Inhaltsverzeichnis

Development of
Algorithmic Constructions

08:44:29
Deutsch
29.Mar 2024

Polynom = x^2-37x+5

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) = 5 = 5
f(1) = 31 = 31
f(2) = 65 = 5*13
f(3) = 97 = 97
f(4) = 127 = 127
f(5) = 155 = 5*31
f(6) = 181 = 181
f(7) = 205 = 5*41
f(8) = 227 = 227
f(9) = 247 = 13*19
f(10) = 265 = 5*53
f(11) = 281 = 281
f(12) = 295 = 5*59
f(13) = 307 = 307
f(14) = 317 = 317
f(15) = 325 = 5*5*13
f(16) = 331 = 331
f(17) = 335 = 5*67
f(18) = 337 = 337
f(19) = 337 = 337
f(20) = 335 = 5*67
f(21) = 331 = 331
f(22) = 325 = 5*5*13
f(23) = 317 = 317
f(24) = 307 = 307
f(25) = 295 = 5*59
f(26) = 281 = 281
f(27) = 265 = 5*53
f(28) = 247 = 13*19
f(29) = 227 = 227
f(30) = 205 = 5*41
f(31) = 181 = 181
f(32) = 155 = 5*31
f(33) = 127 = 127
f(34) = 97 = 97
f(35) = 65 = 5*13
f(36) = 31 = 31
f(37) = 5 = 5
f(38) = 43 = 43
f(39) = 83 = 83
f(40) = 125 = 5*5*5
f(41) = 169 = 13*13
f(42) = 215 = 5*43
f(43) = 263 = 263
f(44) = 313 = 313
f(45) = 365 = 5*73
f(46) = 419 = 419
f(47) = 475 = 5*5*19
f(48) = 533 = 13*41
f(49) = 593 = 593
f(50) = 655 = 5*131
f(51) = 719 = 719
f(52) = 785 = 5*157
f(53) = 853 = 853
f(54) = 923 = 13*71
f(55) = 995 = 5*199
f(56) = 1069 = 1069
f(57) = 1145 = 5*229
f(58) = 1223 = 1223
f(59) = 1303 = 1303
f(60) = 1385 = 5*277
f(61) = 1469 = 13*113
f(62) = 1555 = 5*311
f(63) = 1643 = 31*53
f(64) = 1733 = 1733
f(65) = 1825 = 5*5*73
f(66) = 1919 = 19*101
f(67) = 2015 = 5*13*31
f(68) = 2113 = 2113
f(69) = 2213 = 2213
f(70) = 2315 = 5*463
f(71) = 2419 = 41*59
f(72) = 2525 = 5*5*101
f(73) = 2633 = 2633
f(74) = 2743 = 13*211
f(75) = 2855 = 5*571
f(76) = 2969 = 2969
f(77) = 3085 = 5*617
f(78) = 3203 = 3203
f(79) = 3323 = 3323
f(80) = 3445 = 5*13*53
f(81) = 3569 = 43*83
f(82) = 3695 = 5*739
f(83) = 3823 = 3823
f(84) = 3953 = 59*67
f(85) = 4085 = 5*19*43
f(86) = 4219 = 4219
f(87) = 4355 = 5*13*67
f(88) = 4493 = 4493
f(89) = 4633 = 41*113
f(90) = 4775 = 5*5*191
f(91) = 4919 = 4919
f(92) = 5065 = 5*1013
f(93) = 5213 = 13*401
f(94) = 5363 = 31*173
f(95) = 5515 = 5*1103
f(96) = 5669 = 5669
f(97) = 5825 = 5*5*233
f(98) = 5983 = 31*193
f(99) = 6143 = 6143
f(100) = 6305 = 5*13*97

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^2-37x+5

f(0)=5
f(1)=31
f(2)=13
f(3)=97
f(4)=127
f(5)=1
f(6)=181
f(7)=41
f(8)=227
f(9)=19
f(10)=53
f(11)=281
f(12)=59
f(13)=307
f(14)=317
f(15)=1
f(16)=331
f(17)=67
f(18)=337
f(19)=1
f(20)=1
f(21)=1
f(22)=1
f(23)=1
f(24)=1
f(25)=1
f(26)=1
f(27)=1
f(28)=1
f(29)=1
f(30)=1
f(31)=1
f(32)=1
f(33)=1
f(34)=1
f(35)=1
f(36)=1
f(37)=1
f(38)=43
f(39)=83
f(40)=1
f(41)=1
f(42)=1
f(43)=263
f(44)=313
f(45)=73
f(46)=419
f(47)=1
f(48)=1
f(49)=593
f(50)=131
f(51)=719
f(52)=157
f(53)=853
f(54)=71
f(55)=199
f(56)=1069
f(57)=229
f(58)=1223
f(59)=1303
f(60)=277
f(61)=113
f(62)=311
f(63)=1
f(64)=1733
f(65)=1
f(66)=101
f(67)=1
f(68)=2113
f(69)=2213
f(70)=463
f(71)=1
f(72)=1
f(73)=2633
f(74)=211
f(75)=571
f(76)=2969
f(77)=617
f(78)=3203
f(79)=3323
f(80)=1
f(81)=1
f(82)=739
f(83)=3823
f(84)=1
f(85)=1
f(86)=4219
f(87)=1
f(88)=4493
f(89)=1
f(90)=191
f(91)=4919
f(92)=1013
f(93)=401
f(94)=173
f(95)=1103
f(96)=5669
f(97)=233
f(98)=193
f(99)=6143

b) Substitution of the polynom
The polynom f(x)=x^2-37x+5 could be written as f(y)= y^2-337.25 with x=y+18.5

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-18.5
f'(x)>2x-38 with x > 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

5, 31, 13, 97, 127, 1, 181, 41, 227, 19, 53, 281, 59, 307, 317, 1, 331, 67, 337, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 43, 83, 1, 1, 1, 263, 313, 73, 419, 1, 1, 593, 131, 719, 157, 853, 71, 199, 1069, 229, 1223, 1303, 277, 113, 311, 1, 1733, 1, 101, 1, 2113, 2213, 463, 1, 1, 2633, 211, 571, 2969, 617, 3203, 3323, 1, 1, 739, 3823, 1, 1, 4219, 1, 4493, 1, 191, 4919, 1013, 401, 173, 1103, 5669, 233, 193, 6143, 1, 6469, 1327, 6803, 367, 1429, 563, 1499, 7673, 7853, 1607, 8219, 1, 661, 8783, 359, 1, 1873, 1, 751, 1993, 10169, 1, 557, 251, 1, 863, 2287, 271, 383, 1, 1, 1, 241, 13003, 2647, 13469, 2741, 1, 1091, 577, 14669, 1, 257, 15413, 1, 15919, 647, 16433, 16693, 3391, 1, 269, 433, 1, 3659, 599, 3769, 1471, 19403, 1, 1051, 4051, 20543, 1, 1, 21419, 1, 22013, 421, 4523, 1, 929, 1, 1, 4831, 24469, 4957, 1931, 25423, 1, 1, 5279, 26723, 2081, 5477, 523, 1, 28393, 487, 1163, 1, 5953, 30113, 743, 6163, 439, 1, 31883, 1697, 6521, 32969, 1, 33703, 2621, 1, 34819, 7039, 35573, 1, 1, 503, 1, 37493, 881, 1531, 38669, 601, 1, 39863, 8053, 607, 1, 3191, 41893, 8461, 42719, 8627, 449, 43973, 683, 44819, 9049, 45673, 46103, 1, 3613, 499, 47843, 911, 1949, 49169, 9923, 3851, 50513, 10193, 51419, 1, 887, 1, 10651, 53719, 10837, 1, 1, 11119, 1, 1, 1, 57503, 11597, 991, 907, 59443, 821, 2417, 60919, 1, 613, 4801, 12583, 63419, 2557, 64433, 643, 1, 1609, 13297, 67003, 67523, 1, 1, 1063, 69623, 1, 1, 1, 1, 1, 1, 587, 1, 1, 75013, 1, 1171, 1783, 3089, 77783, 1, 1, 6113, 16007, 80603, 81173, 16349, 1, 1, 6421, 84053, 16927, 2749, 1, 4547, 6691, 1, 88169, 1, 89363, 89963, 1, 7013, 3671, 92383, 92993, 1, 94219, 1459, 1801, 1, 1, 1, 1031, 98573, 1, 1, 100469, 1, 1433, 2381, 1, 103669, 673, 2441, 105613, 1, 3449, 1, 108233, 108893, 21911, 5801, 1, 8581, 112223, 1, 1, 1, 2803, 115603, 1789, 116969, 23531, 118343, 119033, 4789, 1, 24223, 1, 1213, 1297, 1493, 997, 1, 1, 1, 3109, 827, 128903, 1, 1, 2473, 1, 132523, 133253, 1, 1, 27091, 136193, 7207, 5507, 1427, 2141, 773, 140663, 28283, 142169, 5717, 2711, 1, 1, 145969, 29347, 147503, 4783, 2293, 1, 30119, 1, 2143, 1, 153719, 2377, 1871, 1229, 1, 157669, 1, 12251, 2389, 32173, 1, 1, 1, 164093, 1, 165719, 1753, 1481, 3911, 1, 13063, 34129, 171473, 3251, 1117, 173969, 34961, 1, 5693, 1, 178169, 35803, 1373, 13901, 36313, 9601, 7331, 184133, 184993, 37171, 1, 37517, 947, 2281, 38039, 1217, 2953, 4703, 193703, 38917, 195469, 1, 197243, 15241, 1, 6449, 40163, 4691, 202613, 1, 1, 1, 206233, 1, 41611, 2069, 3229, 210803, 211723, 1, 5209, 42899, 1, 1, 43457, 3257, 1, 2269, 3299, 1, 222919, 44773, 224813, 1, 45343, 1, 1, 229583, 5623, 46301, 7499, 46687, 1, 4441, 47269, 237319, 47659, 239273, 18481, 48247, 1, 1, 2161, 245183, 1, 19013, 49633, 1291, 2579, 1, 252169, 1, 1619, 255193, 51241, 257219, 51647, 8363, 20021, 52259, 1, 1699, 1, 265403, 4099, 267469, 1, 269543, 270583, 1, 1, 4211, 4657, 275813, 55373, 277919, 11159, 1657, 1553, 56431, 3989, 56857, 4259, 6661, 4423, 1, 1093, 6761, 9413, 3083, 22613, 59011, 1, 297233, 11933, 299419, 60103, 23201, 1151, 60763, 3019, 12241, 1, 1, 61871, 4253, 1, 312703, 1, 62989, 1, 63439, 318323, 319453, 1, 1, 4967, 1153, 325133, 1, 327419, 65713, 6221, 1, 66403, 1579, 1, 17657, 336643, 5197, 7883, 1, 341303, 8353, 68729, 344819, 5323, 347173, 348353, 1319, 350719, 70381, 1, 1279, 1, 3677, 3767, 359063, 1, 1, 11699, 1, 5449, 366293, 2371, 1, 1, 371153, 5101, 74719, 374819, 75209, 29021, 378503, 1, 20051, 76441, 1249, 1, 1, 387169, 1, 389663, 9091, 1913, 1, 15787, 1, 5441, 79691, 399719, 1, 402253, 3571, 4261, 13099, 1, 408623, 31531, 82237, 6991, 1, 1, 416333, 1, 1669, 1, 3319, 1, 1, 425419, 1, 428033, 1, 86131, 1, 1, 1, 435923, 1, 438569, 1, 1, 1423, 6829, 10859, 1, 447893, 1979, 1, 34763, 1, 1, 1, 91463, 458669, 18401, 35491, 8731, 92821, 465469, 1, 5641, 1, 1, 472319, 1, 475073, 1, 1, 1, 5059, 4969, 1, 19391, 9173, 1, 15773, 1, 1667, 493169, 1, 495983, 38261, 99761, 11633, 2447, 503053, 504473, 1, 26701, 101749, 8647, 511603, 102607, 1, 7937, 517343, 4591, 20809, 521669, 104623, 40351, 2297, 1, 1, 4243, 531833, 17203, 1, 536219, 2029, 539153, 8069, 1, 41813, 1627, 546523, 548003, 109897, 550969, 110491, 42611, 7823, 22277, 4397, 1, 13693, 1, 1, 565919, 22697, 1, 1, 1567, 1, 114997, 2897, 1, 3739, 581069, 8963, 2221, 585653, 117437, 1, 118051, 31147, 45641, 4759, 596419, 2027, 599513, 1, 1, 2593, 24229, 1, 608843, 122081, 611969, 9439, 615103, 616673, 2333, 619819, 1, 1, 15233, 1, 20249, 2927, 630893, 632483, 1951, 14783, 127453, 1, 640463, 128413, 1, 1, 646883, 9679, 130021, 34301, 3187, 1, 1, 131639, 659819, 1, 1847, 51131, 1877, 5099, 1, 1, 672883, 26981, 4001, 4373, 679463, 681113, 7187, 684419, 2111, 9421, 689393, 3371, 1, 1, 696053, 1, 3253, 2909, 140549, 13291, 16421, 10889, 709469, 142231, 712843, 37607, 28649, 717919, 11071, 1, 1, 1, 726419, 1, 1811, 7243, 2767, 7577, 1, 738403, 740123, 1, 1, 149059, 39317, 1787, 150097, 4451, 150791, 11279, 757433, 30367, 1, 152533, 1, 766163, 1, 3361, 1, 17981, 59611, 5011, 778469, 1, 782003, 1, 157109, 1, 1, 2543, 19333, 2693, 1, 12277, 1, 3119, 6427, 3547, 1, 3833, 62351, 162473, 1, 1, 817783, 819593, 12637, 823219, 1, 15601, 828673, 166099, 26849, 1, 835973, 2729, 5417, 4649, 1, 65011, 846983, 1, 1, 170503, 854363, 856213, 1, 859919, 34471, 863633, 865493, 4231, 66863, 2099, 872953, 874823, 2617, 14891, 176089, 1, 1, 177217, 3301, 5741, 891743, 1, 1, 28949, 1, 901213, 15307, 181003, 69763, 36353, 1, 2879, 1, 916469, 1, 48437, 922223, 184829, 1, 185599, 2971, 1, 186757, 935719, 187531, 21851, 1, 2903, 23059, 189473, 1, 1, 1, 2063, 1, 959083, 2677,

6. Sequence of the polynom (only primes)

5, 31, 13, 97, 127, 181, 41, 227, 19, 53, 281, 59, 307, 317, 331, 67, 337, 43, 83, 263, 313, 73, 419, 593, 131, 719, 157, 853, 71, 199, 1069, 229, 1223, 1303, 277, 113, 311, 1733, 101, 2113, 2213, 463, 2633, 211, 571, 2969, 617, 3203, 3323, 739, 3823, 4219, 4493, 191, 4919, 1013, 401, 173, 1103, 5669, 233, 193, 6143, 6469, 1327, 6803, 367, 1429, 563, 1499, 7673, 7853, 1607, 8219, 661, 8783, 359, 1873, 751, 1993, 10169, 557, 251, 863, 2287, 271, 383, 241, 13003, 2647, 13469, 2741, 1091, 577, 14669, 257, 15413, 15919, 647, 16433, 16693, 3391, 269, 433, 3659, 599, 3769, 1471, 19403, 1051, 4051, 20543, 21419, 22013, 421, 4523, 929, 4831, 24469, 4957, 1931, 25423, 5279, 26723, 2081, 5477, 523, 28393, 487, 1163, 5953, 30113, 743, 6163, 439, 31883, 1697, 6521, 32969, 33703, 2621, 34819, 7039, 35573, 503, 37493, 881, 1531, 38669, 601, 39863, 8053, 607, 3191, 41893, 8461, 42719, 8627, 449, 43973, 683, 44819, 9049, 45673, 46103, 3613, 499, 47843, 911, 1949, 49169, 9923, 3851, 50513, 10193, 51419, 887, 10651, 53719, 10837, 11119, 57503, 11597, 991, 907, 59443, 821, 2417, 60919, 613, 4801, 12583, 63419, 2557, 64433, 643, 1609, 13297, 67003, 67523, 1063, 69623, 587, 75013, 1171, 1783, 3089, 77783, 6113, 16007, 80603, 81173, 16349, 6421, 84053, 16927, 2749, 4547, 6691, 88169, 89363, 89963, 7013, 3671, 92383, 92993, 94219, 1459, 1801, 1031, 98573, 100469, 1433, 2381, 103669, 673, 2441, 105613, 3449, 108233, 108893, 21911, 5801, 8581, 112223, 2803, 115603, 1789, 116969, 23531, 118343, 119033, 4789, 24223, 1213, 1297, 1493, 997, 3109, 827, 128903, 2473, 132523, 133253, 27091, 136193, 7207, 5507, 1427, 2141, 773, 140663, 28283, 142169, 5717, 2711, 145969, 29347, 147503, 4783, 2293, 30119, 2143, 153719, 2377, 1871, 1229, 157669, 12251, 2389, 32173, 164093, 165719, 1753, 1481, 3911, 13063, 34129, 171473, 3251, 1117, 173969, 34961, 5693, 178169, 35803, 1373, 13901, 36313, 9601, 7331, 184133, 184993, 37171, 37517, 947, 2281, 38039, 1217, 2953, 4703, 193703, 38917, 195469, 197243, 15241, 6449, 40163, 4691, 202613, 206233, 41611, 2069, 3229, 210803, 211723, 5209, 42899, 43457, 3257, 2269, 3299, 222919, 44773, 224813, 45343, 229583, 5623, 46301, 7499, 46687, 4441, 47269, 237319, 47659, 239273, 18481, 48247, 2161, 245183, 19013, 49633, 1291, 2579, 252169, 1619, 255193, 51241, 257219, 51647, 8363, 20021, 52259, 1699, 265403, 4099, 267469, 269543, 270583, 4211, 4657, 275813, 55373, 277919, 11159, 1657, 1553, 56431, 3989, 56857, 4259, 6661, 4423, 1093, 6761, 9413, 3083, 22613, 59011, 297233, 11933, 299419, 60103, 23201, 1151, 60763, 3019, 12241, 61871, 4253, 312703, 62989, 63439, 318323, 319453, 4967, 1153, 325133, 327419, 65713, 6221, 66403, 1579, 17657, 336643, 5197, 7883, 341303, 8353, 68729, 344819, 5323, 347173, 348353, 1319, 350719, 70381, 1279, 3677, 3767, 359063, 11699, 5449, 366293, 2371, 371153, 5101, 74719, 374819, 75209, 29021, 378503, 20051, 76441, 1249, 387169, 389663, 9091, 1913, 15787, 5441, 79691, 399719, 402253, 3571, 4261, 13099, 408623, 31531, 82237, 6991, 416333, 1669, 3319, 425419, 428033, 86131, 435923, 438569, 1423, 6829, 10859, 447893, 1979, 34763, 91463, 458669, 18401, 35491, 8731, 92821, 465469, 5641, 472319, 475073, 5059, 4969, 19391, 9173, 15773, 1667, 493169, 495983, 38261, 99761, 11633, 2447, 503053, 504473, 26701, 101749, 8647, 511603, 102607, 7937, 517343, 4591, 20809, 521669, 104623, 40351, 2297, 4243, 531833, 17203, 536219, 2029, 539153, 8069, 41813, 1627, 546523, 548003, 109897, 550969, 110491, 42611, 7823, 22277, 4397, 13693, 565919, 22697, 1567, 114997, 2897, 3739, 581069, 8963, 2221, 585653, 117437, 118051, 31147, 45641, 4759, 596419, 2027, 599513, 2593, 24229, 608843, 122081, 611969, 9439, 615103, 616673, 2333, 619819, 15233, 20249, 2927, 630893, 632483, 1951, 14783, 127453, 640463, 128413, 646883, 9679, 130021, 34301, 3187, 131639, 659819, 1847, 51131, 1877, 5099, 672883, 26981, 4001, 4373, 679463, 681113, 7187, 684419, 2111, 9421, 689393, 3371, 696053, 3253, 2909, 140549, 13291, 16421, 10889, 709469, 142231, 712843, 37607, 28649, 717919, 11071, 726419, 1811, 7243, 2767, 7577, 738403, 740123, 149059, 39317, 1787, 150097, 4451, 150791, 11279, 757433, 30367, 152533, 766163, 3361, 17981, 59611, 5011, 778469, 782003, 157109, 2543, 19333, 2693, 12277, 3119, 6427, 3547, 3833, 62351, 162473, 817783, 819593, 12637, 823219, 15601, 828673, 166099, 26849, 835973, 2729, 5417, 4649, 65011, 846983, 170503, 854363, 856213, 859919, 34471, 863633, 865493, 4231, 66863, 2099, 872953, 874823, 2617, 14891, 176089, 177217, 3301, 5741, 891743, 28949, 901213, 15307, 181003, 69763, 36353, 2879, 916469, 48437, 922223, 184829, 185599, 2971, 186757, 935719, 187531, 21851, 2903, 23059, 189473, 2063, 959083, 2677,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-37x+5 and
the reducible primes which appear as divisor for the first time
p | x^2-37x+5 and p < x^2-37x+5

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)
11010731.0000000.7000001.0000000.0000000.0000000.000000
21005736210.5700000.3600000.5700005.7000005.1428577.000000
31.0006932434500.6930000.2430000.69300012.1578956.75000021.428572
410.0007.1061.7635.3430.7106000.1763000.71060010.2539687.25514411.873333
5100.00071.34613.17358.1730.7134600.1317300.71346010.0402487.47192310.887704
61.000.000710.247107.757602.4900.7102470.1077570.7102479.9549668.18014110.356867
710.000.0007.077.012911.9786.165.0340.7077010.0911980.7077019.9641568.46328410.232592
8100.000.00070.567.2887.893.58662.673.7020.7056730.0789360.7056739.9713398.65545710.165995
91.000.000.000704.160.91269.665.265634.495.6470.7041610.0696650.7041619.9785748.82555310.123794
1010.000.000.0007.029.723.480623.400.2496.406.323.2310.7029720.0623400.7029729.9831218.94850910.096718


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
245411.2500001.0000000.2500001.6666672.0000001.000000
388621.0000000.7500000.2500001.6000001.5000002.000000
416151140.9375000.6875000.2500001.8750001.8333332.000000
532171250.5312500.3750000.1562501.1333331.0909091.250000
6643524110.5468750.3750000.1718752.0588242.0000002.200000
71288043370.6250000.3359380.2890622.2857141.7916673.363636
825617178930.6679690.3046880.3632812.1375001.8139542.513514
95123461372090.6757810.2675780.4082032.0233921.7564102.247312
101.0247122484640.6953120.2421880.4531252.0578031.8102192.220096
112.0481.4434589850.7045900.2236330.4809572.0266851.8467742.122845
124.0962.9108222.0880.7104490.2006840.5097662.0166321.7947602.119797
138.1925.8231.4894.3340.7108150.1817630.5290532.0010311.8114362.075670
1416.38411.6822.6789.0040.7130130.1634520.5495612.0061821.7985222.077527
1532.76823.4074.84118.5660.7143250.1477360.5665892.0036811.8076922.061972
1665.53646.8149.04437.7700.7143250.1380000.5763242.0000001.8682092.034364
17131.07293.54416.87076.6740.7136840.1287080.5849761.9982061.8653252.030024
18262.144186.68531.523155.1620.7121470.1202510.5918961.9956921.8685832.023659
19524.288373.05259.356313.6960.7115400.1132130.5983281.9982971.8829432.021732
201.048.576744.788112.597632.1910.7102850.1073810.6029041.9964721.8969782.015298
212.097.1521.488.033213.2271.274.8060.7095490.1016750.6078751.9979281.8937182.016489
224.194.3042.972.459405.5682.566.8910.7086890.0966950.6119951.9975761.9020482.013554
238.388.6085.938.318773.5245.164.7940.7079030.0922110.6156911.9977801.9072612.012082
2416.777.21611.864.3811.477.41310.386.9680.7071720.0880610.6191121.9979361.9099772.011110
2533.554.43223.707.6642.827.11820.880.5460.7065430.0842550.6222891.9982221.9135602.010264
2667.108.86447.376.7585.421.38641.955.3720.7059690.0807850.6251841.9983731.9176372.009304
27134.217.72894.688.63710.416.49684.272.1410.7054850.0776090.6278761.9986311.9213712.008614
28268.435.456189.243.38220.046.727169.196.6550.7049870.0746800.6303071.9985861.9245172.007741
29536.870.912378.239.74838.629.391339.610.3570.7045260.0719530.6325741.9986951.9269672.007193
301.073.741.824756.043.18674.529.722681.513.4640.7041200.0694110.6347091.9988471.9293532.006751
312.147.483.6481.511.269.540143.991.6721.367.277.8680.7037400.0670510.6366881.9989191.9320032.006238
324.294.967.2963.020.998.443278.488.2012.742.510.2420.7033810.0648410.6385401.9989811.9340572.005818
338.589.934.5926.039.114.894539.234.7285.499.880.1660.7030450.0627750.6402701.9990461.9362932.005418
3417.179.869.18412.072.827.4901.045.200.64311.027.626.8470.7027310.0608390.6418921.9991061.9383042.005067


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
122110011
244311012
386421122
41611742333
53212843333
6642412125667
712843162710111111
825678275117202219
9512137459232333735
101.0242488016857656462
112.048458152306106113124115
124.096822271551199210203210
138.1921.489493996358383376372
1416.3842.6789061.772657686680655
1532.7684.8411.6113.2301.1881.2211.2281.204
1665.5369.0442.9796.0652.2622.2772.2452.260
17131.07216.8705.60511.2654.2454.2664.2114.148
18262.14431.52310.47621.0477.9197.9267.8867.792
19524.28859.35619.77039.58614.88114.95814.78914.728
201.048.576112.59737.63874.95928.19828.28228.08828.029
212.097.152213.22771.181142.04653.51253.49153.12453.100
224.194.304405.568135.028270.540101.901101.386101.216101.065
238.388.608773.524257.968515.556194.025193.434193.033193.032
2416.777.2161.477.413492.257985.156370.094369.202369.311368.806
2533.554.4322.827.118942.0331.885.085707.321706.632707.155706.010
2667.108.8645.421.3861.806.1053.615.2811.356.9201.354.7311.355.4331.354.302
27134.217.72810.416.4963.471.3376.945.1592.606.5122.603.3622.603.8572.602.765
28268.435.45620.046.7276.679.62413.367.1035.014.4415.010.4065.011.1175.010.763
29536.870.91238.629.39112.874.79125.754.6009.659.8399.656.0609.657.3929.656.100
301.073.741.82474.529.72224.842.81649.686.90618.630.77518.634.27618.632.67818.631.993
312.147.483.648143.991.67247.999.76895.991.90435.999.79236.001.01435.997.15935.993.707
324.294.967.296278.488.20192.835.896185.652.30569.619.36769.628.53969.619.99869.620.297
338.589.934.592539.234.728179.751.819359.482.909134.810.874134.812.742134.803.798134.807.314
3417.179.869.1841.045.200.643348.408.544696.792.099261.297.544261.305.325261.306.354261.291.420


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
241100010
382111010
4164131120
5325231220
66411651352
712837191869913
825693474618251931
951220910910048535355
101.024464236228105129119111
112.048985497488223271252239
124.0962.0881.0721.016503529553503
138.1924.3342.2362.0981.0821.1201.0711.061
1416.3849.0044.6924.3122.1882.2692.2922.255
1532.76818.5669.5768.9904.5474.6404.6984.681
1665.53637.77019.43018.3409.3779.4559.4949.444
17131.07276.67439.26537.40919.03019.19119.22619.227
18262.144155.16279.34875.81438.64939.02838.73438.751
19524.288313.696160.148153.54878.27378.45378.32578.645
201.048.576632.191322.355309.836157.757158.286157.848158.300
212.097.1521.274.806648.946625.860318.609318.990318.722318.485
224.194.3042.566.8911.304.6511.262.240641.698641.472641.390642.331
238.388.6085.164.7942.624.0972.540.6971.291.4751.290.7781.290.4571.292.084
2416.777.21610.386.9685.273.1385.113.8302.596.5482.596.1672.596.7002.597.553
2533.554.43220.880.54610.591.90710.288.6395.219.8475.219.4625.219.4445.221.793
2667.108.86441.955.37221.266.43720.688.93510.490.24510.488.83110.487.62810.488.668
27134.217.72884.272.14142.690.88741.581.25421.070.89321.067.16021.066.45221.067.636
28268.435.456169.196.65585.664.57983.532.07642.293.82542.302.99142.302.06042.297.779
29536.870.912339.610.357171.859.315167.751.04284.905.02284.908.02884.896.23384.901.074
301.073.741.824681.513.464344.707.505336.805.959170.374.028170.393.846170.371.536170.374.054
312.147.483.6481.367.277.868691.267.976676.009.892341.828.675341.835.949341.814.021341.799.223
324.294.967.2962.742.510.2421.386.021.4231.356.488.819685.656.871685.631.983685.613.049685.608.339
338.589.934.5925.499.880.1662.778.506.7762.721.373.3901.375.007.9491.374.968.4301.374.964.3851.374.939.402
3417.179.869.18411.027.626.8475.569.087.3205.458.539.5272.756.955.0502.756.945.3682.756.864.7482.756.861.681


8. Check for existing Integer Sequences by OEIS

Found in Database : 5, 31, 13, 97, 127, 1, 181, 41, 227, 19, 53, 281, 59, 307, 317, 1, 331, 67, 337, 1,
Found in Database : 5, 31, 13, 97, 127, 181, 41, 227, 19, 53, 281, 59, 307, 317, 331, 67, 337, 43, 83,
Found in Database : 5, 13, 19, 31, 41, 43, 53, 59, 67, 71, 73, 83, 97, 101, 113, 127, 131,