Inhaltsverzeichnis

Development of
Algorithmic Constructions

11:55:15
Deutsch
29.Mar 2024

Polynom = x^2+78x-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) = 29 = 29
f(2) = 23 = 23
f(3) = 53 = 53
f(4) = 191 = 191
f(5) = 139 = 139
f(6) = 367 = 367
f(7) = 229 = 229
f(8) = 551 = 19*29
f(9) = 323 = 17*19
f(10) = 743 = 743
f(11) = 421 = 421
f(12) = 943 = 23*41
f(13) = 523 = 523
f(14) = 1151 = 1151
f(15) = 629 = 17*37
f(16) = 1367 = 1367
f(17) = 739 = 739
f(18) = 1591 = 37*43
f(19) = 853 = 853
f(20) = 1823 = 1823
f(21) = 971 = 971
f(22) = 2063 = 2063
f(23) = 1093 = 1093
f(24) = 2311 = 2311
f(25) = 1219 = 23*53
f(26) = 2567 = 17*151
f(27) = 1349 = 19*71
f(28) = 2831 = 19*149
f(29) = 1483 = 1483
f(30) = 3103 = 29*107
f(31) = 1621 = 1621
f(32) = 3383 = 17*199
f(33) = 1763 = 41*43
f(34) = 3671 = 3671
f(35) = 1909 = 23*83
f(36) = 3967 = 3967
f(37) = 2059 = 29*71
f(38) = 4271 = 4271
f(39) = 2213 = 2213
f(40) = 4583 = 4583
f(41) = 2371 = 2371
f(42) = 4903 = 4903
f(43) = 2533 = 17*149
f(44) = 5231 = 5231
f(45) = 2699 = 2699
f(46) = 5567 = 19*293
f(47) = 2869 = 19*151
f(48) = 5911 = 23*257
f(49) = 3043 = 17*179
f(50) = 6263 = 6263
f(51) = 3221 = 3221
f(52) = 6623 = 37*179
f(53) = 3403 = 41*83
f(54) = 6991 = 6991
f(55) = 3589 = 37*97
f(56) = 7367 = 53*139
f(57) = 3779 = 3779
f(58) = 7751 = 23*337
f(59) = 3973 = 29*137
f(60) = 8143 = 17*479
f(61) = 4171 = 43*97
f(62) = 8543 = 8543
f(63) = 4373 = 4373
f(64) = 8951 = 8951
f(65) = 4579 = 19*241
f(66) = 9367 = 17*19*29
f(67) = 4789 = 4789
f(68) = 9791 = 9791
f(69) = 5003 = 5003
f(70) = 10223 = 10223
f(71) = 5221 = 23*227
f(72) = 10663 = 10663
f(73) = 5443 = 5443
f(74) = 11111 = 41*271
f(75) = 5669 = 5669
f(76) = 11567 = 43*269
f(77) = 5899 = 17*347
f(78) = 12031 = 53*227
f(79) = 6133 = 6133
f(80) = 12503 = 12503
f(81) = 6371 = 23*277
f(82) = 12983 = 12983
f(83) = 6613 = 17*389
f(84) = 13471 = 19*709
f(85) = 6859 = 19*19*19
f(86) = 13967 = 13967
f(87) = 7109 = 7109
f(88) = 14471 = 29*499
f(89) = 7363 = 37*199
f(90) = 14983 = 14983
f(91) = 7621 = 7621
f(92) = 15503 = 37*419
f(93) = 7883 = 7883
f(94) = 16031 = 17*23*41
f(95) = 8149 = 29*281
f(96) = 16567 = 16567
f(97) = 8419 = 8419
f(98) = 17111 = 71*241
f(99) = 8693 = 8693
f(100) = 17663 = 17*1039

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+78x-137

f(0)=137
f(1)=29
f(2)=23
f(3)=53
f(4)=191
f(5)=139
f(6)=367
f(7)=229
f(8)=19
f(9)=17
f(10)=743
f(11)=421
f(12)=41
f(13)=523
f(14)=1151
f(15)=37
f(16)=1367
f(17)=739
f(18)=43
f(19)=853
f(20)=1823
f(21)=971
f(22)=2063
f(23)=1093
f(24)=2311
f(25)=1
f(26)=151
f(27)=71
f(28)=149
f(29)=1483
f(30)=107
f(31)=1621
f(32)=199
f(33)=1
f(34)=3671
f(35)=83
f(36)=3967
f(37)=1
f(38)=4271
f(39)=2213
f(40)=4583
f(41)=2371
f(42)=4903
f(43)=1
f(44)=5231
f(45)=2699
f(46)=293
f(47)=1
f(48)=257
f(49)=179
f(50)=6263
f(51)=3221
f(52)=1
f(53)=1
f(54)=6991
f(55)=97
f(56)=1
f(57)=3779
f(58)=337
f(59)=1
f(60)=479
f(61)=1
f(62)=8543
f(63)=4373
f(64)=8951
f(65)=241
f(66)=1
f(67)=4789
f(68)=9791
f(69)=5003
f(70)=10223
f(71)=227
f(72)=10663
f(73)=5443
f(74)=271
f(75)=5669
f(76)=269
f(77)=347
f(78)=1
f(79)=6133
f(80)=12503
f(81)=277
f(82)=12983
f(83)=389
f(84)=709
f(85)=1
f(86)=13967
f(87)=7109
f(88)=499
f(89)=1
f(90)=14983
f(91)=7621
f(92)=419
f(93)=7883
f(94)=1
f(95)=281
f(96)=16567
f(97)=8419
f(98)=1
f(99)=8693

b) Substitution of the polynom
The polynom f(x)=x^2+78x-137 could be written as f(y)= y^2-1658 with x=y-39

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+39
f'(x)>2x+77

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, 29, 23, 53, 191, 139, 367, 229, 19, 17, 743, 421, 41, 523, 1151, 37, 1367, 739, 43, 853, 1823, 971, 2063, 1093, 2311, 1, 151, 71, 149, 1483, 107, 1621, 199, 1, 3671, 83, 3967, 1, 4271, 2213, 4583, 2371, 4903, 1, 5231, 2699, 293, 1, 257, 179, 6263, 3221, 1, 1, 6991, 97, 1, 3779, 337, 1, 479, 1, 8543, 4373, 8951, 241, 1, 4789, 9791, 5003, 10223, 227, 10663, 5443, 271, 5669, 269, 347, 1, 6133, 12503, 277, 12983, 389, 709, 1, 13967, 7109, 499, 1, 14983, 7621, 419, 7883, 1, 281, 16567, 8419, 1, 8693, 1039, 8971, 18223, 487, 1, 9539, 181, 9829, 1, 1, 20543, 613, 21143, 10723, 21751, 1, 22367, 1, 1, 1, 23623, 11971, 1277, 647, 859, 12619, 691, 563, 1543, 359, 26903, 1, 27583, 13963, 1663, 349, 1, 1, 29671, 15013, 1321, 809, 1637, 15733, 1, 947, 1123, 383, 33311, 16843, 1481, 1013, 1, 607, 35591, 17989, 887, 18379, 1, 18773, 1997, 1009, 1, 1, 39551, 19979, 1091, 20389, 2423, 1, 42023, 21221, 42863, 941, 43711, 761, 1087, 1, 251, 1, 2437, 23371, 1627, 23813, 907, 1427, 2129, 24709, 49871, 25163, 1181, 25621, 1, 26083, 52631, 1, 1, 659, 1, 1447, 1499, 1, 3319, 769, 1979, 673, 58367, 29429, 59351, 1301, 60343, 1049, 61343, 1, 62351, 593, 63367, 1, 3389, 1, 65423, 32971, 66463, 33493, 67511, 34019, 68567, 34549, 1, 35083, 4159, 1, 3121, 1, 72871, 36709, 1, 1, 1831, 37813, 1, 38371, 3361, 38933, 733, 39499, 317, 2357, 1877, 1, 1153, 41221, 83023, 2459, 84191, 1, 4493, 42979, 2111, 43573, 87743, 44171, 3067, 44773, 5303, 1973, 91367, 45989, 1747, 1607, 5519, 47221, 95063, 47843, 1, 2551, 2269, 1327, 4297, 1213, 100103, 2963, 101383, 1, 1237, 51659, 103967, 1, 1, 52963, 2011, 1, 107903, 2857, 5749, 54949, 110567, 55619, 1, 1373, 1, 2477, 1381, 57653, 6823, 1, 1, 59029, 118751, 59723, 379, 1, 6397, 3217, 3323, 3637, 124367, 62539, 125791, 1471, 1, 1, 128663, 64693, 5657, 65419, 883, 2281, 881, 66883, 134503, 3559, 1, 1, 1, 69109, 138967, 69859, 8263, 1, 3463, 1, 143503, 1361, 145031, 1, 1511, 3203, 4003, 1, 7877, 1, 1559, 76003, 152791, 4517, 5323, 3373, 1, 433, 157543, 1931, 3701, 79973, 160751, 80779, 9551, 983, 163991, 4337, 1, 83221, 9839, 1, 3187, 84869, 1, 1993, 5939, 86533, 7561, 2131, 2473, 5189, 1, 1, 937, 89909, 1, 1, 1, 91621, 2593, 4021, 185831, 1, 3539, 94219, 189311, 95093, 11239, 95971, 4703, 4211, 194591, 2273, 11551, 1, 10429, 99523, 1117, 1, 201743, 101323, 7019, 102229, 8929, 6067, 207191, 104053, 4861, 104971, 1, 6229, 1, 2887, 491, 1, 216431, 1, 1097, 109621, 1, 110563, 13063, 111509, 7723, 112459, 225871, 4931, 13399, 114371, 229703, 1, 231631, 6121, 1, 117269, 5477, 1, 2861, 7013, 239423, 1693, 241391, 121189, 243367, 7187, 2293, 3329, 247343, 124171, 1, 1, 13229, 1, 253367, 127189, 1, 128203, 1879, 129221, 1, 1, 15383, 1, 263567, 132299, 541, 1, 5051, 134371, 269783, 7127, 1, 1, 273967, 1, 1, 138563, 1867, 1, 280303, 140683, 9739, 6163, 7691, 569, 286711, 3889, 1913, 4999, 1, 7687, 7151, 147139, 6869, 148229, 1, 149323, 299743, 1, 301943, 151523, 304151, 152629, 306367, 153739, 13417, 9109, 310823, 8209, 16477, 5417, 4441, 1, 587, 159349, 319831, 160483, 1, 7027, 1429, 1, 1, 163909, 1, 165059, 1, 1, 1, 1, 19759, 168533, 338231, 4139, 3511, 1, 2503, 4001, 345263, 2087, 11987, 10259, 15217, 3313, 352367, 176779, 354751, 1, 18797, 1289, 1399, 1, 15737, 1697, 8887, 182789, 366791, 184003, 1, 185221, 371663, 5039, 374111, 187669, 1303, 1, 19949, 10007, 1583, 6599, 383983, 192613, 386471, 193859, 1, 1, 13499, 1, 1, 197621, 9221, 11699, 399031, 2819, 401567, 10601, 21269, 202693, 17681, 203971, 1787, 205253, 24223, 2909, 414367, 1, 7867, 209123, 1, 210421, 3037, 211723, 1579, 213029, 1, 1, 429991, 215653, 10061, 12763, 435263, 9491, 2939, 219619, 440567, 1, 443231, 1, 2953, 223621, 1531, 9781, 5437, 1, 23893, 2347, 26863, 1, 459383, 2153, 462103, 2389, 1, 1, 701, 6337, 11471, 1, 1, 1571, 475823, 1, 25189, 1, 20929, 1, 484151, 242773, 486943, 1, 489743, 5711, 492551, 246979, 6977, 248389, 1, 10861, 1783, 251221, 1, 13297, 26669, 8761, 7177, 255499, 1, 11171, 2053, 6983, 2713, 259813, 14083, 1907, 2633, 262709, 526871, 1, 1, 265621, 1, 14057, 2339, 15797, 538567, 9311, 541511, 3271, 3917, 272971, 23801, 274453, 18979, 275939, 757, 277429, 556351, 6803, 559343, 14759, 1741, 281923, 10667, 12323, 568367, 3433, 15443, 286453, 574423, 1, 577463, 17029, 1, 12653, 20123, 292549, 586631, 17299, 1, 15559, 1759, 10247, 6143, 298709, 598967, 4229, 26177, 301813, 1, 303371, 608303, 304933, 4463, 5783, 36151, 4339, 1, 1, 797, 2239, 2749, 1, 627191, 1, 1, 18587, 17123, 1, 1, 13877, 22067, 18869, 14957, 1801, 811, 6113, 1, 17137, 1, 1, 656063, 2207, 38783, 330469, 662567, 332099, 821, 3119, 39359, 2221, 8101, 337013, 1, 338659, 827, 17911, 1, 341963, 829, 1, 1, 1, 30097, 9377, 695567, 20507, 13187, 350293, 8461, 351971, 4673, 353653, 708991, 355339, 37493, 1, 1, 358723, 719143, 360421, 17623, 12487, 42703, 363829, 729367, 1, 10321, 1, 1, 1, 5399, 370693, 743111, 1, 39293, 374149, 3769, 10159, 1, 1, 1, 1, 2179, 8863, 763967, 3947, 767471, 2767, 33521, 386371, 1, 388133, 40949, 20521, 781567, 1, 46183, 13567, 18341, 7457, 792223, 1, 795791, 3727, 1, 400579, 802951, 23669, 806543, 1, 1871, 929, 42829, 1, 22091, 409589, 820991, 11119, 2297, 14249, 1, 1, 831911, 1, 2137, 418699, 1, 5923, 2659, 3083, 2621, 1, 850271, 2141, 1, 1, 857671, 6053, 6197, 431621, 865103, 1, 21191, 435349, 872567, 437219, 876311, 1, 30347, 23209, 1, 11969, 16747, 444739, 24091, 15401, 895151, 19501, 52879, 450421, 3259, 1, 22111, 454229, 53551, 1, 914191, 458053, 2543, 1, 1, 2017, 2417, 1, 17539, 27397, 9623, 16127, 2237, 1, 1, 27739, 9743, 11549, 1, 3191, 3391, 25127, 1, 479371, 960703, 13009, 1, 21013, 968567, 2711, 11717, 1, 1, 1, 980423, 3253, 984391, 1, 988367, 9343, 1, 1, 14033, 29363, 9349, 501173, 23357, 17351, 12149, 29717, 44017, 1033, 1016423, 509221, 1, 511243, 1024511, 513269, 25087, 1, 1, 1, 28019, 519371, 1040783, 1, 61463, 523459, 4621, 18121, 1053071, 527563, 1057183, 23027, 1, 4969, 36739, 31397, 1, 28201, 1073711, 2143,

6. Sequence of the polynom (only primes)

137, 29, 23, 53, 191, 139, 367, 229, 19, 17, 743, 421, 41, 523, 1151, 37, 1367, 739, 43, 853, 1823, 971, 2063, 1093, 2311, 151, 71, 149, 1483, 107, 1621, 199, 3671, 83, 3967, 4271, 2213, 4583, 2371, 4903, 5231, 2699, 293, 257, 179, 6263, 3221, 6991, 97, 3779, 337, 479, 8543, 4373, 8951, 241, 4789, 9791, 5003, 10223, 227, 10663, 5443, 271, 5669, 269, 347, 6133, 12503, 277, 12983, 389, 709, 13967, 7109, 499, 14983, 7621, 419, 7883, 281, 16567, 8419, 8693, 1039, 8971, 18223, 487, 9539, 181, 9829, 20543, 613, 21143, 10723, 21751, 22367, 23623, 11971, 1277, 647, 859, 12619, 691, 563, 1543, 359, 26903, 27583, 13963, 1663, 349, 29671, 15013, 1321, 809, 1637, 15733, 947, 1123, 383, 33311, 16843, 1481, 1013, 607, 35591, 17989, 887, 18379, 18773, 1997, 1009, 39551, 19979, 1091, 20389, 2423, 42023, 21221, 42863, 941, 43711, 761, 1087, 251, 2437, 23371, 1627, 23813, 907, 1427, 2129, 24709, 49871, 25163, 1181, 25621, 26083, 52631, 659, 1447, 1499, 3319, 769, 1979, 673, 58367, 29429, 59351, 1301, 60343, 1049, 61343, 62351, 593, 63367, 3389, 65423, 32971, 66463, 33493, 67511, 34019, 68567, 34549, 35083, 4159, 3121, 72871, 36709, 1831, 37813, 38371, 3361, 38933, 733, 39499, 317, 2357, 1877, 1153, 41221, 83023, 2459, 84191, 4493, 42979, 2111, 43573, 87743, 44171, 3067, 44773, 5303, 1973, 91367, 45989, 1747, 1607, 5519, 47221, 95063, 47843, 2551, 2269, 1327, 4297, 1213, 100103, 2963, 101383, 1237, 51659, 103967, 52963, 2011, 107903, 2857, 5749, 54949, 110567, 55619, 1373, 2477, 1381, 57653, 6823, 59029, 118751, 59723, 379, 6397, 3217, 3323, 3637, 124367, 62539, 125791, 1471, 128663, 64693, 5657, 65419, 883, 2281, 881, 66883, 134503, 3559, 69109, 138967, 69859, 8263, 3463, 143503, 1361, 145031, 1511, 3203, 4003, 7877, 1559, 76003, 152791, 4517, 5323, 3373, 433, 157543, 1931, 3701, 79973, 160751, 80779, 9551, 983, 163991, 4337, 83221, 9839, 3187, 84869, 1993, 5939, 86533, 7561, 2131, 2473, 5189, 937, 89909, 91621, 2593, 4021, 185831, 3539, 94219, 189311, 95093, 11239, 95971, 4703, 4211, 194591, 2273, 11551, 10429, 99523, 1117, 201743, 101323, 7019, 102229, 8929, 6067, 207191, 104053, 4861, 104971, 6229, 2887, 491, 216431, 1097, 109621, 110563, 13063, 111509, 7723, 112459, 225871, 4931, 13399, 114371, 229703, 231631, 6121, 117269, 5477, 2861, 7013, 239423, 1693, 241391, 121189, 243367, 7187, 2293, 3329, 247343, 124171, 13229, 253367, 127189, 128203, 1879, 129221, 15383, 263567, 132299, 541, 5051, 134371, 269783, 7127, 273967, 138563, 1867, 280303, 140683, 9739, 6163, 7691, 569, 286711, 3889, 1913, 4999, 7687, 7151, 147139, 6869, 148229, 149323, 299743, 301943, 151523, 304151, 152629, 306367, 153739, 13417, 9109, 310823, 8209, 16477, 5417, 4441, 587, 159349, 319831, 160483, 7027, 1429, 163909, 165059, 19759, 168533, 338231, 4139, 3511, 2503, 4001, 345263, 2087, 11987, 10259, 15217, 3313, 352367, 176779, 354751, 18797, 1289, 1399, 15737, 1697, 8887, 182789, 366791, 184003, 185221, 371663, 5039, 374111, 187669, 1303, 19949, 10007, 1583, 6599, 383983, 192613, 386471, 193859, 13499, 197621, 9221, 11699, 399031, 2819, 401567, 10601, 21269, 202693, 17681, 203971, 1787, 205253, 24223, 2909, 414367, 7867, 209123, 210421, 3037, 211723, 1579, 213029, 429991, 215653, 10061, 12763, 435263, 9491, 2939, 219619, 440567, 443231, 2953, 223621, 1531, 9781, 5437, 23893, 2347, 26863, 459383, 2153, 462103, 2389, 701, 6337, 11471, 1571, 475823, 25189, 20929, 484151, 242773, 486943, 489743, 5711, 492551, 246979, 6977, 248389, 10861, 1783, 251221, 13297, 26669, 8761, 7177, 255499, 11171, 2053, 6983, 2713, 259813, 14083, 1907, 2633, 262709, 526871, 265621, 14057, 2339, 15797, 538567, 9311, 541511, 3271, 3917, 272971, 23801, 274453, 18979, 275939, 757, 277429, 556351, 6803, 559343, 14759, 1741, 281923, 10667, 12323, 568367, 3433, 15443, 286453, 574423, 577463, 17029, 12653, 20123, 292549, 586631, 17299, 15559, 1759, 10247, 6143, 298709, 598967, 4229, 26177, 301813, 303371, 608303, 304933, 4463, 5783, 36151, 4339, 797, 2239, 2749, 627191, 18587, 17123, 13877, 22067, 18869, 14957, 1801, 811, 6113, 17137, 656063, 2207, 38783, 330469, 662567, 332099, 821, 3119, 39359, 2221, 8101, 337013, 338659, 827, 17911, 341963, 829, 30097, 9377, 695567, 20507, 13187, 350293, 8461, 351971, 4673, 353653, 708991, 355339, 37493, 358723, 719143, 360421, 17623, 12487, 42703, 363829, 729367, 10321, 5399, 370693, 743111, 39293, 374149, 3769, 10159, 2179, 8863, 763967, 3947, 767471, 2767, 33521, 386371, 388133, 40949, 20521, 781567, 46183, 13567, 18341, 7457, 792223, 795791, 3727, 400579, 802951, 23669, 806543, 1871, 929, 42829, 22091, 409589, 820991, 11119, 2297, 14249, 831911, 2137, 418699, 5923, 2659, 3083, 2621, 850271, 2141, 857671, 6053, 6197, 431621, 865103, 21191, 435349, 872567, 437219, 876311, 30347, 23209, 11969, 16747, 444739, 24091, 15401, 895151, 19501, 52879, 450421, 3259, 22111, 454229, 53551, 914191, 458053, 2543, 2017, 2417, 17539, 27397, 9623, 16127, 2237, 27739, 9743, 11549, 3191, 3391, 25127, 479371, 960703, 13009, 21013, 968567, 2711, 11717, 980423, 3253, 984391, 988367, 9343, 14033, 29363, 9349, 501173, 23357, 17351, 12149, 29717, 44017, 1033, 1016423, 509221, 511243, 1024511, 513269, 25087, 28019, 519371, 1040783, 61463, 523459, 4621, 18121, 1053071, 527563, 1057183, 23027, 4969, 36739, 31397, 28201, 1073711, 2143,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+78x-137 and
the reducible primes which appear as divisor for the first time
p | x^2+78x-137 and p < x^2+78x-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)
11010911.0000000.9000001.000000
21008356270.8300000.5600000.8300008.3000006.22222227.000000
31.0007663683980.7660000.3680000.7660009.2289156.57142914.740741
410.0007.3402.7054.6350.7340000.2705000.7340009.5822467.35054311.645729
5100.00072.59421.02851.5660.7259400.2102800.7259409.8901917.77375211.125351
61.000.000720.505171.386549.1190.7205050.1713860.7205059.9251328.15037110.648858
710.000.0007.164.0051.446.9435.717.0620.7164010.1446940.7164019.9430338.44259710.411335
8100.000.00071.334.14112.511.84058.822.3010.7133410.1251180.7133419.9572998.64708510.288903
91.000.000.000710.951.527110.279.008600.672.5190.7109520.1102790.7109529.9664978.81397210.211646
1010.000.000.0007.090.681.365985.894.2696.104.787.0960.7090680.0985890.7090689.9735098.94000010.163254


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)
123301.5000001.5000000.000000
245501.2500001.2500000.0000001.6666671.666667-nan
389811.1250001.0000000.1250001.8000001.600000inf
416161331.0000000.8125000.1875001.7777781.6250003.000000
532312290.9687500.6875000.2812501.9375001.6923083.000000
6645338150.8281250.5937500.2343751.7096771.7272731.666667
712810368350.8046880.5312500.2734381.9433961.7894742.333333
8256202120820.7890620.4687500.3203121.9611651.7647062.342857
95123972151820.7753910.4199220.3554691.9653471.7916672.219512
101.0247843764080.7656250.3671880.3984381.9748111.7488372.241758
112.0481.5456768690.7543950.3300780.4243161.9706631.7978722.129902
124.0963.0341.2371.7970.7407230.3020020.4387211.9637541.8298822.067894
138.1926.0372.2973.7400.7369380.2803960.4565431.9897821.8569122.081247
1416.38411.9944.1987.7960.7320560.2562260.4758301.9867481.8276012.084492
1532.76823.8947.74516.1490.7291870.2363590.4928281.9921631.8449262.071447
1665.53647.59514.42033.1750.7262420.2200320.5062101.9919231.8618462.054307
17131.07295.09726.76768.3300.7255330.2042160.5213171.9980461.8562412.059684
18262.144189.72450.188139.5360.7237400.1914520.5322881.9950581.8749952.042090
19524.288378.39894.805283.5930.7217370.1808260.5409111.9944661.8889972.032400
201.048.576755.433178.965576.4680.7204370.1706740.5497631.9963981.8877172.032730
212.097.1521.507.836339.3401.168.4960.7189920.1618100.5571821.9959891.8961252.026992
224.194.3043.010.353645.0712.365.2820.7177240.1537970.5639271.9964721.9009582.024211
238.388.6086.012.1561.228.1354.784.0210.7167050.1464050.5703001.9971601.9038762.022601
2416.777.21612.006.9332.344.3859.662.5480.7156690.1397360.5759331.9971091.9088982.019754
2533.554.43223.980.7704.485.64819.495.1220.7146830.1336830.5810001.9972441.9133582.017596
2667.108.86447.903.2118.597.43839.305.7730.7138130.1281120.5857021.9975681.9166552.016185
27134.217.72895.695.43716.509.86979.185.5680.7129870.1230080.5899781.9976831.9203242.014604
28268.435.456191.190.34631.752.632159.437.7140.7122400.1182880.5939521.9979051.9232522.013469
29536.870.912382.002.20961.169.781320.832.4280.7115350.1139380.5975971.9980201.9264482.012274
301.073.741.824763.306.288117.980.853645.325.4350.7108840.1098780.6010061.9981731.9287442.011410
312.147.483.6481.525.313.918227.854.8901.297.459.0280.7102800.1061030.6041761.9982991.9312872.010550
324.294.967.2963.048.186.811440.566.1992.607.620.6120.7097110.1025770.6071341.9984001.9335382.009790
338.589.934.5926.091.822.557852.832.9765.238.989.5810.7091810.0992830.6098991.9985071.9357662.009107
3417.179.869.18412.175.056.4371.652.606.94310.522.449.4940.7086820.0961940.6124871.9985901.9377852.008488
3534.359.738.36824.333.987.0653.205.557.87921.128.429.1860.7082120.0932940.6149181.9986751.9396982.007938
3668.719.476.73648.637.720.4036.223.448.40442.414.271.9990.7077720.0905630.6172091.9987571.9414562.007450


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
123031011
245051022
388351133
41613581246
5322211111579
664381523181019
71286831371171733
825612059611293357
95122151091061585898
101.024376192184194105176
112.0486763483281173177325
124.0961.2376306071303335598
138.1922.2971.1711.12615886081.100
1416.3844.1982.1092.08911.0671.0902.040
1532.7687.7453.8953.85011.9642.0023.778
1665.53614.4207.2117.20913.6633.6517.105
17131.07226.76713.39313.37416.8376.74213.187
18262.14450.18825.24124.947112.77212.72424.691
19524.28894.80547.66847.137124.09623.96146.747
201.048.576178.96589.74589.220145.34745.30788.310
212.097.152339.340170.429168.911185.89085.816167.633
224.194.304645.071323.672321.3991163.208162.873318.989
238.388.6081.228.135616.521611.6141310.496310.117607.521
2416.777.2162.344.3851.176.9171.167.4681592.572592.6561.159.156
2533.554.4324.485.6482.251.1522.234.49611.133.9511.133.5922.218.104
2667.108.8648.597.4384.314.2614.283.17712.173.2732.170.6564.253.508
27134.217.72816.509.8698.281.7308.228.13914.168.6184.168.2308.173.020
28268.435.45631.752.63215.925.89715.826.73518.012.4428.015.16015.725.029
29536.870.91261.169.78130.678.03030.491.751115.431.67815.434.88330.303.219
301.073.741.824117.980.85359.166.37758.814.476129.753.70129.760.73858.466.413
312.147.483.648227.854.890114.251.471113.603.419157.453.10857.450.198112.951.583
324.294.967.296440.566.199220.900.142219.666.0571111.056.804111.053.674218.455.720
338.589.934.592852.832.976427.556.328425.276.6481214.907.934214.933.065422.991.976
3417.179.869.1841.652.606.943828.448.138824.158.8051416.336.489416.378.989819.891.464
3534.359.738.3683.205.557.8791.606.804.3731.598.753.5061807.391.830807.446.8641.590.719.184
3668.719.476.7366.223.448.4043.119.283.1873.104.165.21711.567.223.8881.567.245.4713.088.979.044


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
240000000
381100100
4163211110
5329541323
66415693534
712835171851299
825682374518222220
9512182929038474948
101.0244082012079298105113
112.048869432437186220224239
124.0961.797899898401436470490
138.1923.7401.8581.882874906980980
1416.3847.7963.8393.9571.8321.9102.0022.052
1532.76816.1498.0968.0533.8493.9804.1354.185
1665.53633.17516.71116.4648.0798.1868.4388.472
17131.07268.33034.28234.04816.64016.87017.21217.608
18262.144139.53669.75569.78134.04534.60535.13335.753
19524.288283.593141.671141.92269.35170.20271.58972.451
201.048.576576.468287.912288.556141.444142.744145.310146.970
212.097.1521.168.496583.592584.904287.062289.485294.608297.341
224.194.3042.365.2821.181.1881.184.094581.738585.958595.812601.774
238.388.6084.784.0212.391.0192.393.0021.177.4311.186.3931.205.0021.215.195
2416.777.2169.662.5484.830.5024.832.0462.380.7252.397.2882.431.1542.453.381
2533.554.43219.495.1229.745.8379.749.2854.806.2524.839.9484.904.0254.944.897
2667.108.86439.305.77319.644.45019.661.3239.698.9069.761.8509.885.4559.959.562
27134.217.72879.185.56839.584.17739.601.39119.549.15719.675.49019.907.07320.053.848
28268.435.456159.437.71479.704.61679.733.09839.377.29639.626.24940.077.50940.356.660
29536.870.912320.832.428160.387.563160.444.86579.285.07979.755.17780.626.44781.165.725
301.073.741.824645.325.435322.610.072322.715.363159.562.365160.449.343162.149.932163.163.795
312.147.483.6481.297.459.028648.626.857648.832.171320.936.678322.653.588325.966.393327.902.369
324.294.967.2962.607.620.6121.303.621.7061.303.998.906645.263.283648.629.190654.992.381658.735.758
338.589.934.5925.238.989.5812.619.198.5542.619.791.0271.296.871.0461.303.428.0121.315.722.5021.322.968.021
3417.179.869.18410.522.449.4945.260.744.6125.261.704.8822.605.687.7732.618.361.2862.642.171.7932.656.228.642
3534.359.738.36821.128.429.18610.563.358.50910.565.070.6775.233.801.5815.258.341.8535.304.494.6885.331.791.064
3668.719.476.73642.414.271.99921.205.544.42421.208.727.57510.509.820.46810.557.521.51910.646.984.39310.699.945.619


8. Check for existing Integer Sequences by OEIS

Found in Database : 137, 29, 23, 53, 191, 139, 367, 229, 19, 17, 743, 421, 41, 523, 1151, 37, 1367, 739, 43, 853,
Found in Database : 137, 29, 23, 53, 191, 139, 367, 229, 19, 17, 743, 421, 41, 523, 1151, 37, 1367, 739, 43, 853, 1823, 971, 2063, 1093, 2311, 151, 71, 149, 1483, 107, 1621, 199, 3671, 83, 3967, 4271, 2213,
Found in Database : 17, 19, 23, 29, 37, 41, 43, 53, 71, 83, 97, 107, 137, 139, 149,