437, 879863, 55109, 883627, 1, 887399, 1, 891179, 55817, 8689, 28027, 898763, 112583, 31123, 1, 1, 1, 910199, 1, 914027, 114493, 1, 114973, 1, 57727, 925559, 3623, 929419, 2477, 1, 116903, 1, 29347, 15427, 58937, 13309, 118361, 948839, 1, 2699, 59669, 15683, 1033, 960587, 120319, 964519, 120811, 968459, 1, 972407, 60899, 1, 4217, 980327, 122789, 984299, 61643, 988279, 15473, 992267, 3359, 996263, 124783, 5237, 31321, 1004279, 1, 8923, 2687, 9461, 1, 1, 63649, 1020407, 1, 1, 128311, 1, 128819, 3299, 1, 35747, 64919, 1040747, 130349, 1044839, 1, 3169, 65687, 9319, 8243, 1057163, 132403, 1061287, 2179, 11971, 33359, 1, 66977, 2833, 4637, 1077863, 2213, 1082027, 67757, 1086199, 1, 18481, 136559, 1094567, 1, 3533, 1, 1102967, 69067, 23557, 138661, 1111399, 1, 1115627, 69859,

6. Sequence of the polynom (only primes)

137, 107, 29, 359, 61, 619, 47, 887, 1163, 163, 1447, 199, 37, 59, 2039, 2347, 313, 2663, 353, 103, 197, 3319, 109, 3659, 479, 4007, 523, 4363, 71, 307, 5099, 661, 5479, 709, 5867, 379, 6263, 101, 113, 859, 7079, 911, 7499, 241, 7927, 509, 8363, 8807, 1129, 593, 9719, 311, 167, 1303, 10663, 157, 89, 743, 1549, 12647, 1613, 13163, 839, 13687, 1811, 14759, 1879, 15307, 487, 547, 1009, 16427, 2089, 191, 2161, 17579, 1117, 491, 577, 647, 2383, 181, 2459, 19979, 317, 20599, 1307, 21227, 2693, 21863, 1427, 23159, 367, 23819, 3019, 521, 25163, 797, 25847, 1637, 26539, 3361, 27239, 3449, 27947, 28663, 907, 29387, 3719, 30119, 30859, 31607, 1999, 32363, 4093, 211, 2143, 34679, 1223, 4483, 36263, 4583, 1171, 37879, 2393, 38699, 4889, 4993, 223, 2549, 463, 1301, 257, 5419, 43787, 691, 757, 2819, 1231, 5749, 46439, 5861, 47339, 48247, 761, 233, 6203, 50087, 1609, 277, 6673, 883, 6793, 503, 3457, 55799, 1759, 56779, 7159, 947, 7283, 58763, 1013, 3767, 60779, 2131, 7789, 62827, 63863, 1381, 8179, 929, 8311, 2311, 2111, 68087, 4289, 69163, 8713, 8849, 71339, 4493, 677, 2281, 73547, 9403, 75787, 1193, 76919, 78059, 9829, 769, 9973, 80363, 5059, 81527, 1283, 82699, 2267, 10559, 331, 2677, 86263, 1861, 1249, 11161, 89899, 5657, 91127, 92363, 93607, 11779, 3271, 373, 347, 6047, 97387, 12253, 98663, 12413, 1123, 6287, 3491, 102539, 12899, 919, 13063, 1021, 3307, 106487, 787, 109159, 13729, 1873, 6949, 111863, 3517, 113227, 114599, 14411, 115979, 1823, 1051, 120167, 121579, 7643, 2617, 1933, 124427, 15643, 125863, 15823, 2087, 4001, 389, 8093, 1217, 16369, 131687, 16553, 8369, 4643, 4231, 136139, 137639, 17299, 401, 1093, 140663, 8839, 4903, 143719, 18061, 145259, 9127, 146807, 1153, 1667, 18839, 151499, 4759, 3257, 154667, 19433, 157867, 1579, 5009, 2269, 162727, 20443, 164363, 1523, 10427, 431, 21061, 169319, 21269, 1049, 10739, 172663, 2711, 174347, 439, 22111, 177739, 5581, 6247, 182887, 11593, 1187, 5851, 188107, 23623, 6547, 191627, 5227, 12143, 2749, 24509, 457, 24733, 4229, 12479, 1201, 461, 25411, 3461, 25639, 206027, 1907, 13049, 209707, 211559, 26561, 467, 13397, 217163, 220939, 3467, 222839, 226663, 228587, 14347, 3779, 3617, 232459, 29179, 29423, 7417, 3907, 14957, 2333, 30161, 242279, 8423, 15329, 2767, 7727, 248267, 31159, 250279, 1607, 1979, 1291, 15959, 256363, 32173, 258407, 32429, 1439, 262519, 7151, 33203, 266663, 268747, 8431, 1361, 16993, 1429, 1181, 275047, 34513, 277163, 17389, 8761, 281419, 35311, 2753, 285707, 4481, 287863, 18059, 290027, 36389, 6217, 601, 10151, 4177, 4651, 1789, 37483, 300967, 2683, 10531, 19157, 307627, 38593, 1901, 38873, 312107, 19577, 314359, 9859, 39719, 3583, 321163, 1259, 1787, 20287, 5521, 1409, 2903, 41149, 330347, 20719, 3109, 334987, 42019, 7177, 1459, 339659, 10651, 4817, 344363, 9371, 349099, 21893, 351479, 353867, 44383, 356263, 44683, 358667, 5623, 12451, 22643, 45589, 2671, 45893, 368363, 23099, 5813, 46811, 1559, 47119, 1993, 23869, 383147, 48049, 385639, 24337, 390647, 3607, 395687, 398219, 3121, 400759, 25127, 8581, 405863, 50893, 408427, 410999, 3221, 413579, 51859, 416167, 52183, 418763, 13127, 1997, 26417, 53161, 1913, 429227, 431863, 13537, 14983, 437159, 11887, 442487, 27739, 2237, 55813, 15443, 56149, 2287, 453239, 7103, 57163, 458663, 57503, 2069, 14461, 464119, 466859, 6653, 8053, 14891, 477899, 2063, 60259, 483467, 4721, 2099, 1657, 4597, 494699, 1949, 500363, 62723, 503207, 63079, 15859, 508919, 511787, 64153, 17747, 64513, 517547, 8821, 65599, 8627, 65963, 8291, 11321, 33347, 535019, 67061, 8819, 67429, 540907, 14699, 8521, 5309, 68539, 3373, 6211, 17321, 555767, 3559, 561767, 564779, 35393, 17791, 570827, 2467, 573863, 4211, 4519, 36343, 583019, 821, 586087, 73453, 589163, 36919, 12601, 4639, 595339, 74611, 598439, 2027, 20743, 5351, 37889, 1693, 610919, 76561, 3677, 21283, 620363, 77743, 2251, 78139, 5857, 9817, 1987, 633067, 79333, 636263, 5659, 1723, 10067, 2791, 6427, 81343, 11057, 2551, 41077, 17807, 82561, 2129, 2861, 665387, 4259, 20947, 671947, 84199, 675239, 14437, 2657, 2221, 42719, 2843, 85853, 9697, 86269, 23971, 698507, 87523, 701863, 87943, 11953, 22091, 708599, 24551, 89209, 6563, 89633, 11783, 722167, 22621, 725579, 90911, 8191, 12007, 11471, 2351, 46099, 739307, 742759, 3209, 15877, 46747, 2719, 10657, 2017, 4219, 47837, 767147, 770663, 96553, 5651, 48497, 4657, 24359, 4793, 97879, 21211, 98323, 788363, 6173, 791927, 27431, 99661, 7331, 100109, 802667, 806263, 4111, 28051, 7933, 4297, 828007, 2207, 831659, 835319, 26161, 7841, 17929, 105563, 4253, 9551, 853739, 106949, 14533, 861163, 53939, 864887, 4799, 2377, 109279, 876107, 27437, 879863, 55109, 883627, 887399, 891179, 55817, 8689, 28027, 898763, 112583, 31123, 910199, 914027, 114493, 114973, 57727, 925559, 3623, 929419, 2477, 116903, 29347, 15427, 58937, 13309, 118361, 948839, 2699, 59669, 15683, 1033, 960587, 120319, 964519, 120811, 968459, 972407, 60899, 4217, 980327, 122789, 984299, 61643, 988279, 15473, 992267, 3359, 996263, 124783, 5237, 31321, 1004279, 8923, 2687, 9461, 63649, 1020407, 128311, 128819, 3299, 35747, 64919, 1040747, 130349, 1044839, 3169, 65687, 9319, 8243, 1057163, 132403, 1061287, 2179, 11971, 33359, 66977, 2833, 4637, 1077863, 2213, 1082027, 67757, 1086199, 18481, 136559, 1094567, 3533, 1102967, 69067, 23557, 138661, 1111399, 1115627, 69859,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+120x-137 and
the reducible primes which appear as divisor for the first time
p | x^2+120x-137 and p < x^2+120x-137

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)
1109630.9000000.6000000.9000000.0000000.0000000.000000
21008837510.8800000.3700000.8800009.7777786.16666717.000000
31.0007702505200.7700000.2500000.7700008.7500006.75675710.196078
410.0007.3581.8735.4850.7358000.1873000.7358009.5558447.49200010.548077
5100.00072.41314.80457.6090.7241300.1480400.7241309.8413977.90389710.503008
61.000.000719.007119.963599.0440.7190070.1199630.7190079.9292538.10341810.398445
710.000.0007.152.1621.013.4316.138.7310.7152160.1013430.7152169.9472778.44786310.247546
8100.000.00071.225.0458.782.36562.442.6800.7122500.0878240.7122509.9585348.66597310.171920
91.000.000.000709.986.36977.486.879632.499.4900.7099860.0774870.7099869.9682128.82300910.129282
1010.000.000.0007.081.884.477693.496.5076.388.387.9700.7081880.0693500.7081889.9746778.94985710.100225


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)
122201.0000001.0000000.0000000.0000000.0000000.000000
244311.0000000.7500000.2500002.0000001.500000inf
388531.0000000.6250000.3750002.0000001.6666673.000000
41615960.9375000.5625000.3750001.8750001.8000002.000000
5322915140.9062500.4687500.4375001.9333331.6666672.333333
6645626300.8750000.4062500.4687501.9310341.7333332.142857
712811049610.8593750.3828120.4765621.9642861.8846152.033333
8256205801250.8007810.3125000.4882811.8636361.6326532.049180
95124001402600.7812500.2734380.5078121.9512201.7500002.080000
101.0247892555340.7705080.2490230.5214841.9725001.8214292.053846
112.0481.5404751.0650.7519530.2319340.5200201.9518381.8627451.994382
124.0963.0358652.1700.7409670.2111820.5297851.9707791.8210532.037559
138.1926.0241.5664.4580.7353520.1911620.5441891.9848431.8104052.054378
1416.38412.0222.8829.1400.7337650.1759030.5578611.9956841.8403582.050247
1532.76823.9165.41218.5040.7298580.1651610.5646971.9893531.8778632.024508
1665.53647.57310.15037.4230.7259060.1548770.5710301.9891701.8754622.022428
17131.07294.82218.85475.9680.7234340.1438450.5795901.9931891.8575372.029982
18262.144189.22835.321153.9070.7218480.1347390.5871091.9956131.8733962.025945
19524.288377.60666.379311.2270.7202260.1266080.5936181.9955081.8793072.022176
201.048.576753.834125.282628.5520.7189120.1194780.5994341.9963511.8873742.019593
212.097.1521.505.209237.1201.268.0890.7177400.1130680.6046721.9967381.8926902.017477
224.194.3043.005.629451.0162.554.6130.7165980.1075310.6090671.9968181.9020582.014538
238.388.6086.001.955859.8295.142.1260.7154890.1025000.6129891.9969051.9064272.012879
2416.777.21611.986.6621.643.65710.343.0050.7144610.0979700.6164911.9971261.9116092.011426
2533.554.43223.942.5573.145.56020.796.9970.7135440.0937450.6197991.9974331.9137572.010731
2667.108.86447.828.7636.032.08141.796.6820.7127040.0898850.6228191.9976471.9176492.009746
27134.217.72895.553.43111.588.34183.965.0900.7119290.0863400.6255891.9978241.9211182.008894
28268.435.456190.912.42022.299.531168.612.8890.7112040.0830720.6281321.9979651.9243082.008131
29536.870.912381.471.42142.965.744338.505.6770.7105460.0800300.6305161.9981491.9267552.007591
301.073.741.824762.277.51982.901.022679.376.4970.7099260.0772080.6327191.9982561.9294682.006987
312.147.483.6481.523.299.996160.167.2871.363.132.7090.7093420.0745840.6347581.9983541.9320302.006447
324.294.967.2963.044.275.215309.797.3362.734.477.8790.7088010.0721300.6366701.9984741.9342112.006025
338.589.934.5926.084.208.987599.871.8275.484.337.1600.7082950.0698340.6384611.9985741.9363362.005625
3417.179.869.18412.160.256.3581.162.719.45910.997.536.8990.7078200.0676790.6401411.9986581.9382802.005263


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
122021100
243031101
385141202
4169271404
532155101707
66426818111014
7128491435123025
8256802456140039
95121405189172067
101.0242558017511290125
112.04847515432112400234
124.09686528757814320432
138.1921.5665191.04717930772
1416.3842.8829651.91711.45901.422
1532.7685.4121.8113.60112.72302.688
1665.53610.1503.3736.77715.07405.075
17131.07218.8546.27312.58119.41409.439
18262.14435.32111.79423.527117.698017.622
19524.28866.37922.09144.288133.261033.117
201.048.576125.28241.78183.501162.746062.535
212.097.152237.12078.962158.1581118.7460118.373
224.194.304451.016150.437300.5791225.5430225.472
238.388.608859.829286.884572.9451430.0880429.740
2416.777.2161.643.657547.5181.096.1391821.5950822.061
2533.554.4323.145.5601.047.9372.097.62311.572.99301.572.566
2667.108.8646.032.0812.009.6764.022.40513.016.31703.015.763
27134.217.72811.588.3413.861.5707.726.77115.794.63505.793.705
28268.435.45622.299.5317.431.31814.868.213111.149.240011.150.290
29536.870.91242.965.74414.319.66828.646.076121.485.614021.480.129
301.073.741.82482.901.02227.637.28855.263.734141.454.733041.446.288
312.147.483.648160.167.28753.396.980106.770.307180.086.140080.081.146
324.294.967.296309.797.336103.271.767206.525.5691154.906.3660154.890.969
338.589.934.592599.871.827199.962.227399.909.6001299.926.6060299.945.220
3417.179.869.1841.162.719.459387.579.271775.140.1881581.349.0690581.370.389


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
241010010
383120021
4166330222
53214772345
6643016146699
712861303113141618
8256125685731303133
951226013612463696167
101.024534287247123148126137
112.0481.065568497255289254267
124.0962.1701.1631.007516565533556
138.1924.4582.3732.0851.0691.1621.0921.135
1416.3849.1404.8294.3112.2082.3272.2802.325
1532.76818.5049.7188.7864.5204.6804.5674.737
1665.53637.42319.42617.9979.1919.4939.2559.484
17131.07275.96839.34536.62318.69719.31618.80419.151
18262.144153.90779.35874.54938.11238.94538.10338.747
19524.288311.227160.215151.01276.96978.56577.26678.427
201.048.576628.552323.804304.748155.376158.388155.822158.966
212.097.1521.268.089652.163615.926313.859319.570314.264320.396
224.194.3042.554.6131.310.2211.244.392631.938644.118633.468645.089
238.388.6085.142.1262.633.5302.508.5961.272.9841.296.3711.275.4751.297.296
2416.777.21610.343.0055.290.9825.052.0232.562.3872.607.3162.564.5532.608.749
2533.554.43220.796.99710.626.53310.170.4645.156.2215.240.5395.157.4875.242.750
2667.108.86441.796.68221.335.92120.460.76110.368.63310.529.49810.369.37410.529.177
27134.217.72883.965.09042.817.25141.147.83920.839.04021.142.80220.839.52821.143.720
28268.435.456168.612.88985.913.32882.699.56141.860.78442.447.01441.863.62542.441.466
29536.870.912338.505.677172.356.770166.148.90784.067.75985.191.08184.070.65385.176.184
301.073.741.824679.376.497345.683.714333.692.783168.774.607170.915.623168.778.247170.908.020
312.147.483.6481.363.132.709693.147.226669.985.483338.724.166342.842.022338.722.228342.844.293
324.294.967.2962.734.477.8791.389.596.0151.344.881.864679.648.598687.597.429679.649.817687.582.035
338.589.934.5925.484.337.1602.785.393.7462.698.943.4141.363.402.4861.378.800.5531.363.376.7341.378.757.387
3417.179.869.18410.997.536.8995.582.416.1625.415.120.7372.734.447.1022.764.335.6762.734.454.9942.764.299.127


8. Check for existing Integer Sequences by OEIS

Found in Database : 137, 1, 107, 29, 359, 61, 619, 47, 887, 1, 1163, 163, 1447, 199, 37, 59, 2039, 1, 2347, 313,
Found in Database : 137, 107, 29, 359, 61, 619, 47, 887, 1163, 163, 1447, 199, 37, 59, 2039, 2347, 313, 2663, 353, 103, 197, 3319, 109, 3659, 479, 4007, 523, 4363, 71, 307, 5099, 661, 5479, 709, 5867, 379,
Found in Database : 29, 37, 47, 59, 61, 71, 89, 101, 103, 107, 109, 113, 137,