Inhaltsverzeichnis

Development of
Algorithmic Constructions

16:49:24
Deutsch
29.Mar 2024

Polynom = x^2+19x-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) = 17 = 17
f(2) = 39 = 3*13
f(3) = 63 = 3*3*7
f(4) = 89 = 89
f(5) = 117 = 3*3*13
f(6) = 147 = 3*7*7
f(7) = 179 = 179
f(8) = 213 = 3*71
f(9) = 249 = 3*83
f(10) = 287 = 7*41
f(11) = 327 = 3*109
f(12) = 369 = 3*3*41
f(13) = 413 = 7*59
f(14) = 459 = 3*3*3*17
f(15) = 507 = 3*13*13
f(16) = 557 = 557
f(17) = 609 = 3*7*29
f(18) = 663 = 3*13*17
f(19) = 719 = 719
f(20) = 777 = 3*7*37
f(21) = 837 = 3*3*3*31
f(22) = 899 = 29*31
f(23) = 963 = 3*3*107
f(24) = 1029 = 3*7*7*7
f(25) = 1097 = 1097
f(26) = 1167 = 3*389
f(27) = 1239 = 3*7*59
f(28) = 1313 = 13*101
f(29) = 1389 = 3*463
f(30) = 1467 = 3*3*163
f(31) = 1547 = 7*13*17
f(32) = 1629 = 3*3*181
f(33) = 1713 = 3*571
f(34) = 1799 = 7*257
f(35) = 1887 = 3*17*37
f(36) = 1977 = 3*659
f(37) = 2069 = 2069
f(38) = 2163 = 3*7*103
f(39) = 2259 = 3*3*251
f(40) = 2357 = 2357
f(41) = 2457 = 3*3*3*7*13
f(42) = 2559 = 3*853
f(43) = 2663 = 2663
f(44) = 2769 = 3*13*71
f(45) = 2877 = 3*7*137
f(46) = 2987 = 29*103
f(47) = 3099 = 3*1033
f(48) = 3213 = 3*3*3*7*17
f(49) = 3329 = 3329
f(50) = 3447 = 3*3*383
f(51) = 3567 = 3*29*41
f(52) = 3689 = 7*17*31
f(53) = 3813 = 3*31*41
f(54) = 3939 = 3*13*101
f(55) = 4067 = 7*7*83
f(56) = 4197 = 3*1399
f(57) = 4329 = 3*3*13*37
f(58) = 4463 = 4463
f(59) = 4599 = 3*3*7*73
f(60) = 4737 = 3*1579
f(61) = 4877 = 4877
f(62) = 5019 = 3*7*239
f(63) = 5163 = 3*1721
f(64) = 5309 = 5309
f(65) = 5457 = 3*17*107
f(66) = 5607 = 3*3*7*89
f(67) = 5759 = 13*443
f(68) = 5913 = 3*3*3*3*73
f(69) = 6069 = 3*7*17*17
f(70) = 6227 = 13*479
f(71) = 6387 = 3*2129
f(72) = 6549 = 3*37*59
f(73) = 6713 = 7*7*137
f(74) = 6879 = 3*2293
f(75) = 7047 = 3*3*3*3*3*29
f(76) = 7217 = 7*1031
f(77) = 7389 = 3*3*821
f(78) = 7563 = 3*2521
f(79) = 7739 = 71*109
f(80) = 7917 = 3*7*13*29
f(81) = 8097 = 3*2699
f(82) = 8279 = 17*487
f(83) = 8463 = 3*7*13*31
f(84) = 8649 = 3*3*31*31
f(85) = 8837 = 8837
f(86) = 9027 = 3*3*17*59
f(87) = 9219 = 3*7*439
f(88) = 9413 = 9413
f(89) = 9609 = 3*3203
f(90) = 9807 = 3*7*467
f(91) = 10007 = 10007
f(92) = 10209 = 3*41*83
f(93) = 10413 = 3*3*13*89
f(94) = 10619 = 7*37*41
f(95) = 10827 = 3*3*3*401
f(96) = 11037 = 3*13*283
f(97) = 11249 = 7*1607
f(98) = 11463 = 3*3821
f(99) = 11679 = 3*17*229
f(100) = 11897 = 11897

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+19x-3

f(0)=3
f(1)=17
f(2)=13
f(3)=7
f(4)=89
f(5)=1
f(6)=1
f(7)=179
f(8)=71
f(9)=83
f(10)=41
f(11)=109
f(12)=1
f(13)=59
f(14)=1
f(15)=1
f(16)=557
f(17)=29
f(18)=1
f(19)=719
f(20)=37
f(21)=31
f(22)=1
f(23)=107
f(24)=1
f(25)=1097
f(26)=389
f(27)=1
f(28)=101
f(29)=463
f(30)=163
f(31)=1
f(32)=181
f(33)=571
f(34)=257
f(35)=1
f(36)=659
f(37)=2069
f(38)=103
f(39)=251
f(40)=2357
f(41)=1
f(42)=853
f(43)=2663
f(44)=1
f(45)=137
f(46)=1
f(47)=1033
f(48)=1
f(49)=3329
f(50)=383
f(51)=1
f(52)=1
f(53)=1
f(54)=1
f(55)=1
f(56)=1399
f(57)=1
f(58)=4463
f(59)=73
f(60)=1579
f(61)=4877
f(62)=239
f(63)=1721
f(64)=5309
f(65)=1
f(66)=1
f(67)=443
f(68)=1
f(69)=1
f(70)=479
f(71)=2129
f(72)=1
f(73)=1
f(74)=2293
f(75)=1
f(76)=1031
f(77)=821
f(78)=2521
f(79)=1
f(80)=1
f(81)=2699
f(82)=487
f(83)=1
f(84)=1
f(85)=8837
f(86)=1
f(87)=439
f(88)=9413
f(89)=3203
f(90)=467
f(91)=10007
f(92)=1
f(93)=1
f(94)=1
f(95)=401
f(96)=283
f(97)=1607
f(98)=3821
f(99)=229

b) Substitution of the polynom
The polynom f(x)=x^2+19x-3 could be written as f(y)= y^2-93.25 with x=y-9.5

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+9.5
f'(x)>2x+18

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, 17, 13, 7, 89, 1, 1, 179, 71, 83, 41, 109, 1, 59, 1, 1, 557, 29, 1, 719, 37, 31, 1, 107, 1, 1097, 389, 1, 101, 463, 163, 1, 181, 571, 257, 1, 659, 2069, 103, 251, 2357, 1, 853, 2663, 1, 137, 1, 1033, 1, 3329, 383, 1, 1, 1, 1, 1, 1399, 1, 4463, 73, 1579, 4877, 239, 1721, 5309, 1, 1, 443, 1, 1, 479, 2129, 1, 1, 2293, 1, 1031, 821, 2521, 1, 1, 2699, 487, 1, 1, 8837, 1, 439, 9413, 3203, 467, 10007, 1, 1, 1, 401, 283, 1607, 3821, 229, 11897, 577, 457, 739, 1, 4339, 1019, 4493, 653, 1, 4729, 1, 14669, 1657, 1, 1, 307, 5303, 2309, 421, 1, 16937, 1, 5821, 17729, 857, 6089, 18539, 6271, 1, 1, 1, 1, 1, 6833, 1, 3011, 419, 1, 3137, 2473, 1, 22859, 1103, 7823, 1, 1, 2711, 1901, 1, 1, 25667, 8663, 1, 1567, 1, 337, 3947, 1, 9433, 4091, 743, 1, 29663, 1429, 3373, 30707, 1, 1, 31769, 10709, 1, 1, 11071, 1, 409, 1, 373, 5009, 11813, 11939, 5171, 1, 1, 1, 599, 1, 2963, 1, 13103, 673, 461, 643, 1, 4591, 1987, 1453, 1091, 14321, 6197, 1123, 1, 911, 1669, 1, 1, 2207, 503, 1523, 2269, 1, 48527, 5441, 1, 683, 1, 2417, 1249, 1013, 5791, 1, 1, 1049, 593, 18143, 631, 4259, 2659, 6257, 521, 1, 19249, 58229, 19571, 2819, 3511, 1543, 1, 1973, 1, 20719, 8951, 569, 21221, 1, 21559, 7243, 65699, 1051, 1, 67247, 1, 1, 68813, 797, 1109, 1, 1, 1, 71999, 24179, 24359, 809, 1, 2767, 827, 8423, 25453, 76913, 1, 839, 883, 3769, 1, 80279, 691, 3877, 1, 1, 1, 83717, 28099, 9431, 1, 3187, 28879, 733, 1, 2267, 89009, 1, 1, 1279, 1451, 829, 92627, 31079, 1, 94463, 773, 1, 1, 10771, 1913, 1, 32939, 33149, 1, 1, 11261, 101987, 1, 1187, 103913, 1, 35069, 2861, 2731, 1, 787, 709, 5197, 1861, 36821, 2179, 15971, 37489, 967, 1, 12721, 2953, 1, 1, 1, 117899, 5647, 1, 7057, 1, 5779, 1, 1, 5879, 9551, 1, 4651, 18041, 1, 42571, 2621, 43049, 1, 130589, 1, 863, 1289, 1, 1, 1, 45233, 1, 137177, 1483, 1, 139409, 1, 3613, 1, 1283, 3671, 1, 2837, 5387, 146213, 2333, 2897, 148517, 7109, 50021, 1, 50539, 1, 11783, 17107, 7369, 155537, 1, 1, 1327, 52903, 1, 22901, 1, 1741, 1, 1, 54779, 1, 1129, 1, 167597, 18713, 8059, 170063, 56963, 1, 1583, 57793, 1489, 1471, 19543, 58909, 25367, 59471, 59753, 1, 1231, 20201, 14051, 971, 1499, 14249, 1, 1, 1, 2029, 1, 5147, 1, 64063, 27581, 4973, 1, 1, 1, 21943, 198377, 3163, 66721, 11827, 1, 67619, 5507, 4013, 1, 206477, 7681, 1, 209213, 1, 1, 1, 2447, 7919, 30677, 1, 1, 1, 1, 73133, 997, 1, 24691, 5443, 1, 1531, 226007, 75653, 10853, 228869, 1, 25643, 33107, 1, 77893, 33521, 78539, 4639, 237563, 1, 2957, 8293, 3833, 6217, 1777, 1381, 1, 7949, 82471, 3943, 1, 27823, 1, 1, 4969, 84809, 1, 1171, 28607, 1, 1373, 2347, 6379, 12503, 2143, 3727, 1213, 1, 267713, 2297, 1, 1, 6971, 90971, 5591, 1, 1, 1277, 1, 1, 280277, 13397, 1, 283463, 1, 2441, 1, 10657, 1, 289889, 96989, 13907, 293129, 1, 10937, 3257, 33053, 99523, 1, 3457, 1, 1, 14479, 1, 1, 1, 102829, 309599, 1, 1, 1, 8053, 5003, 1, 3919, 106189, 1, 1, 1, 1, 108079, 1, 19207, 1, 8431, 329957, 15767, 110753, 11497, 1, 1, 336887, 37561, 1, 26183, 1, 1609, 3779, 115021, 1, 1601, 12911, 116593, 350963, 1, 117779, 8647, 1303, 13219, 358109, 1, 17167, 21277, 1, 2477, 365327, 7187, 1409, 52709, 41131, 1, 53231, 4297, 1, 1, 17977, 42083, 379979, 2017, 7499, 12377, 4139, 1, 1753, 129553, 2063, 30089, 1, 131221, 56417, 132059, 1, 56957, 133321, 1, 9817, 1, 1, 406313, 1493, 8017, 14143, 137143, 6551, 4099, 1709, 1, 417869, 139721, 10781, 60251, 4549, 1, 1, 47441, 142759, 429587, 1, 144071, 11717, 20707, 48463, 1, 48757, 20959, 1, 1433, 21149, 445463, 148933, 1, 9173, 1, 150721, 1, 1, 3709, 457559, 1, 17047, 461627, 7349, 4993, 1, 155693, 22307, 7963, 9239, 1, 473939, 52813, 1, 1, 2251, 5527, 68891, 4357, 53891, 1, 1, 1, 2903, 3347, 164471, 2239, 1, 2633, 1, 3271, 23899, 503297, 4547, 168713, 2339, 1, 56713, 73121, 1, 171571, 12589, 1, 10177, 1, 1, 58151, 30871, 1, 1933, 1, 2131, 1949, 9043, 178333, 1, 76847, 1, 180289, 77477, 10663, 181763, 1, 26107, 3593, 1, 1, 1, 555677, 185723, 1, 560159, 187219, 1, 19471, 1613, 189223, 4783, 14633, 6577, 1999, 11279, 1, 578267, 9203, 193771, 8209, 27827, 1, 587429, 1, 1, 16001, 1, 1667, 596663, 199403, 6449, 1, 1951, 67157, 6659, 22501, 7001, 1, 29153, 1877, 16631, 1, 22907, 6967, 1, 1, 624797, 16061, 1759, 1, 16183, 1, 90617, 70657, 1, 13043, 5209, 1, 1, 1, 5531, 1, 1, 1, 653579, 1, 31277, 7933, 220021, 1, 1, 73883, 222193, 1, 223283, 223829, 1, 6079, 4421, 6337, 10789, 1, 40177, 32603, 228773, 687977, 17683, 10973, 692963, 1979, 1, 697967, 7523, 7541, 3463, 1, 26161, 2467, 78857, 1, 1, 1, 18371, 718163, 1, 80173, 723257, 1, 34603, 728369, 1, 34847, 3319, 245071, 2213, 8117, 1, 247363, 1801, 1, 249089, 1, 5107, 1, 24329, 1, 252559, 1, 1, 2137, 764657, 255469, 1, 45289, 1, 2503, 1, 258983, 1, 111497, 1, 6701, 1, 1, 2459, 791117, 2221, 264893, 7307, 266083, 1, 1, 3079, 5479, 2003, 8699, 270269, 4003, 271471, 1, 1, 1, 1, 1, 3023, 275699, 828917, 1, 92507, 1, 30971, 5701, 839879, 7583, 40169, 845387, 282409, 1, 121559, 2311, 1, 4219, 286103, 286721, 50707, 1, 1, 10453, 1, 290443, 67169, 291689, 41759, 67601, 1, 14009, 12457, 1931, 296071, 127157, 4073, 1, 1, 23017, 33317, 901457, 1, 301753, 907163, 1, 1, 3637, 304933, 14551, 54037, 3527, 1, 29819, 1, 23801, 1, 3491, 1, 133709, 11579, 8467, 2237, 44939, 3121, 72893, 1, 1, 1, 1, 45589, 56431, 5431, 45869, 9371, 11117, 1, 138731, 8317, 325021, 4813, 1, 326999, 1, 1, 109661, 988937, 1, 19469, 994913, 332303, 3659, 1000907, 3067, 1, 1, 112103, 8219, 3911, 338321, 338993,

6. Sequence of the polynom (only primes)

3, 17, 13, 7, 89, 179, 71, 83, 41, 109, 59, 557, 29, 719, 37, 31, 107, 1097, 389, 101, 463, 163, 181, 571, 257, 659, 2069, 103, 251, 2357, 853, 2663, 137, 1033, 3329, 383, 1399, 4463, 73, 1579, 4877, 239, 1721, 5309, 443, 479, 2129, 2293, 1031, 821, 2521, 2699, 487, 8837, 439, 9413, 3203, 467, 10007, 401, 283, 1607, 3821, 229, 11897, 577, 457, 739, 4339, 1019, 4493, 653, 4729, 14669, 1657, 307, 5303, 2309, 421, 16937, 5821, 17729, 857, 6089, 18539, 6271, 6833, 3011, 419, 3137, 2473, 22859, 1103, 7823, 2711, 1901, 25667, 8663, 1567, 337, 3947, 9433, 4091, 743, 29663, 1429, 3373, 30707, 31769, 10709, 11071, 409, 373, 5009, 11813, 11939, 5171, 599, 2963, 13103, 673, 461, 643, 4591, 1987, 1453, 1091, 14321, 6197, 1123, 911, 1669, 2207, 503, 1523, 2269, 48527, 5441, 683, 2417, 1249, 1013, 5791, 1049, 593, 18143, 631, 4259, 2659, 6257, 521, 19249, 58229, 19571, 2819, 3511, 1543, 1973, 20719, 8951, 569, 21221, 21559, 7243, 65699, 1051, 67247, 68813, 797, 1109, 71999, 24179, 24359, 809, 2767, 827, 8423, 25453, 76913, 839, 883, 3769, 80279, 691, 3877, 83717, 28099, 9431, 3187, 28879, 733, 2267, 89009, 1279, 1451, 829, 92627, 31079, 94463, 773, 10771, 1913, 32939, 33149, 11261, 101987, 1187, 103913, 35069, 2861, 2731, 787, 709, 5197, 1861, 36821, 2179, 15971, 37489, 967, 12721, 2953, 117899, 5647, 7057, 5779, 5879, 9551, 4651, 18041, 42571, 2621, 43049, 130589, 863, 1289, 45233, 137177, 1483, 139409, 3613, 1283, 3671, 2837, 5387, 146213, 2333, 2897, 148517, 7109, 50021, 50539, 11783, 17107, 7369, 155537, 1327, 52903, 22901, 1741, 54779, 1129, 167597, 18713, 8059, 170063, 56963, 1583, 57793, 1489, 1471, 19543, 58909, 25367, 59471, 59753, 1231, 20201, 14051, 971, 1499, 14249, 2029, 5147, 64063, 27581, 4973, 21943, 198377, 3163, 66721, 11827, 67619, 5507, 4013, 206477, 7681, 209213, 2447, 7919, 30677, 73133, 997, 24691, 5443, 1531, 226007, 75653, 10853, 228869, 25643, 33107, 77893, 33521, 78539, 4639, 237563, 2957, 8293, 3833, 6217, 1777, 1381, 7949, 82471, 3943, 27823, 4969, 84809, 1171, 28607, 1373, 2347, 6379, 12503, 2143, 3727, 1213, 267713, 2297, 6971, 90971, 5591, 1277, 280277, 13397, 283463, 2441, 10657, 289889, 96989, 13907, 293129, 10937, 3257, 33053, 99523, 3457, 14479, 102829, 309599, 8053, 5003, 3919, 106189, 108079, 19207, 8431, 329957, 15767, 110753, 11497, 336887, 37561, 26183, 1609, 3779, 115021, 1601, 12911, 116593, 350963, 117779, 8647, 1303, 13219, 358109, 17167, 21277, 2477, 365327, 7187, 1409, 52709, 41131, 53231, 4297, 17977, 42083, 379979, 2017, 7499, 12377, 4139, 1753, 129553, 2063, 30089, 131221, 56417, 132059, 56957, 133321, 9817, 406313, 1493, 8017, 14143, 137143, 6551, 4099, 1709, 417869, 139721, 10781, 60251, 4549, 47441, 142759, 429587, 144071, 11717, 20707, 48463, 48757, 20959, 1433, 21149, 445463, 148933, 9173, 150721, 3709, 457559, 17047, 461627, 7349, 4993, 155693, 22307, 7963, 9239, 473939, 52813, 2251, 5527, 68891, 4357, 53891, 2903, 3347, 164471, 2239, 2633, 3271, 23899, 503297, 4547, 168713, 2339, 56713, 73121, 171571, 12589, 10177, 58151, 30871, 1933, 2131, 1949, 9043, 178333, 76847, 180289, 77477, 10663, 181763, 26107, 3593, 555677, 185723, 560159, 187219, 19471, 1613, 189223, 4783, 14633, 6577, 1999, 11279, 578267, 9203, 193771, 8209, 27827, 587429, 16001, 1667, 596663, 199403, 6449, 1951, 67157, 6659, 22501, 7001, 29153, 1877, 16631, 22907, 6967, 624797, 16061, 1759, 16183, 90617, 70657, 13043, 5209, 5531, 653579, 31277, 7933, 220021, 73883, 222193, 223283, 223829, 6079, 4421, 6337, 10789, 40177, 32603, 228773, 687977, 17683, 10973, 692963, 1979, 697967, 7523, 7541, 3463, 26161, 2467, 78857, 18371, 718163, 80173, 723257, 34603, 728369, 34847, 3319, 245071, 2213, 8117, 247363, 1801, 249089, 5107, 24329, 252559, 2137, 764657, 255469, 45289, 2503, 258983, 111497, 6701, 2459, 791117, 2221, 264893, 7307, 266083, 3079, 5479, 2003, 8699, 270269, 4003, 271471, 3023, 275699, 828917, 92507, 30971, 5701, 839879, 7583, 40169, 845387, 282409, 121559, 2311, 4219, 286103, 286721, 50707, 10453, 290443, 67169, 291689, 41759, 67601, 14009, 12457, 1931, 296071, 127157, 4073, 23017, 33317, 901457, 301753, 907163, 3637, 304933, 14551, 54037, 3527, 29819, 23801, 3491, 133709, 11579, 8467, 2237, 44939, 3121, 72893, 45589, 56431, 5431, 45869, 9371, 11117, 138731, 8317, 325021, 4813, 326999, 109661, 988937, 19469, 994913, 332303, 3659, 1000907, 3067, 112103, 8219, 3911, 338321, 338993,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+19x-3 and
the reducible primes which appear as divisor for the first time
p | x^2+19x-3 and p < x^2+19x-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)
1109450.9000000.4000000.5000000.0000000.0000000.000000
21006118430.6100000.1800000.4300006.7777784.5000008.600000
31.0006661005660.6660000.1000000.56600010.9180335.55555513.162790
410.0006.7156596.0560.6715000.0659000.60560010.0825826.59000010.699647
5100.00067.7615.10762.6540.6776100.0510700.62654010.0909907.74962010.345773
61.000.000680.55641.512639.0440.6805560.0415120.63904410.0434768.12845110.199573
710.000.0006.820.782351.5276.469.2550.6820780.0351530.64692610.0223678.46808110.123333
8100.000.00068.336.1923.050.45365.285.7390.6833620.0305050.65285710.0188218.67772010.091694
91.000.000.000684.326.20626.910.114657.416.0920.6843260.0269100.65741610.0141118.82167810.069826
1010.000.000.0006.851.406.123240.869.8586.610.536.2650.6851410.0240870.66105410.0119018.95090410.055331


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
387430.8750000.5000000.3750001.4000001.3333331.500000
41612570.7500000.3125000.4375001.7142861.2500002.333333
532207130.6250000.2187500.4062501.6666671.4000001.857143
6644014260.6250000.2187500.4062502.0000002.0000002.000000
71288222600.6406250.1718750.4687502.0500001.5714292.307692
8256166331330.6484380.1289060.5195312.0243901.5000002.216667
9512337582790.6582030.1132810.5449222.0301201.7575762.097744
101.0246811025790.6650390.0996090.5654302.0207721.7586212.075269
112.0481.3691791.1900.6684570.0874020.5810552.0102791.7549022.055268
124.0962.7573092.4480.6730960.0754390.5976562.0138791.7262572.057143
138.1925.5065574.9490.6721190.0679930.6041261.9970981.8025892.021650
1416.38411.0571.02210.0350.6748660.0623780.6124882.0081731.8348292.027682
1532.76822.1401.87820.2620.6756590.0573120.6183472.0023521.8375732.019133
1665.53644.3903.50140.8890.6773380.0534210.6239172.0049681.8642172.018014
17131.07288.8646.51182.3530.6779790.0496750.6283042.0018921.8597542.014062
18262.144177.96912.262165.7070.6788980.0467760.6321222.0027121.8832742.012155
19524.288356.43323.134333.2990.6798420.0441250.6357172.0027811.8866422.011375
201.048.576713.67043.386670.2840.6806090.0413760.6392332.0022561.8754212.011059
212.097.1521.428.24282.3141.345.9280.6810390.0392500.6417882.0012641.8972482.007997
224.194.3042.858.944156.4452.702.4990.6816250.0372990.6443262.0017221.9005882.007908
238.388.6085.720.924298.5405.422.3840.6819870.0355890.6463992.0010621.9082752.006433
2416.777.21611.449.043570.99410.878.0490.6824160.0340340.6483822.0012581.9126212.006138
2533.554.43222.911.0681.092.82921.818.2390.6828030.0325690.6502342.0011341.9139062.005713
2667.108.86445.845.4742.096.42643.749.0480.6831510.0312390.6519122.0010191.9183482.005159
27134.217.72891.735.3604.025.49087.709.8700.6834820.0299920.6534892.0009691.9201682.004841
28268.435.456183.552.8807.744.499175.808.3810.6837880.0288510.6549372.0008961.9238652.004431
29536.870.912367.260.70214.922.552352.338.1500.6840760.0277950.6562812.0008441.9268582.004103
301.073.741.824734.820.49628.788.658706.031.8380.6843550.0268120.6575432.0008141.9292052.003847
312.147.483.6481.470.200.09255.628.9551.414.571.1370.6846150.0259040.6587112.0007611.9323222.003551
324.294.967.2962.941.456.725107.604.3312.833.852.3940.6848610.0250540.6598082.0007191.9343222.003330
338.589.934.5925.884.893.195208.350.7485.676.542.4470.6850920.0242550.6608372.0006731.9362672.003119
3417.179.869.18411.773.555.929403.844.19411.369.711.7350.6853110.0235070.6618042.0006411.9382902.002929
3534.359.738.36823.554.244.096783.552.60122.770.691.4950.6855190.0228040.6627142.0006061.9402352.002750
3668.719.476.73647.121.985.8581.521.582.50245.600.403.3560.6857150.0221420.6635732.0005731.9419022.002592


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
122011100
243022100
384032200
4165042210
5327063211
664140134253
7128220217384
82563303197107
95125805617111812
101.024102010030232722
112.048179017751454538
124.096309030783758269
138.1925570555146133149129
1416.3841.02201.020263261264234
1532.7681.87801.876483473478444
1665.5363.50103.499875880888858
17131.0726.51106.5091.6081.6061.6591.638
18262.14412.262012.2603.0133.0773.0943.078
19524.28823.134023.1325.7545.7875.8215.772
201.048.57643.386043.38410.84510.89810.78410.859
212.097.15282.314082.31220.58820.54820.56520.613
224.194.304156.4450156.44339.21438.86739.23239.132
238.388.608298.5400298.53874.78774.41574.58774.751
2416.777.216570.9940570.992142.881142.296142.744143.073
2533.554.4321.092.82901.092.827273.551273.050273.002273.226
2667.108.8642.096.42602.096.424524.392523.353524.154524.527
27134.217.7284.025.49004.025.4881.006.3101.006.1771.006.6741.006.329
28268.435.4567.744.49907.744.4971.937.4511.936.1221.934.8151.936.111
29536.870.91214.922.552014.922.5503.732.9803.729.2053.729.6213.730.746
301.073.741.82428.788.658028.788.6567.198.0117.197.2197.197.9517.195.477
312.147.483.64855.628.955055.628.95313.906.32613.906.91713.906.95113.908.761
324.294.967.296107.604.3310107.604.32926.900.58626.904.54526.898.67426.900.526
338.589.934.592208.350.7480208.350.74652.089.01752.083.99052.086.59752.091.144
3417.179.869.184403.844.1940403.844.192100.965.940100.958.443100.956.682100.963.129
3534.359.738.368783.552.6010783.552.599195.888.803195.886.608195.882.143195.895.047
3668.719.476.7361.521.582.50201.521.582.500380.399.112380.396.834380.392.610380.393.946


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
121100010
242200011
383210012
4167341222
53213671453
6642612145867
712860283214171514
8256133557832363035
951227912615361806870
101.024579266313134160142143
112.0481.190564626287308297298
124.0962.4481.1711.277592636606614
138.1924.9492.3722.5771.1971.2761.2311.245
1416.38410.0354.8265.2092.4612.5452.5032.526
1532.76820.2629.84310.4195.0625.1215.0155.064
1665.53640.88919.93620.95310.22010.20810.19410.267
17131.07282.35340.18942.16420.64620.55720.41320.737
18262.144165.70781.09084.61741.26341.40441.34341.697
19524.288333.299163.271170.02882.85583.30383.39683.745
201.048.576670.284328.336341.948167.029167.627167.676167.952
212.097.1521.345.928659.862686.066335.917336.626336.711336.674
224.194.3042.702.4991.326.0241.376.475674.926675.928676.222675.423
238.388.6085.422.3842.663.8882.758.4961.355.3681.355.4621.355.6351.355.919
2416.777.21610.878.0495.348.4645.529.5852.717.6672.720.8822.719.4962.720.004
2533.554.43221.818.23910.738.24311.079.9965.452.1835.455.2975.455.8105.454.949
2667.108.86443.749.04821.547.42322.201.62510.937.11010.936.96610.938.01410.936.958
27134.217.72887.709.87043.226.99244.482.87821.926.74721.927.20921.929.60521.926.309
28268.435.456175.808.38186.695.45189.112.93043.947.35143.954.28843.953.69343.953.049
29536.870.912352.338.150173.833.585178.504.56588.081.30688.075.11288.089.57088.092.162
301.073.741.824706.031.838348.504.203357.527.635176.505.472176.497.655176.500.890176.527.821
312.147.483.6481.414.571.137698.570.232716.000.905353.642.695353.637.773353.622.138353.668.531
324.294.967.2962.833.852.3941.400.053.6091.433.798.785708.474.486708.457.502708.452.353708.468.053
338.589.934.5925.676.542.4472.805.593.8632.870.948.5841.419.158.3521.419.119.4531.419.126.4881.419.138.154
3417.179.869.18411.369.711.7355.621.509.9585.748.201.7772.842.415.5942.842.445.6382.842.429.5152.842.420.988
3534.359.738.36822.770.691.49511.262.529.19111.508.162.3045.692.642.5815.692.724.1505.692.667.1605.692.657.604
3668.719.476.73645.600.403.35622.561.917.20323.038.486.15311.400.084.73511.400.191.38111.400.082.60511.400.044.635


8. Check for existing Integer Sequences by OEIS

Found in Database : 3, 17, 13, 7, 89, 1, 1, 179, 71, 83, 41, 109, 1, 59, 1, 1, 557, 29, 1, 719,
Found in Database : 3, 17, 13, 7, 89, 179, 71, 83, 41, 109, 59, 557, 29, 719, 37, 31, 107, 1097, 389, 101, 463, 163, 181, 571, 257, 659, 2069, 103, 251,
Found in Database : 3, 7, 13, 17, 29, 31, 37, 41, 59, 71, 73, 83, 89, 101, 103, 107, 109, 137,