Inhaltsverzeichnis

Development of
Algorithmic Constructions

13:37:20
Deutsch
29.Mar 2024

Polynom = x^2+49x-3

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) = 3 = 3
f(1) = 47 = 47
f(2) = 99 = 3*3*11
f(3) = 153 = 3*3*17
f(4) = 209 = 11*19
f(5) = 267 = 3*89
f(6) = 327 = 3*109
f(7) = 389 = 389
f(8) = 453 = 3*151
f(9) = 519 = 3*173
f(10) = 587 = 587
f(11) = 657 = 3*3*73
f(12) = 729 = 3*3*3*3*3*3
f(13) = 803 = 11*73
f(14) = 879 = 3*293
f(15) = 957 = 3*11*29
f(16) = 1037 = 17*61
f(17) = 1119 = 3*373
f(18) = 1203 = 3*401
f(19) = 1289 = 1289
f(20) = 1377 = 3*3*3*3*17
f(21) = 1467 = 3*3*163
f(22) = 1559 = 1559
f(23) = 1653 = 3*19*29
f(24) = 1749 = 3*11*53
f(25) = 1847 = 1847
f(26) = 1947 = 3*11*59
f(27) = 2049 = 3*683
f(28) = 2153 = 2153
f(29) = 2259 = 3*3*251
f(30) = 2367 = 3*3*263
f(31) = 2477 = 2477
f(32) = 2589 = 3*863
f(33) = 2703 = 3*17*53
f(34) = 2819 = 2819
f(35) = 2937 = 3*11*89
f(36) = 3057 = 3*1019
f(37) = 3179 = 11*17*17
f(38) = 3303 = 3*3*367
f(39) = 3429 = 3*3*3*127
f(40) = 3557 = 3557
f(41) = 3687 = 3*1229
f(42) = 3819 = 3*19*67
f(43) = 3953 = 59*67
f(44) = 4089 = 3*29*47
f(45) = 4227 = 3*1409
f(46) = 4367 = 11*397
f(47) = 4509 = 3*3*3*167
f(48) = 4653 = 3*3*11*47
f(49) = 4799 = 4799
f(50) = 4947 = 3*17*97
f(51) = 5097 = 3*1699
f(52) = 5249 = 29*181
f(53) = 5403 = 3*1801
f(54) = 5559 = 3*17*109
f(55) = 5717 = 5717
f(56) = 5877 = 3*3*653
f(57) = 6039 = 3*3*11*61
f(58) = 6203 = 6203
f(59) = 6369 = 3*11*193
f(60) = 6537 = 3*2179
f(61) = 6707 = 19*353
f(62) = 6879 = 3*2293
f(63) = 7053 = 3*2351
f(64) = 7229 = 7229
f(65) = 7407 = 3*3*823
f(66) = 7587 = 3*3*3*281
f(67) = 7769 = 17*457
f(68) = 7953 = 3*11*241
f(69) = 8139 = 3*2713
f(70) = 8327 = 11*757
f(71) = 8517 = 3*17*167
f(72) = 8709 = 3*2903
f(73) = 8903 = 29*307
f(74) = 9099 = 3*3*3*337
f(75) = 9297 = 3*3*1033
f(76) = 9497 = 9497
f(77) = 9699 = 3*53*61
f(78) = 9903 = 3*3301
f(79) = 10109 = 11*919
f(80) = 10317 = 3*19*181
f(81) = 10527 = 3*11*11*29
f(82) = 10739 = 10739
f(83) = 10953 = 3*3*1217
f(84) = 11169 = 3*3*17*73
f(85) = 11387 = 59*193
f(86) = 11607 = 3*53*73
f(87) = 11829 = 3*3943
f(88) = 12053 = 17*709
f(89) = 12279 = 3*4093
f(90) = 12507 = 3*11*379
f(91) = 12737 = 47*271
f(92) = 12969 = 3*3*11*131
f(93) = 13203 = 3*3*3*3*163
f(94) = 13439 = 89*151
f(95) = 13677 = 3*47*97
f(96) = 13917 = 3*4639
f(97) = 14159 = 14159
f(98) = 14403 = 3*4801
f(99) = 14649 = 3*19*257
f(100) = 14897 = 14897

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+49x-3

f(0)=3
f(1)=47
f(2)=11
f(3)=17
f(4)=19
f(5)=89
f(6)=109
f(7)=389
f(8)=151
f(9)=173
f(10)=587
f(11)=73
f(12)=1
f(13)=1
f(14)=293
f(15)=29
f(16)=61
f(17)=373
f(18)=401
f(19)=1289
f(20)=1
f(21)=163
f(22)=1559
f(23)=1
f(24)=53
f(25)=1847
f(26)=59
f(27)=683
f(28)=2153
f(29)=251
f(30)=263
f(31)=2477
f(32)=863
f(33)=1
f(34)=2819
f(35)=1
f(36)=1019
f(37)=1
f(38)=367
f(39)=127
f(40)=3557
f(41)=1229
f(42)=67
f(43)=1
f(44)=1
f(45)=1409
f(46)=397
f(47)=167
f(48)=1
f(49)=4799
f(50)=97
f(51)=1699
f(52)=181
f(53)=1801
f(54)=1
f(55)=5717
f(56)=653
f(57)=1
f(58)=6203
f(59)=193
f(60)=2179
f(61)=353
f(62)=2293
f(63)=2351
f(64)=7229
f(65)=823
f(66)=281
f(67)=457
f(68)=241
f(69)=2713
f(70)=757
f(71)=1
f(72)=2903
f(73)=307
f(74)=337
f(75)=1033
f(76)=9497
f(77)=1
f(78)=3301
f(79)=919
f(80)=1
f(81)=1
f(82)=10739
f(83)=1217
f(84)=1
f(85)=1
f(86)=1
f(87)=3943
f(88)=709
f(89)=4093
f(90)=379
f(91)=271
f(92)=131
f(93)=1
f(94)=1
f(95)=1
f(96)=4639
f(97)=14159
f(98)=4801
f(99)=257

b) Substitution of the polynom
The polynom f(x)=x^2+49x-3 could be written as f(y)= y^2-603.25 with x=y-24.5

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+24.5
f'(x)>2x+48

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

3, 47, 11, 17, 19, 89, 109, 389, 151, 173, 587, 73, 1, 1, 293, 29, 61, 373, 401, 1289, 1, 163, 1559, 1, 53, 1847, 59, 683, 2153, 251, 263, 2477, 863, 1, 2819, 1, 1019, 1, 367, 127, 3557, 1229, 67, 1, 1, 1409, 397, 167, 1, 4799, 97, 1699, 181, 1801, 1, 5717, 653, 1, 6203, 193, 2179, 353, 2293, 2351, 7229, 823, 281, 457, 241, 2713, 757, 1, 2903, 307, 337, 1033, 9497, 1, 3301, 919, 1, 1, 10739, 1217, 1, 1, 1, 3943, 709, 4093, 379, 271, 131, 1, 1, 1, 4639, 14159, 4801, 257, 14897, 1, 1, 1423, 5303, 317, 16427, 5563, 5651, 1, 1, 1973, 149, 6101, 563, 1, 6379, 6473, 1, 2221, 751, 157, 409, 641, 1, 659, 7349, 22349, 839, 2551, 439, 1, 7963, 24203, 743, 487, 2287, 1, 1, 1, 8819, 8929, 577, 9151, 1, 2557, 3163, 1, 29153, 9833, 9949, 30197, 599, 10301, 31259, 1171, 1, 443, 991, 1, 1153, 191, 11393, 1, 3881, 3923, 1, 1093, 419, 197, 12409, 12539, 199, 1, 479, 2063, 1, 1, 3673, 223, 1249, 41627, 1, 1, 1, 1, 857, 44129, 14851, 1, 2671, 463, 1, 46703, 827, 15859, 48017, 557, 16301, 809, 503, 1861, 1, 1, 17203, 52067, 17509, 1039, 53453, 1997, 6043, 4987, 18443, 1, 1, 18919, 19079, 57719, 1, 6521, 1, 19889, 1823, 1, 1, 20549, 2143, 6961, 2339, 63659, 1, 21559, 1, 1, 22073, 6067, 1, 1, 1, 22943, 1, 4111, 23473, 1, 1, 1, 733, 2521, 24551, 24733, 74747, 1321, 1, 76403, 1, 1, 1, 2383, 1553, 79769, 1, 1, 1381, 1013, 9181, 83207, 2539, 28123, 7723, 28513, 1511, 5101, 1, 9767, 88499, 1747, 1031, 8209, 1, 1, 1, 10303, 3457, 1999, 1087, 31729, 643, 32143, 1, 1, 331, 10993, 5857, 1, 33613, 101477, 34039, 34253, 1951, 1051, 11633, 1, 1, 1, 107309, 1, 36209, 1, 1, 1, 1, 1, 3413, 3907, 37993, 38219, 115337, 4297, 12967, 117389, 39359, 1, 7027, 1, 40283, 121547, 1, 719, 123653, 41453, 887, 125777, 3833, 1, 1, 14293, 1597, 130079, 739, 2579, 132257, 1, 1, 1, 1669, 1373, 7193, 1, 46051, 138899, 46549, 883, 1, 15767, 1, 829, 1, 48313, 145703, 48823, 2887, 1663, 1, 1, 150329, 1, 1, 13879, 51151, 51413, 155027, 1, 17401, 1, 52733, 52999, 1, 1, 1, 162209, 1, 1, 2699, 421, 2917, 167087, 1, 5113, 169553, 1721, 1, 1933, 1987, 57901, 10267, 1103, 1, 3001, 1, 19861, 1, 60149, 1, 9587, 1151, 1, 184727, 1213, 20717, 17029, 1, 1, 189929, 63601, 1, 192557, 21493, 2399, 195203, 1, 1, 6823, 1, 1091, 1, 1, 22483, 203249, 4003, 2357, 205967, 6269, 69263, 18973, 23291, 1, 211457, 70793, 1, 3631, 71719, 1, 1, 1, 1, 1, 73589, 1, 4201, 1021, 1, 3089, 8387, 2297, 228359, 6949, 1301, 4363, 1, 77723, 12323, 1, 1543, 237053, 7213, 79669, 21817, 1709, 80651, 1, 27103, 1, 245909, 82301, 82633, 1, 83299, 7603, 251903, 1, 1, 254927, 85313, 1, 4229, 1, 7879, 9001, 2647, 29231, 2423, 88379, 1, 267203, 89413, 89759, 15901, 2741, 1, 24859, 91499, 1733, 276599, 92551, 1523, 279767, 3467, 31321, 1, 5569, 1, 4271, 1429, 5653, 1, 547, 1117, 15401, 2083, 8933, 295877, 8999, 1361, 1, 1, 11161, 17791, 101183, 1, 305759, 1, 1, 28099, 11489, 34591, 3221, 5501, 3617, 315803, 1187, 106019, 29017, 35591, 1, 322589, 1, 108289, 1, 1, 109433, 1367, 1, 1, 332903, 1, 5881, 1279, 1907, 112901, 2281, 1, 1, 343379, 1, 6779, 1, 4001, 116423, 350453, 1, 1, 354017, 1, 118801, 1, 601, 10909, 21247, 1, 1, 3761, 7177, 1, 368453, 1, 11239, 372107, 1, 41617, 375779, 125669, 126079, 2417, 6679, 7489, 1, 1, 1, 2069, 1, 1, 390647, 130633, 4519, 394409, 43963, 1, 1, 2833, 12143, 401987, 7907, 1, 1579, 15077, 1, 1, 136973, 12491, 1, 12569, 1, 1, 1, 1609, 1889, 2309, 141283, 1, 12923, 142589, 2053, 1, 1, 2503, 144779, 145219, 436979, 1, 146543, 40087, 16381, 4481, 444953, 148763, 1, 6701, 1, 150551, 26647, 2657, 4603, 457049, 1, 1, 9811, 154159, 154613, 5227, 1, 1, 16183, 1, 157351, 1, 158269, 9337, 7829, 1, 1, 6599, 161039, 161503, 1, 1, 1, 490097, 1, 54767, 29077, 1, 165709, 17191, 9803, 15193, 502787, 1, 1, 26687, 169493, 5861, 511337, 2897, 171401, 515639, 1741, 3389, 47269, 1, 1, 30841, 1, 3739, 1, 58901, 59063, 1, 9377, 1, 1, 1, 6211, 8087, 1, 20177, 3271, 182579, 1, 550703, 1, 184559, 555167, 1, 62017, 559649, 187049, 9871, 564149, 1, 1949, 3041, 63353, 63521, 3847, 1, 1, 577757, 193093, 193601, 1, 2237, 1, 586919, 2687, 196663, 1, 197689, 1, 1, 1, 6053, 1, 18253, 4283, 1, 202339, 202859, 1, 67967, 68141, 3121, 18679, 205993, 1, 1, 207569, 11779, 1, 1, 1, 210209, 210739, 1, 211801, 1, 12049, 1, 2459, 643403, 3209, 3217, 2243, 1, 1, 1, 1, 72923, 657929, 219851, 3613, 1, 1, 222023, 11317, 6761, 24851, 1, 1, 1, 2207, 4817, 1787, 2113, 25339, 1, 62497, 1, 1, 692453, 3793, 3931, 697457, 77681, 77867, 9623, 13807, 1, 6491, 21491, 1, 1777, 4177, 8839, 717653, 239783, 240349, 7451, 1, 4567, 66169, 9007, 1, 43117, 244901, 245473, 25453, 1, 247193, 6143, 82781, 1, 748487, 250073, 5333, 753689, 251809, 3767, 1, 1, 1, 12527, 23209, 4337, 45259, 257053, 257639, 26713, 1, 86467, 41051, 23689, 261169, 71389, 262351, 9067, 790607, 5179, 88241, 1, 4507, 1, 1, 5051, 24391, 4663, 89833, 30011, 1, 1, 271903, 3181, 5153, 1, 48409, 2777, 3167, 828407, 1, 1, 11423, 5927, 279179, 839369, 1, 1993, 1, 282239, 14887, 850403, 284083, 16747, 855947, 95311, 10613, 1, 287789, 1, 6619, 1, 1, 872687, 1, 97381, 13109, 1, 26729, 46523, 1, 295901, 889589, 99053, 5839, 30871, 3083, 299683, 1, 1, 301583, 1, 1, 33721, 14957, 304763, 305401, 1, 16141, 1, 1, 2017, 9371, 2273, 6607, 311173, 32257, 1, 313109, 941267, 104801, 9547, 2683, 28759, 1, 2879, 318301, 16787, 56401, 106753, 1, 964703, 2663, 4423, 88237, 4441, 1, 4957, 36241, 108943, 1, 1, 328813, 1, 330139, 1, 1, 110711, 110933, 1000397, 334133, 6317, 1, 1, 1, 1012439, 10247, 1, 1, 340169, 1, 21799, 20129, 342869, 1030637, 1, 114967, 1, 346259, 346939, 1, 1, 2141,

6. Sequence of the polynom (only primes)

3, 47, 11, 17, 19, 89, 109, 389, 151, 173, 587, 73, 293, 29, 61, 373, 401, 1289, 163, 1559, 53, 1847, 59, 683, 2153, 251, 263, 2477, 863, 2819, 1019, 367, 127, 3557, 1229, 67, 1409, 397, 167, 4799, 97, 1699, 181, 1801, 5717, 653, 6203, 193, 2179, 353, 2293, 2351, 7229, 823, 281, 457, 241, 2713, 757, 2903, 307, 337, 1033, 9497, 3301, 919, 10739, 1217, 3943, 709, 4093, 379, 271, 131, 4639, 14159, 4801, 257, 14897, 1423, 5303, 317, 16427, 5563, 5651, 1973, 149, 6101, 563, 6379, 6473, 2221, 751, 157, 409, 641, 659, 7349, 22349, 839, 2551, 439, 7963, 24203, 743, 487, 2287, 8819, 8929, 577, 9151, 2557, 3163, 29153, 9833, 9949, 30197, 599, 10301, 31259, 1171, 443, 991, 1153, 191, 11393, 3881, 3923, 1093, 419, 197, 12409, 12539, 199, 479, 2063, 3673, 223, 1249, 41627, 857, 44129, 14851, 2671, 463, 46703, 827, 15859, 48017, 557, 16301, 809, 503, 1861, 17203, 52067, 17509, 1039, 53453, 1997, 6043, 4987, 18443, 18919, 19079, 57719, 6521, 19889, 1823, 20549, 2143, 6961, 2339, 63659, 21559, 22073, 6067, 22943, 4111, 23473, 733, 2521, 24551, 24733, 74747, 1321, 76403, 2383, 1553, 79769, 1381, 1013, 9181, 83207, 2539, 28123, 7723, 28513, 1511, 5101, 9767, 88499, 1747, 1031, 8209, 10303, 3457, 1999, 1087, 31729, 643, 32143, 331, 10993, 5857, 33613, 101477, 34039, 34253, 1951, 1051, 11633, 107309, 36209, 3413, 3907, 37993, 38219, 115337, 4297, 12967, 117389, 39359, 7027, 40283, 121547, 719, 123653, 41453, 887, 125777, 3833, 14293, 1597, 130079, 739, 2579, 132257, 1669, 1373, 7193, 46051, 138899, 46549, 883, 15767, 829, 48313, 145703, 48823, 2887, 1663, 150329, 13879, 51151, 51413, 155027, 17401, 52733, 52999, 162209, 2699, 421, 2917, 167087, 5113, 169553, 1721, 1933, 1987, 57901, 10267, 1103, 3001, 19861, 60149, 9587, 1151, 184727, 1213, 20717, 17029, 189929, 63601, 192557, 21493, 2399, 195203, 6823, 1091, 22483, 203249, 4003, 2357, 205967, 6269, 69263, 18973, 23291, 211457, 70793, 3631, 71719, 73589, 4201, 1021, 3089, 8387, 2297, 228359, 6949, 1301, 4363, 77723, 12323, 1543, 237053, 7213, 79669, 21817, 1709, 80651, 27103, 245909, 82301, 82633, 83299, 7603, 251903, 254927, 85313, 4229, 7879, 9001, 2647, 29231, 2423, 88379, 267203, 89413, 89759, 15901, 2741, 24859, 91499, 1733, 276599, 92551, 1523, 279767, 3467, 31321, 5569, 4271, 1429, 5653, 547, 1117, 15401, 2083, 8933, 295877, 8999, 1361, 11161, 17791, 101183, 305759, 28099, 11489, 34591, 3221, 5501, 3617, 315803, 1187, 106019, 29017, 35591, 322589, 108289, 109433, 1367, 332903, 5881, 1279, 1907, 112901, 2281, 343379, 6779, 4001, 116423, 350453, 354017, 118801, 601, 10909, 21247, 3761, 7177, 368453, 11239, 372107, 41617, 375779, 125669, 126079, 2417, 6679, 7489, 2069, 390647, 130633, 4519, 394409, 43963, 2833, 12143, 401987, 7907, 1579, 15077, 136973, 12491, 12569, 1609, 1889, 2309, 141283, 12923, 142589, 2053, 2503, 144779, 145219, 436979, 146543, 40087, 16381, 4481, 444953, 148763, 6701, 150551, 26647, 2657, 4603, 457049, 9811, 154159, 154613, 5227, 16183, 157351, 158269, 9337, 7829, 6599, 161039, 161503, 490097, 54767, 29077, 165709, 17191, 9803, 15193, 502787, 26687, 169493, 5861, 511337, 2897, 171401, 515639, 1741, 3389, 47269, 30841, 3739, 58901, 59063, 9377, 6211, 8087, 20177, 3271, 182579, 550703, 184559, 555167, 62017, 559649, 187049, 9871, 564149, 1949, 3041, 63353, 63521, 3847, 577757, 193093, 193601, 2237, 586919, 2687, 196663, 197689, 6053, 18253, 4283, 202339, 202859, 67967, 68141, 3121, 18679, 205993, 207569, 11779, 210209, 210739, 211801, 12049, 2459, 643403, 3209, 3217, 2243, 72923, 657929, 219851, 3613, 222023, 11317, 6761, 24851, 2207, 4817, 1787, 2113, 25339, 62497, 692453, 3793, 3931, 697457, 77681, 77867, 9623, 13807, 6491, 21491, 1777, 4177, 8839, 717653, 239783, 240349, 7451, 4567, 66169, 9007, 43117, 244901, 245473, 25453, 247193, 6143, 82781, 748487, 250073, 5333, 753689, 251809, 3767, 12527, 23209, 4337, 45259, 257053, 257639, 26713, 86467, 41051, 23689, 261169, 71389, 262351, 9067, 790607, 5179, 88241, 4507, 5051, 24391, 4663, 89833, 30011, 271903, 3181, 5153, 48409, 2777, 3167, 828407, 11423, 5927, 279179, 839369, 1993, 282239, 14887, 850403, 284083, 16747, 855947, 95311, 10613, 287789, 6619, 872687, 97381, 13109, 26729, 46523, 295901, 889589, 99053, 5839, 30871, 3083, 299683, 301583, 33721, 14957, 304763, 305401, 16141, 2017, 9371, 2273, 6607, 311173, 32257, 313109, 941267, 104801, 9547, 2683, 28759, 2879, 318301, 16787, 56401, 106753, 964703, 2663, 4423, 88237, 4441, 4957, 36241, 108943, 328813, 330139, 110711, 110933, 1000397, 334133, 6317, 1012439, 10247, 340169, 21799, 20129, 342869, 1030637, 114967, 346259, 346939, 2141,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+49x-3 and
the reducible primes which appear as divisor for the first time
p | x^2+49x-3 and p < x^2+49x-3

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)
11011561.1000000.5000001.1000000.0000000.0000000.000000
21007521540.7500000.2100000.7500006.8181824.2000009.000000
31.0006811175640.6810000.1170000.6810009.0800005.57142910.444445
410.0006.8728466.0260.6872000.0846000.68720010.0910437.23076910.684397
5100.00068.9306.58262.3480.6893000.0658200.68930010.0305597.78014210.346498
61.000.000690.20953.175637.0340.6902090.0531750.69020910.0131878.07885210.217393
710.000.0006.903.779448.4996.455.2800.6903780.0448500.69037810.0024478.43439610.133337
8100.000.00069.056.4263.886.90965.169.5170.6905640.0388690.69056410.0026998.66648310.095536
91.000.000.000690.775.68534.288.468656.487.2170.6907760.0342880.69077610.0030628.82152610.073532
1010.000.000.0006.909.513.439306.887.9966.602.625.4430.6909510.0306890.69095110.0025428.95018110.057508


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
389451.1250000.5000000.6250001.8000001.3333332.500000
41614590.8750000.3125000.5625001.5555561.2500001.800000
5322810180.8750000.3125000.5625002.0000002.0000002.000000
6645017330.7812500.2656250.5156251.7857141.7000001.833333
71289423710.7343750.1796880.5546881.8800001.3529412.151515
8256177371400.6914060.1445310.5468751.8829791.6086961.971831
9512353712820.6894530.1386720.5507811.9943501.9189192.014286
101.0246991205790.6826170.1171880.5654301.9801701.6901412.053191
112.0481.4022191.1830.6845700.1069340.5776372.0057231.8250002.043178
124.0962.7954032.3920.6823730.0983890.5839841.9935811.8401832.021978
138.1925.6177204.8970.6856690.0878910.5977782.0096601.7866002.047241
1416.38411.2871.3009.9870.6889040.0793460.6095582.0094361.8055562.039412
1532.76822.5212.42720.0940.6872860.0740660.6132201.9953041.8669232.012016
1665.53645.1704.47740.6930.6892400.0683140.6209262.0056841.8446642.025132
17131.07290.3398.42181.9180.6892320.0642470.6249851.9999781.8809472.013073
18262.144180.82215.722165.1000.6897810.0599750.6298072.0015941.8669992.015430
19524.288361.73029.533332.1970.6899450.0563300.6336152.0004761.8784512.012096
201.048.576723.66355.507668.1560.6901390.0529360.6372032.0005611.8794912.011325
212.097.1521.447.414105.2281.342.1860.6901810.0501770.6400042.0001221.8957612.008791
224.194.3042.894.817199.9742.694.8430.6901780.0476780.6425011.9999921.9003882.007802
238.388.6085.791.217380.8745.410.3430.6903670.0454040.6449632.0005471.9046182.007665
2416.777.21611.582.579727.61510.854.9640.6903760.0433690.6470062.0000251.9103822.006336
2533.554.43223.166.9711.392.09221.774.8790.6904300.0414880.6489422.0001571.9132262.005984
2667.108.86446.339.1632.670.14543.669.0180.6905070.0397880.6507192.0002251.9180812.005477
27134.217.72892.690.1745.129.67887.560.4960.6905960.0382190.6523772.0002561.9211232.005094
28268.435.456185.397.1649.869.251175.527.9130.6906580.0367660.6538922.0001811.9239512.004647
29536.870.912370.830.85219.015.221351.815.6310.6907260.0354190.6553082.0001971.9267142.004328
301.073.741.824741.720.64136.683.121705.037.5200.6907810.0341640.6566172.0001591.9291452.003997
312.147.483.6481.483.560.14970.880.0471.412.680.1020.6908370.0330060.6578302.0001601.9322252.003695
324.294.967.2962.967.360.727137.099.5142.830.261.2130.6908930.0319210.6589722.0001621.9342472.003469
338.589.934.5925.935.145.048265.458.8815.669.686.1670.6909420.0309030.6600382.0001431.9362492.003238
3417.179.869.18411.871.087.965514.517.23711.356.570.7280.6909880.0299490.6610392.0001341.9382182.003033


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
122010101
243021101
384031111
4165041211
53210093223
664170153464
7128230215675
82563703581397
95127106918191618
101.024120011830312831
112.048219021762554953
124.096403040110610294101
138.1927200718182181181176
1416.3841.30001.298323337329311
1532.7682.42702.425602615616594
1665.5364.47704.4751.1311.1291.1271.090
17131.0728.42108.4192.1402.1262.1192.036
18262.14415.722015.7204.0083.9453.9443.825
19524.28829.533029.5317.4677.4577.3527.257
201.048.57655.507055.50513.93113.93113.84513.800
212.097.152105.2280105.22626.33026.38026.24626.272
224.194.304199.9740199.97249.87249.99150.09950.012
238.388.608380.8740380.87295.11495.08395.19195.486
2416.777.216727.6150727.613181.581182.142181.914181.978
2533.554.4321.392.09201.392.090347.604348.307348.064348.117
2667.108.8642.670.14502.670.143667.543667.366667.321667.915
27134.217.7285.129.67805.129.6761.282.7871.282.0221.282.0881.282.781
28268.435.4569.869.25109.869.2492.466.9592.466.5982.467.2362.468.458
29536.870.91219.015.221019.015.2194.754.4984.753.0414.752.7314.754.951
301.073.741.82436.683.121036.683.1199.168.4809.173.5779.169.3839.171.681
312.147.483.64870.880.047070.880.04517.718.49517.720.81517.717.59417.723.143
324.294.967.296137.099.5140137.099.51234.267.23934.275.57334.275.95134.280.751
338.589.934.592265.458.8810265.458.87966.354.62666.364.98466.367.24066.372.031
3417.179.869.184514.517.2370514.517.235128.617.841128.633.810128.630.478128.635.108


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
121010100
242021100
385232111
4169453141
532186124563
66433141988116
712871373420152016
8256140766438323238
951228216012267667475
101.024579307272152136141150
112.0481.183617566306276292309
124.0962.3921.2491.143609573591619
138.1924.8972.5482.3491.2361.2161.2171.228
1416.3849.9875.1824.8052.5342.4802.4802.493
1532.76820.09410.3349.7605.0095.0465.0794.960
1665.53640.69321.06219.63110.16210.16810.25210.111
17131.07281.91842.29239.62620.44320.50120.46820.506
18262.144165.10085.07080.03041.15741.22041.19741.526
19524.288332.197170.800161.39782.72182.99283.13483.350
201.048.576668.156342.819325.337166.709167.074167.086167.287
212.097.1521.342.186687.290654.896335.281335.651335.221336.033
224.194.3042.694.8431.378.5521.316.291672.810673.993673.851674.189
238.388.6085.410.3432.765.0562.645.2871.352.6891.352.5351.352.5671.352.552
2416.777.21610.854.9645.539.7195.315.2452.715.3932.712.9892.712.7332.713.849
2533.554.43221.774.87911.100.70810.674.1715.443.7245.445.2555.443.2795.442.621
2667.108.86443.669.01822.241.64821.427.37010.918.10010.916.96810.915.83810.918.112
27134.217.72887.560.49644.567.45742.993.03921.894.43621.885.63721.890.36721.890.056
28268.435.456175.527.91389.277.98486.249.92943.887.51343.869.83843.883.89543.886.667
29536.870.912351.815.631178.813.363173.002.26887.961.34087.942.24887.949.96087.962.083
301.073.741.824705.037.520358.116.643346.920.877176.271.132176.237.464176.253.724176.275.200
312.147.483.6481.412.680.102717.160.909695.519.193353.174.401353.140.745353.191.898353.173.058
324.294.967.2962.830.261.2131.436.023.4291.394.237.784707.573.920707.542.948707.573.758707.570.587
338.589.934.5925.669.686.1672.875.249.5512.794.436.6161.417.403.9051.417.407.8421.417.435.3591.417.439.061
3417.179.869.18411.356.570.7285.756.539.6095.600.031.1192.839.065.8052.839.119.7542.839.162.1442.839.223.025


8. Check for existing Integer Sequences by OEIS

Found in Database : 3, 47, 11, 17, 19, 89, 109, 389, 151, 173, 587, 73, 1, 1, 293, 29, 61, 373, 401, 1289,
Found in Database : 3, 47, 11, 17, 19, 89, 109, 389, 151, 173, 587, 73, 293, 29, 61, 373, 401, 1289, 163, 1559, 53, 1847, 59, 683, 2153, 251, 263, 2477, 863, 2819, 1019, 367, 127,
Found in Database : 3, 11, 17, 19, 29, 47, 53, 59, 61, 67, 73, 89, 97, 109, 127, 131, 149,