Inhaltsverzeichnis

Development of
Algorithmic Constructions

16:41:48
Deutsch
29.Mar 2024

Polynom = x^2-88x+43

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) = 43 = 43
f(1) = 11 = 11
f(2) = 129 = 3*43
f(3) = 53 = 53
f(4) = 293 = 293
f(5) = 93 = 3*31
f(6) = 449 = 449
f(7) = 131 = 131
f(8) = 597 = 3*199
f(9) = 167 = 167
f(10) = 737 = 11*67
f(11) = 201 = 3*67
f(12) = 869 = 11*79
f(13) = 233 = 233
f(14) = 993 = 3*331
f(15) = 263 = 263
f(16) = 1109 = 1109
f(17) = 291 = 3*97
f(18) = 1217 = 1217
f(19) = 317 = 317
f(20) = 1317 = 3*439
f(21) = 341 = 11*31
f(22) = 1409 = 1409
f(23) = 363 = 3*11*11
f(24) = 1493 = 1493
f(25) = 383 = 383
f(26) = 1569 = 3*523
f(27) = 401 = 401
f(28) = 1637 = 1637
f(29) = 417 = 3*139
f(30) = 1697 = 1697
f(31) = 431 = 431
f(32) = 1749 = 3*11*53
f(33) = 443 = 443
f(34) = 1793 = 11*163
f(35) = 453 = 3*151
f(36) = 1829 = 31*59
f(37) = 461 = 461
f(38) = 1857 = 3*619
f(39) = 467 = 467
f(40) = 1877 = 1877
f(41) = 471 = 3*157
f(42) = 1889 = 1889
f(43) = 473 = 11*43
f(44) = 1893 = 3*631
f(45) = 473 = 11*43
f(46) = 1889 = 1889
f(47) = 471 = 3*157
f(48) = 1877 = 1877
f(49) = 467 = 467
f(50) = 1857 = 3*619
f(51) = 461 = 461
f(52) = 1829 = 31*59
f(53) = 453 = 3*151
f(54) = 1793 = 11*163
f(55) = 443 = 443
f(56) = 1749 = 3*11*53
f(57) = 431 = 431
f(58) = 1697 = 1697
f(59) = 417 = 3*139
f(60) = 1637 = 1637
f(61) = 401 = 401
f(62) = 1569 = 3*523
f(63) = 383 = 383
f(64) = 1493 = 1493
f(65) = 363 = 3*11*11
f(66) = 1409 = 1409
f(67) = 341 = 11*31
f(68) = 1317 = 3*439
f(69) = 317 = 317
f(70) = 1217 = 1217
f(71) = 291 = 3*97
f(72) = 1109 = 1109
f(73) = 263 = 263
f(74) = 993 = 3*331
f(75) = 233 = 233
f(76) = 869 = 11*79
f(77) = 201 = 3*67
f(78) = 737 = 11*67
f(79) = 167 = 167
f(80) = 597 = 3*199
f(81) = 131 = 131
f(82) = 449 = 449
f(83) = 93 = 3*31
f(84) = 293 = 293
f(85) = 53 = 53
f(86) = 129 = 3*43
f(87) = 11 = 11
f(88) = 43 = 43
f(89) = 33 = 3*11
f(90) = 223 = 223
f(91) = 79 = 79
f(92) = 411 = 3*137
f(93) = 127 = 127
f(94) = 607 = 607
f(95) = 177 = 3*59
f(96) = 811 = 811
f(97) = 229 = 229
f(98) = 1023 = 3*11*31
f(99) = 283 = 283
f(100) = 1243 = 11*113

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^2-88x+43

f(0)=43
f(1)=11
f(2)=3
f(3)=53
f(4)=293
f(5)=31
f(6)=449
f(7)=131
f(8)=199
f(9)=167
f(10)=67
f(11)=1
f(12)=79
f(13)=233
f(14)=331
f(15)=263
f(16)=1109
f(17)=97
f(18)=1217
f(19)=317
f(20)=439
f(21)=1
f(22)=1409
f(23)=1
f(24)=1493
f(25)=383
f(26)=523
f(27)=401
f(28)=1637
f(29)=139
f(30)=1697
f(31)=431
f(32)=1
f(33)=443
f(34)=163
f(35)=151
f(36)=59
f(37)=461
f(38)=619
f(39)=467
f(40)=1877
f(41)=157
f(42)=1889
f(43)=1
f(44)=631
f(45)=1
f(46)=1
f(47)=1
f(48)=1
f(49)=1
f(50)=1
f(51)=1
f(52)=1
f(53)=1
f(54)=1
f(55)=1
f(56)=1
f(57)=1
f(58)=1
f(59)=1
f(60)=1
f(61)=1
f(62)=1
f(63)=1
f(64)=1
f(65)=1
f(66)=1
f(67)=1
f(68)=1
f(69)=1
f(70)=1
f(71)=1
f(72)=1
f(73)=1
f(74)=1
f(75)=1
f(76)=1
f(77)=1
f(78)=1
f(79)=1
f(80)=1
f(81)=1
f(82)=1
f(83)=1
f(84)=1
f(85)=1
f(86)=1
f(87)=1
f(88)=1
f(89)=1
f(90)=223
f(91)=1
f(92)=137
f(93)=127
f(94)=607
f(95)=1
f(96)=811
f(97)=229
f(98)=1
f(99)=283

b) Substitution of the polynom
The polynom f(x)=x^2-88x+43 could be written as f(y)= y^2-1893 with x=y+44

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-44
f'(x)>2x-89 with x > 44

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

43, 11, 3, 53, 293, 31, 449, 131, 199, 167, 67, 1, 79, 233, 331, 263, 1109, 97, 1217, 317, 439, 1, 1409, 1, 1493, 383, 523, 401, 1637, 139, 1697, 431, 1, 443, 163, 151, 59, 461, 619, 467, 1877, 157, 1889, 1, 631, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 223, 1, 137, 127, 607, 1, 811, 229, 1, 283, 113, 1, 1471, 397, 569, 457, 1951, 173, 2203, 1, 821, 1, 2731, 239, 1, 787, 1097, 859, 3583, 311, 353, 1009, 1, 1087, 4507, 389, 4831, 1249, 1721, 1, 5503, 1, 5851, 1, 2069, 1597, 6571, 563, 1, 1783, 2441, 1879, 701, 659, 1, 1, 2837, 2179, 8923, 761, 9343, 2389, 3257, 227, 1, 1, 10651, 2719, 3701, 2833, 373, 983, 1, 3067, 379, 3187, 1181, 1103, 1, 3433, 1, 3559, 337, 1229, 349, 347, 1, 359, 16063, 1361, 16603, 4219, 5717, 4357, 17707, 1499, 1, 4639, 571, 4783, 19423, 1, 20011, 5077, 6869, 5227, 21211, 1, 1, 503, 7481, 5689, 23071, 1949, 1, 6007, 8117, 1, 2273, 2111, 2333, 1, 1, 1, 1, 1, 27691, 1, 9461, 653, 1, 1, 1, 7537, 10169, 7717, 31231, 2633, 31963, 1, 991, 8269, 3041, 2819, 433, 8647, 11657, 8839, 1153, 3011, 36523, 839, 12437, 857, 719, 3209, 1, 9829, 13241, 1, 40543, 3413, 3761, 1, 1279, 10657, 43051, 3623, 1021, 11083, 1, 11299, 45631, 1, 46507, 1, 15797, 11959, 911, 1, 1, 12409, 1, 12637, 4637, 4289, 4721, 13099, 1, 1, 1, 4523, 54751, 13807, 599, 1277, 56671, 1, 977, 14533, 19541, 14779, 59611, 5009, 60607, 15277, 1867, 1, 5693, 5261, 2053, 1, 21557, 1, 65707, 5519, 66751, 1, 1, 1553, 68863, 5783, 69931, 1, 23669, 577, 72091, 6053, 6653, 18433, 2251, 1, 75391, 6329, 76507, 19267, 1, 1, 997, 601, 79903, 1, 27017, 20407, 82207, 6899, 1, 1, 28181, 21283, 7793, 7193, 7901, 1, 683, 1, 1, 1, 90523, 22783, 1, 2099, 1, 709, 94207, 1, 31817, 24019, 96703, 8111, 1, 1, 1, 24967, 9137, 8429, 743, 25609, 34361, 25933, 104383, 8753, 105691, 2417, 673, 2447, 853, 1, 109663, 27583, 1, 27919, 1, 9419, 10337, 28597, 1, 1, 116443, 1, 1997, 29629, 1, 967, 120607, 919, 122011, 2789, 41141, 31033, 953, 10463, 126271, 1, 42569, 1, 1, 1, 1, 32833, 44021, 33199, 1993, 1, 135007, 33937, 45497, 3119, 137983, 1051, 139483, 35059, 46997, 35437, 4597, 11939, 144031, 1, 1, 36583, 1, 12323, 3457, 37357, 50069, 37747, 151771, 12713, 153343, 1, 1, 3539, 156511, 13109, 1, 39727, 1, 40129, 161323, 1, 14813, 40939, 4987, 1, 166207, 1, 3167, 42169, 56501, 1, 171163, 1303, 1, 3947, 58169, 1, 176191, 14753, 177883, 44683, 1931, 1, 16481, 1, 1, 1069, 1, 46399, 937, 1, 1, 47269, 63317, 4337, 191707, 1459, 1, 48589, 1, 49033, 197023, 16493, 198811, 49927, 6079, 50377, 18401, 16943, 1, 51283, 1, 1669, 2143, 1, 209707, 4787, 1, 1, 3617, 1, 1289, 54049, 1, 54517, 1, 18329, 1, 1789, 1, 55933, 224683, 18803, 1, 1, 1291, 57367, 230431, 1753, 232363, 5303, 78101, 1, 7621, 1, 238207, 59797, 1, 60289, 22013, 20261, 22193, 1, 82037, 1, 1811, 20759, 3733, 1, 2711, 1, 3217, 1933, 2267, 1, 86069, 1, 2683, 21773, 262303, 1531, 8011, 1543, 1, 1, 268507, 1, 90197, 1151, 272683, 22811, 274783, 6269, 92297, 6317, 279007, 23339, 281131, 70549, 94421, 2293, 1709, 23873, 26141, 1223, 8779, 1, 5507, 24413, 294043, 73783, 1, 1, 298411, 2269, 9697, 6857, 100937, 75979, 305023, 1, 307243, 1, 2399, 77647, 1, 1, 28541, 1, 105401, 79333, 10273, 26633, 1, 1201, 1607, 1, 1433, 2473, 327583, 82183, 109961, 82759, 332191, 27779, 334507, 2707, 1, 84499, 2803, 1, 2689, 85669, 114617, 86257, 346207, 28949, 348571, 7949, 116981, 1, 2579, 1, 6029, 89227, 1511, 2089, 360511, 1, 32993, 91033, 11071, 91639, 367771, 1, 370207, 92857, 4007, 1, 375103, 2851, 377563, 8609, 1, 95317, 382507, 1, 3407, 1, 2437, 1, 1, 32603, 1, 1, 1, 1, 1, 1, 1747, 100333, 2003, 1, 1, 3079, 407707, 1733, 1, 3319, 9601, 34511, 9661, 104179, 1, 104827, 38237, 35159, 4363, 106129, 141941, 106783, 1, 1, 1, 1, 144569, 9887, 2677, 36473, 5557, 110083, 147221, 110749, 1907, 37139, 40637, 112087, 13627, 112759, 14593, 37811, 1, 2153, 152597, 1453, 3313, 3499, 463231, 10559, 1, 116833, 468703, 1, 471451, 2749, 5099, 118897, 1, 39863, 43613, 120283, 160841, 1, 485311, 40559, 8273, 2309, 163637, 1, 7369, 1, 5119, 124489, 166457, 125197, 3613, 41969, 505051, 1, 15391, 1, 1, 42683, 513631, 128767, 172169, 4177, 519391, 43403, 522283, 11903, 175061, 11969, 528091, 44129, 1, 133117, 4139, 1, 4751, 1, 1583, 135319, 16447, 136057, 2383, 45599, 548671, 1741, 183881, 138283, 3673, 1, 557611, 1, 186869, 140527, 18181, 47093, 10691, 1, 189881, 142789, 1, 1, 1, 144307, 192917, 145069, 3853, 48611, 584863, 4729, 195977, 13397, 8821, 1, 594091, 3463, 1, 1, 600283, 50153, 603391, 151237, 18379, 152017, 1, 1, 612763, 1, 1, 154369, 3943, 51719, 2741, 14177, 208457, 14249, 2221, 52511, 631723, 158329, 6827, 1, 3821, 53309, 1, 1657, 19531, 1, 647743, 1, 650971, 163147, 218069, 1, 657451, 4993, 21313, 15053, 5147, 166399, 1, 1, 670507, 168037, 2843, 1, 61553, 1, 5623, 170509, 1, 5527, 687007, 57389, 3469, 172999, 3919, 15803, 13151, 1, 10453, 1, 1847, 176347, 707071, 59063, 710443, 5743, 1, 1, 1, 1, 720607, 1, 241337, 1, 1, 60761, 730843, 16649, 1, 1, 1, 1, 1987, 1, 248201, 186583, 1, 62483, 6211, 188317, 1, 189187, 758491, 63353, 761983, 1, 8231, 3251, 2029, 5839, 5897, 17597, 258677, 1, 779563, 65111, 1, 1, 262217, 3719, 71837, 2129, 72161, 3371, 265781, 199783, 800923, 1, 804511, 201577, 2777, 1, 1, 6163, 1, 3049, 272981, 6619, 822571, 68699, 2381, 1, 25147, 3923, 75773, 1, 10597, 1, 280277, 1, 844507, 70529, 27361, 19319, 283961, 19403, 2699, 71453, 4967, 1, 287669, 216217, 1, 72383, 79133, 218083, 1, 219019, 1, 1, 881707, 220897, 5569, 2287, 2477, 1, 893023, 1, 298937, 224677, 900607, 75209, 15329, 7309, 2311, 227533,

6. Sequence of the polynom (only primes)

43, 11, 3, 53, 293, 31, 449, 131, 199, 167, 67, 79, 233, 331, 263, 1109, 97, 1217, 317, 439, 1409, 1493, 383, 523, 401, 1637, 139, 1697, 431, 443, 163, 151, 59, 461, 619, 467, 1877, 157, 1889, 631, 223, 137, 127, 607, 811, 229, 283, 113, 1471, 397, 569, 457, 1951, 173, 2203, 821, 2731, 239, 787, 1097, 859, 3583, 311, 353, 1009, 1087, 4507, 389, 4831, 1249, 1721, 5503, 5851, 2069, 1597, 6571, 563, 1783, 2441, 1879, 701, 659, 2837, 2179, 8923, 761, 9343, 2389, 3257, 227, 10651, 2719, 3701, 2833, 373, 983, 3067, 379, 3187, 1181, 1103, 3433, 3559, 337, 1229, 349, 347, 359, 16063, 1361, 16603, 4219, 5717, 4357, 17707, 1499, 4639, 571, 4783, 19423, 20011, 5077, 6869, 5227, 21211, 503, 7481, 5689, 23071, 1949, 6007, 8117, 2273, 2111, 2333, 27691, 9461, 653, 7537, 10169, 7717, 31231, 2633, 31963, 991, 8269, 3041, 2819, 433, 8647, 11657, 8839, 1153, 3011, 36523, 839, 12437, 857, 719, 3209, 9829, 13241, 40543, 3413, 3761, 1279, 10657, 43051, 3623, 1021, 11083, 11299, 45631, 46507, 15797, 11959, 911, 12409, 12637, 4637, 4289, 4721, 13099, 4523, 54751, 13807, 599, 1277, 56671, 977, 14533, 19541, 14779, 59611, 5009, 60607, 15277, 1867, 5693, 5261, 2053, 21557, 65707, 5519, 66751, 1553, 68863, 5783, 69931, 23669, 577, 72091, 6053, 6653, 18433, 2251, 75391, 6329, 76507, 19267, 997, 601, 79903, 27017, 20407, 82207, 6899, 28181, 21283, 7793, 7193, 7901, 683, 90523, 22783, 2099, 709, 94207, 31817, 24019, 96703, 8111, 24967, 9137, 8429, 743, 25609, 34361, 25933, 104383, 8753, 105691, 2417, 673, 2447, 853, 109663, 27583, 27919, 9419, 10337, 28597, 116443, 1997, 29629, 967, 120607, 919, 122011, 2789, 41141, 31033, 953, 10463, 126271, 42569, 32833, 44021, 33199, 1993, 135007, 33937, 45497, 3119, 137983, 1051, 139483, 35059, 46997, 35437, 4597, 11939, 144031, 36583, 12323, 3457, 37357, 50069, 37747, 151771, 12713, 153343, 3539, 156511, 13109, 39727, 40129, 161323, 14813, 40939, 4987, 166207, 3167, 42169, 56501, 171163, 1303, 3947, 58169, 176191, 14753, 177883, 44683, 1931, 16481, 1069, 46399, 937, 47269, 63317, 4337, 191707, 1459, 48589, 49033, 197023, 16493, 198811, 49927, 6079, 50377, 18401, 16943, 51283, 1669, 2143, 209707, 4787, 3617, 1289, 54049, 54517, 18329, 1789, 55933, 224683, 18803, 1291, 57367, 230431, 1753, 232363, 5303, 78101, 7621, 238207, 59797, 60289, 22013, 20261, 22193, 82037, 1811, 20759, 3733, 2711, 3217, 1933, 2267, 86069, 2683, 21773, 262303, 1531, 8011, 1543, 268507, 90197, 1151, 272683, 22811, 274783, 6269, 92297, 6317, 279007, 23339, 281131, 70549, 94421, 2293, 1709, 23873, 26141, 1223, 8779, 5507, 24413, 294043, 73783, 298411, 2269, 9697, 6857, 100937, 75979, 305023, 307243, 2399, 77647, 28541, 105401, 79333, 10273, 26633, 1201, 1607, 1433, 2473, 327583, 82183, 109961, 82759, 332191, 27779, 334507, 2707, 84499, 2803, 2689, 85669, 114617, 86257, 346207, 28949, 348571, 7949, 116981, 2579, 6029, 89227, 1511, 2089, 360511, 32993, 91033, 11071, 91639, 367771, 370207, 92857, 4007, 375103, 2851, 377563, 8609, 95317, 382507, 3407, 2437, 32603, 1747, 100333, 2003, 3079, 407707, 1733, 3319, 9601, 34511, 9661, 104179, 104827, 38237, 35159, 4363, 106129, 141941, 106783, 144569, 9887, 2677, 36473, 5557, 110083, 147221, 110749, 1907, 37139, 40637, 112087, 13627, 112759, 14593, 37811, 2153, 152597, 1453, 3313, 3499, 463231, 10559, 116833, 468703, 471451, 2749, 5099, 118897, 39863, 43613, 120283, 160841, 485311, 40559, 8273, 2309, 163637, 7369, 5119, 124489, 166457, 125197, 3613, 41969, 505051, 15391, 42683, 513631, 128767, 172169, 4177, 519391, 43403, 522283, 11903, 175061, 11969, 528091, 44129, 133117, 4139, 4751, 1583, 135319, 16447, 136057, 2383, 45599, 548671, 1741, 183881, 138283, 3673, 557611, 186869, 140527, 18181, 47093, 10691, 189881, 142789, 144307, 192917, 145069, 3853, 48611, 584863, 4729, 195977, 13397, 8821, 594091, 3463, 600283, 50153, 603391, 151237, 18379, 152017, 612763, 154369, 3943, 51719, 2741, 14177, 208457, 14249, 2221, 52511, 631723, 158329, 6827, 3821, 53309, 1657, 19531, 647743, 650971, 163147, 218069, 657451, 4993, 21313, 15053, 5147, 166399, 670507, 168037, 2843, 61553, 5623, 170509, 5527, 687007, 57389, 3469, 172999, 3919, 15803, 13151, 10453, 1847, 176347, 707071, 59063, 710443, 5743, 720607, 241337, 60761, 730843, 16649, 1987, 248201, 186583, 62483, 6211, 188317, 189187, 758491, 63353, 761983, 8231, 3251, 2029, 5839, 5897, 17597, 258677, 779563, 65111, 262217, 3719, 71837, 2129, 72161, 3371, 265781, 199783, 800923, 804511, 201577, 2777, 6163, 3049, 272981, 6619, 822571, 68699, 2381, 25147, 3923, 75773, 10597, 280277, 844507, 70529, 27361, 19319, 283961, 19403, 2699, 71453, 4967, 287669, 216217, 72383, 79133, 218083, 219019, 881707, 220897, 5569, 2287, 2477, 893023, 298937, 224677, 900607, 75209, 15329, 7309, 2311, 227533,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-88x+43 and
the reducible primes which appear as divisor for the first time
p | x^2-88x+43 and p < x^2-88x+43

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)
11011471.1000000.4000001.1000000.0000000.0000000.000000
21004415290.4400000.1500000.4400004.0000003.7500004.142857
31.0006991445550.6990000.1440000.69900015.8863649.60000019.137932
410.0007.2101.0616.1490.7210000.1061000.72100010.3147357.36805511.079279
5100.00071.8638.03063.8330.7186300.0803000.7186309.9671297.56833210.381038
61.000.000714.45266.243648.2090.7144520.0662430.7144529.9418618.24943910.154763
710.000.0007.113.903558.8766.555.0270.7113900.0558880.7113909.9571478.43675510.112521
8100.000.00070.895.4184.841.96166.053.4570.7089540.0484200.7089549.9657558.66374910.076763
91.000.000.000707.023.58742.739.892664.283.6950.7070240.0427400.7070249.9727698.82698010.056760
1010.000.000.0007.055.071.022382.480.8906.672.590.1320.7055070.0382480.7055079.9785518.94903810.044790


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
416165111.0000000.3125000.6875001.7777781.2500002.200000
5322910190.9062500.3125000.5937501.8125002.0000001.727273
6643912270.6093750.1875000.4218751.3448281.2000001.421053
71286622440.5156250.1718750.3437501.6923081.8333331.629630
8256159411180.6210940.1601560.4609382.4090911.8636362.681818
9512337822550.6582030.1601560.4980472.1194972.0000002.161017
101.0247161485680.6992190.1445310.5546882.1246291.8048782.227451
112.0481.4592751.1840.7124020.1342770.5781252.0377091.8581082.084507
124.0962.9354952.4400.7165530.1208500.5957032.0116521.8000002.060811
138.1925.9098925.0170.7213130.1088870.6124272.0132881.8020202.056148
1416.38411.8421.62710.2150.7227780.0993040.6234742.0040621.8239912.036077
1532.76823.6243.00620.6180.7209470.0917360.6292111.9949331.8475722.018404
1665.53647.1885.47241.7160.7200320.0834960.6365361.9974601.8203592.023281
17131.07294.13610.30583.8310.7182010.0786210.6395801.9949141.8832242.009565
18262.144187.97019.364168.6060.7170490.0738680.6431811.9967921.8790882.011261
19524.288375.18436.563338.6210.7156070.0697380.6458681.9959781.8881952.008357
201.048.576749.01269.219679.7930.7143140.0660120.6483011.9963861.8931432.007534
212.097.1521.496.335130.8281.365.5070.7135080.0623840.6511241.9977451.8900592.008710
224.194.3042.988.284249.0552.739.2290.7124620.0593790.6530831.9970691.9036832.006016
238.388.6085.969.437474.2955.495.1420.7116120.0565400.6550721.9976141.9043792.006091
2416.777.21611.924.237906.20511.018.0320.7107400.0540140.6567261.9975481.9106362.005049
2533.554.43223.823.7021.735.69922.088.0030.7100020.0517280.6582741.9979231.9153492.004714
2667.108.86447.603.0113.326.44044.276.5710.7093400.0495680.6597721.9981371.9164842.004553
27134.217.72895.117.7376.389.19688.728.5410.7086820.0476030.6610791.9981451.9207312.003962
28268.435.456190.070.22812.295.279177.774.9490.7080670.0458030.6622631.9982631.9243862.003582
29536.870.912379.836.93623.694.392356.142.5440.7075010.0441340.6633671.9984031.9271132.003334
301.073.741.824759.105.99945.725.084713.380.9150.7069730.0425850.6643881.9985051.9297852.003077
312.147.483.6481.517.158.84488.335.7101.428.823.1340.7064820.0411350.6653481.9986131.9318872.002889
324.294.967.2963.032.353.526170.862.9242.861.490.6020.7060250.0397820.6662431.9987061.9342452.002691
338.589.934.5926.061.037.167330.846.9305.730.190.2370.7055980.0385160.6670821.9987901.9363302.002519
3417.179.869.18412.115.184.099641.280.87811.473.903.2210.7051960.0373270.6678691.9988631.9383012.002360


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
122101100
243111110
384122110
4165132120
53210185140
664121106150
71282211106556
8256413010617513
9512827110635536
101.02414813710670567
112.0482752641061295135
124.0964954841062435241
138.1928928811064445437
1416.3841.6271.6161068035813
1532.7683.0062.9951061.47951.516
1665.5365.4725.4611062.69452.767
17131.07210.30510.2941065.10355.191
18262.14419.36419.3531069.67759.676
19524.28836.56336.55210618.269518.283
201.048.57669.21969.20810634.565534.643
212.097.152130.828130.81710665.359565.458
224.194.304249.055249.044106124.5955124.449
238.388.608474.295474.284106237.1765237.108
2416.777.216906.205906.194106453.5015452.693
2533.554.4321.735.6991.735.688106868.5315867.157
2667.108.8643.326.4403.326.4291061.663.57351.662.856
27134.217.7286.389.1966.389.1851063.195.01753.194.168
28268.435.45612.295.27912.295.2681066.149.19456.146.074
29536.870.91223.694.39223.694.38110611.848.271511.846.110
301.073.741.82445.725.08445.725.07310622.862.679522.862.394
312.147.483.64888.335.71088.335.69910644.169.792544.165.907
324.294.967.296170.862.924170.862.91310685.433.196585.429.717
338.589.934.592330.846.930330.846.919106165.420.7945165.426.125
3417.179.869.184641.280.878641.280.867106320.647.2685320.633.599


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
242020110
385230212
41611561415
532199103628
664271413310410
71284423211013813
8256118566230253231
951225512113469527460
101.024568272296155117167129
112.0481.184567617321256357250
124.0962.4401.1931.247680509725526
138.1925.0172.4462.5711.3941.1071.4041.112
1416.38410.2155.0085.2072.8122.2892.8282.286
1532.76820.61810.10010.5185.6464.6715.6554.646
1665.53641.71620.42821.28811.4249.48511.4169.391
17131.07283.83141.20042.63122.78819.14622.80419.093
18262.144168.60682.76185.84545.65338.77145.46838.714
19524.288338.621166.332172.28990.76078.46190.90978.491
201.048.576679.793334.638345.155181.794158.099181.733158.167
212.097.1521.365.507673.158692.349363.761318.995364.093318.658
224.194.3042.739.2291.350.4631.388.766727.860641.897727.840641.632
238.388.6085.495.1422.712.2392.782.9031.454.9631.291.9441.456.4421.291.793
2416.777.21611.018.0325.443.2405.574.7922.909.9812.598.8752.911.5192.597.657
2533.554.43222.088.00310.920.44711.167.5565.819.8005.224.0745.823.3125.220.817
2667.108.86444.276.57121.901.72622.374.84511.640.95210.499.15511.642.31610.494.148
27134.217.72888.728.54143.915.90244.812.63923.280.52921.088.13423.281.90421.077.974
28268.435.456177.774.94988.030.65989.744.29046.557.64242.330.83446.562.04442.324.429
29536.870.912356.142.544176.420.933179.721.61193.098.64184.984.76793.093.69784.965.439
301.073.741.824713.380.915353.504.780359.876.135186.183.138170.525.043186.164.125170.508.609
312.147.483.6481.428.823.134708.302.448720.520.686372.308.140342.122.630372.296.099342.096.265
324.294.967.2962.861.490.6021.418.965.5581.442.525.044744.570.442686.217.951744.520.841686.181.368
338.589.934.5925.730.190.2372.842.424.6682.887.765.5691.489.043.0451.376.108.8761.488.922.9661.376.115.350
3417.179.869.18411.473.903.2215.693.083.6975.780.819.5242.977.883.4902.759.132.0812.977.762.8802.759.124.770


8. Check for existing Integer Sequences by OEIS

Found in Database : 43, 11, 3, 53, 293, 31, 449, 131, 199, 167, 67, 1, 79, 233, 331, 263, 1109, 97, 1217, 317,
Found in Database : 43, 11, 3, 53, 293, 31, 449, 131, 199, 167, 67, 79, 233, 331, 263, 1109, 97, 1217, 317, 439, 1409, 1493, 383, 523, 401, 1637, 139, 1697, 431, 443, 163, 151, 59, 461, 619, 467,
Found in Database : 3, 11, 31, 43, 53, 59, 67, 79, 97, 113, 127, 131, 137, 139,