Inhaltsverzeichnis

Development of
Algorithmic Constructions

09:11:16
Deutsch
29.Mar 2024

Polynom = x^2-47

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) = 47 = 47
f(1) = 23 = 23
f(2) = 43 = 43
f(3) = 19 = 19
f(4) = 31 = 31
f(5) = 11 = 11
f(6) = 11 = 11
f(7) = 1 = 1
f(8) = 17 = 17
f(9) = 17 = 17
f(10) = 53 = 53
f(11) = 37 = 37
f(12) = 97 = 97
f(13) = 61 = 61
f(14) = 149 = 149
f(15) = 89 = 89
f(16) = 209 = 11*19
f(17) = 121 = 11*11
f(18) = 277 = 277
f(19) = 157 = 157
f(20) = 353 = 353
f(21) = 197 = 197
f(22) = 437 = 19*23
f(23) = 241 = 241
f(24) = 529 = 23*23
f(25) = 289 = 17*17
f(26) = 629 = 17*37
f(27) = 341 = 11*31
f(28) = 737 = 11*67
f(29) = 397 = 397
f(30) = 853 = 853
f(31) = 457 = 457
f(32) = 977 = 977
f(33) = 521 = 521
f(34) = 1109 = 1109
f(35) = 589 = 19*31
f(36) = 1249 = 1249
f(37) = 661 = 661
f(38) = 1397 = 11*127
f(39) = 737 = 11*67
f(40) = 1553 = 1553
f(41) = 817 = 19*43
f(42) = 1717 = 17*101
f(43) = 901 = 17*53
f(44) = 1889 = 1889
f(45) = 989 = 23*43
f(46) = 2069 = 2069
f(47) = 1081 = 23*47
f(48) = 2257 = 37*61
f(49) = 1177 = 11*107
f(50) = 2453 = 11*223
f(51) = 1277 = 1277
f(52) = 2657 = 2657
f(53) = 1381 = 1381
f(54) = 2869 = 19*151
f(55) = 1489 = 1489
f(56) = 3089 = 3089
f(57) = 1601 = 1601
f(58) = 3317 = 31*107
f(59) = 1717 = 17*101
f(60) = 3553 = 11*17*19
f(61) = 1837 = 11*167
f(62) = 3797 = 3797
f(63) = 1961 = 37*53
f(64) = 4049 = 4049
f(65) = 2089 = 2089
f(66) = 4309 = 31*139
f(67) = 2221 = 2221
f(68) = 4577 = 23*199
f(69) = 2357 = 2357
f(70) = 4853 = 23*211
f(71) = 2497 = 11*227
f(72) = 5137 = 11*467
f(73) = 2641 = 19*139
f(74) = 5429 = 61*89
f(75) = 2789 = 2789
f(76) = 5729 = 17*337
f(77) = 2941 = 17*173
f(78) = 6037 = 6037
f(79) = 3097 = 19*163
f(80) = 6353 = 6353
f(81) = 3257 = 3257
f(82) = 6677 = 11*607
f(83) = 3421 = 11*311
f(84) = 7009 = 43*163
f(85) = 3589 = 37*97
f(86) = 7349 = 7349
f(87) = 3761 = 3761
f(88) = 7697 = 43*179
f(89) = 3937 = 31*127
f(90) = 8053 = 8053
f(91) = 4117 = 23*179
f(92) = 8417 = 19*443
f(93) = 4301 = 11*17*23
f(94) = 8789 = 11*17*47
f(95) = 4489 = 67*67
f(96) = 9169 = 53*173
f(97) = 4681 = 31*151
f(98) = 9557 = 19*503
f(99) = 4877 = 4877
f(100) = 9953 = 37*269

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^2-47

f(0)=47
f(1)=23
f(2)=43
f(3)=19
f(4)=31
f(5)=11
f(6)=1
f(7)=1
f(8)=17
f(9)=1
f(10)=53
f(11)=37
f(12)=97
f(13)=61
f(14)=149
f(15)=89
f(16)=1
f(17)=1
f(18)=277
f(19)=157
f(20)=353
f(21)=197
f(22)=1
f(23)=241
f(24)=1
f(25)=1
f(26)=1
f(27)=1
f(28)=67
f(29)=397
f(30)=853
f(31)=457
f(32)=977
f(33)=521
f(34)=1109
f(35)=1
f(36)=1249
f(37)=661
f(38)=127
f(39)=1
f(40)=1553
f(41)=1
f(42)=101
f(43)=1
f(44)=1889
f(45)=1
f(46)=2069
f(47)=1
f(48)=1
f(49)=107
f(50)=223
f(51)=1277
f(52)=2657
f(53)=1381
f(54)=151
f(55)=1489
f(56)=3089
f(57)=1601
f(58)=1
f(59)=1
f(60)=1
f(61)=167
f(62)=3797
f(63)=1
f(64)=4049
f(65)=2089
f(66)=139
f(67)=2221
f(68)=199
f(69)=2357
f(70)=211
f(71)=227
f(72)=467
f(73)=1
f(74)=1
f(75)=2789
f(76)=337
f(77)=173
f(78)=6037
f(79)=163
f(80)=6353
f(81)=3257
f(82)=607
f(83)=311
f(84)=1
f(85)=1
f(86)=7349
f(87)=3761
f(88)=179
f(89)=1
f(90)=8053
f(91)=1
f(92)=443
f(93)=1
f(94)=1
f(95)=1
f(96)=1
f(97)=1
f(98)=503
f(99)=4877

b) Substitution of the polynom
The polynom f(x)=x^2-47 could be written as f(y)= y^2-47 with x=y+0

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+0
f'(x)>2x-1 with x > 7

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

47, 23, 43, 19, 31, 11, 1, 1, 17, 1, 53, 37, 97, 61, 149, 89, 1, 1, 277, 157, 353, 197, 1, 241, 1, 1, 1, 1, 67, 397, 853, 457, 977, 521, 1109, 1, 1249, 661, 127, 1, 1553, 1, 101, 1, 1889, 1, 2069, 1, 1, 107, 223, 1277, 2657, 1381, 151, 1489, 3089, 1601, 1, 1, 1, 167, 3797, 1, 4049, 2089, 139, 2221, 199, 2357, 211, 227, 467, 1, 1, 2789, 337, 173, 6037, 163, 6353, 3257, 607, 311, 1, 1, 7349, 3761, 179, 1, 8053, 1, 443, 1, 1, 1, 1, 1, 503, 4877, 269, 5077, 10357, 5281, 1, 499, 1, 5701, 11617, 1, 709, 1, 12497, 6361, 563, 599, 1, 359, 13877, 7057, 463, 7297, 401, 7541, 15329, 7789, 1439, 1, 1, 8297, 887, 1, 17377, 8821, 17909, 1, 971, 1, 1, 419, 19553, 1, 20117, 1, 1217, 617, 21269, 10781, 1987, 1, 22453, 367, 23057, 11681, 23669, 631, 1, 12301, 24917, 1, 1, 761, 1, 1, 26849, 1, 27509, 13921, 1483, 1, 1, 1327, 29537, 1, 1, 15289, 1, 15641, 1861, 941, 32353, 1487, 1, 727, 33809, 743, 34549, 919, 751, 17837, 1163, 18217, 3347, 1, 37589, 1117, 1, 19381, 39157, 19777, 39953, 20177, 769, 1871, 3779, 1, 1, 21401, 1879, 21817, 44053, 601, 1, 1, 4159, 2099, 1, 547, 1, 23957, 48353, 787, 929, 24841, 50129, 1, 4639, 25741, 1, 1, 3109, 1, 53777, 733, 54709, 587, 5059, 2551, 56597, 28537, 859, 29017, 1, 29501, 1, 29989, 3557, 1, 1, 30977, 1, 31477, 1, 31981, 2803, 613, 1523, 541, 6047, 1, 1571, 1, 68597, 1, 1, 35089, 70709, 1, 71777, 1, 1, 36697, 691, 1, 75029, 1, 76129, 1667, 77237, 1, 1, 1, 1, 1291, 1, 1097, 81749, 41161, 4363, 41737, 84053, 3847, 1, 42901, 86389, 1, 1307, 2593, 1, 1039, 3911, 2383, 8287, 1, 92369, 46489, 3019, 1, 1789, 47717, 96053, 48337, 653, 4451, 1, 2917, 99809, 50221, 1657, 50857, 5387, 2239, 2801, 2267, 9539, 4799, 5591, 53441, 2029, 1151, 1, 3221, 6481, 1, 111509, 5099, 10259, 1831, 114197, 3023, 115553, 1, 2719, 1, 1, 1, 1, 5471, 7121, 3581, 122453, 1, 1, 1021, 125269, 62989, 839, 5791, 1, 1741, 129553, 1229, 1297, 983, 6971, 3917, 7877, 2927, 1, 1, 1279, 773, 138337, 1, 2087, 70289, 1, 3739, 142837, 1, 1193, 1, 8581, 1, 147409, 1723, 148949, 74861, 150497, 1759, 1, 6947, 6679, 77201, 1, 1, 1, 78781, 158357, 1, 1, 7307, 1, 81181, 163169, 1, 164789, 2671, 166417, 83617, 1031, 84437, 15427, 1, 1, 1, 10177, 5113, 4721, 1, 176353, 4663, 178037, 1, 16339, 90289, 1, 91141, 4259, 91997, 1, 92857, 4339, 1, 1, 8599, 8263, 95461, 1, 96337, 1, 1451, 1129, 98101, 197089, 8999, 1, 99881, 200657, 1, 11909, 5981, 1301, 5399, 206069, 2797, 18899, 9491, 209717, 1, 2377, 1, 213397, 1061, 215249, 108089, 1091, 1, 1171, 3547, 1, 2579, 1, 1153, 224629, 1, 1, 1063, 1093, 10427, 230353, 115657, 10099, 1, 1, 6917, 2207, 2237, 1, 10867, 1, 6343, 1, 121501, 243989, 122489, 1103, 1, 247957, 124477, 1, 1, 14821, 126481, 253969, 1, 255989, 1, 3851, 129517, 13687, 11867, 23827, 131561, 6143, 132589, 14011, 2843, 1, 1, 5101, 135697, 24767, 1, 274529, 1, 276629, 138841, 12119, 1, 12211, 4547, 282977, 12911, 25919, 1, 1, 144161, 289397, 1, 5501, 146317, 293717, 147401, 1, 13499, 2347, 1481, 15803, 1693, 2003, 6599, 17921, 1, 1, 13999, 28099, 3607, 1, 156217, 313553, 3659, 1, 1, 1, 159589, 1, 1, 1979, 9521, 1, 162997, 1181, 1, 14323, 2467, 14423, 15131, 30367, 167597, 336353, 4561, 2273, 1, 3187, 5519, 1, 10133, 2857, 15767, 348053, 174617, 18443, 1, 3637, 176989, 2053, 1, 1, 1, 1, 1, 362357, 1, 1, 9631, 367189, 1, 4153, 1, 1, 1, 1901, 187861, 376949, 1, 12239, 190321, 10321, 1, 1, 1, 1, 3181, 1, 1, 7393, 196541, 2203, 1423, 20887, 2971, 36307, 18211, 401909, 201589, 404449, 11933, 1, 204137, 11069, 205417, 6151, 1, 37699, 9043, 7873, 1, 4157, 11083, 422453, 211877, 425057, 213181, 2287, 1, 430289, 215801, 1, 217117, 435553, 218437, 23063, 219761, 440849, 1, 1753, 2293, 1, 223757, 448853, 13241, 26561, 1783, 454229, 227789, 41539, 1, 14827, 4349, 1, 12203, 465077, 233221, 7669, 234589, 1, 1, 2531, 1, 11071, 1, 12941, 240101, 481589, 241489, 484369, 2729, 1, 1, 1, 1627, 492757, 1, 495569, 1, 1543, 1, 501217, 1, 45823, 252737, 22039, 254161, 1, 255589, 512609, 8291, 515477, 1, 47123, 23627, 30661, 15373, 4127, 13831, 1493, 2969, 2663, 1, 532853, 1, 1, 6247, 538709, 11743, 3323, 11807, 28663, 16061, 1, 274517, 50047, 2281, 29131, 4549, 2309, 278981, 18047, 2777, 1669, 7621, 1, 25771, 51679, 284989, 33617, 1, 24979, 1933, 25111, 2083, 1, 15319, 1, 1, 1, 1, 4243, 1709, 592853, 9587, 13859, 17573, 1, 27299, 1, 301841, 605237, 9787, 9973, 13259, 32183, 13327, 614609, 5813, 1, 28151, 1, 311237, 36709, 18401, 627217, 314401, 1, 16631, 633569, 28871, 1, 6791, 5039, 16883, 20747, 8713, 1, 6113, 1, 1, 3491, 1, 21163, 328837, 1, 1, 34871, 7723, 7481, 333721, 669077, 1, 3217, 336997, 1, 1, 39937, 1, 682229, 14867, 685537, 14939, 5693, 31387, 10331, 1, 695509, 1, 698849, 1, 13249, 18523, 19069, 1, 1, 5303, 712289, 356989, 15227, 358681, 719057, 1, 1, 2851, 1, 33071, 729269, 365489, 732689, 367201, 1, 21701, 739553, 3821, 1, 33851, 1, 374089, 749909, 1, 1, 377557, 756853, 19963, 24527, 16567, 69439, 1, 45137, 1, 770837, 386297, 5197, 388057, 777877, 1, 781409, 1, 71359, 393361, 788497, 395137, 41687, 7489, 2753, 1, 4903, 12919, 1, 36571, 35059, 1, 5827, 9439, 13337, 13151, 817169, 1, 820789, 1, 1, 1, 1, 7829, 831697, 416761, 1, 22031, 839009, 420421, 76607, 1, 846353, 18439, 1, 2713, 2129, 427789, 1, 1, 1, 39227, 78623, 433357, 868577, 435221, 1, 1, 23677, 438961, 879797, 1, 1, 1, 1, 26153, 1, 1, 894869, 23599, 20899, 450277, 2459, 1, 82387, 23899, 910069, 455989, 913889, 14771, 9461, 459817, 1, 1, 84127, 1, 1, 1, 1, 20327, 9277, 469457, 940853, 1, 2617, 1, 86239, 475289, 3541, 1, 1, 2111, 960353, 481157, 9941, 2699, 88019, 1, 5431, 1, 1, 8017, 42611, 1, 984017, 493001, 1, 2647, 2909, 9377, 995957, 498977,

6. Sequence of the polynom (only primes)

47, 23, 43, 19, 31, 11, 17, 53, 37, 97, 61, 149, 89, 277, 157, 353, 197, 241, 67, 397, 853, 457, 977, 521, 1109, 1249, 661, 127, 1553, 101, 1889, 2069, 107, 223, 1277, 2657, 1381, 151, 1489, 3089, 1601, 167, 3797, 4049, 2089, 139, 2221, 199, 2357, 211, 227, 467, 2789, 337, 173, 6037, 163, 6353, 3257, 607, 311, 7349, 3761, 179, 8053, 443, 503, 4877, 269, 5077, 10357, 5281, 499, 5701, 11617, 709, 12497, 6361, 563, 599, 359, 13877, 7057, 463, 7297, 401, 7541, 15329, 7789, 1439, 8297, 887, 17377, 8821, 17909, 971, 419, 19553, 20117, 1217, 617, 21269, 10781, 1987, 22453, 367, 23057, 11681, 23669, 631, 12301, 24917, 761, 26849, 27509, 13921, 1483, 1327, 29537, 15289, 15641, 1861, 941, 32353, 1487, 727, 33809, 743, 34549, 919, 751, 17837, 1163, 18217, 3347, 37589, 1117, 19381, 39157, 19777, 39953, 20177, 769, 1871, 3779, 21401, 1879, 21817, 44053, 601, 4159, 2099, 547, 23957, 48353, 787, 929, 24841, 50129, 4639, 25741, 3109, 53777, 733, 54709, 587, 5059, 2551, 56597, 28537, 859, 29017, 29501, 29989, 3557, 30977, 31477, 31981, 2803, 613, 1523, 541, 6047, 1571, 68597, 35089, 70709, 71777, 36697, 691, 75029, 76129, 1667, 77237, 1291, 1097, 81749, 41161, 4363, 41737, 84053, 3847, 42901, 86389, 1307, 2593, 1039, 3911, 2383, 8287, 92369, 46489, 3019, 1789, 47717, 96053, 48337, 653, 4451, 2917, 99809, 50221, 1657, 50857, 5387, 2239, 2801, 2267, 9539, 4799, 5591, 53441, 2029, 1151, 3221, 6481, 111509, 5099, 10259, 1831, 114197, 3023, 115553, 2719, 5471, 7121, 3581, 122453, 1021, 125269, 62989, 839, 5791, 1741, 129553, 1229, 1297, 983, 6971, 3917, 7877, 2927, 1279, 773, 138337, 2087, 70289, 3739, 142837, 1193, 8581, 147409, 1723, 148949, 74861, 150497, 1759, 6947, 6679, 77201, 78781, 158357, 7307, 81181, 163169, 164789, 2671, 166417, 83617, 1031, 84437, 15427, 10177, 5113, 4721, 176353, 4663, 178037, 16339, 90289, 91141, 4259, 91997, 92857, 4339, 8599, 8263, 95461, 96337, 1451, 1129, 98101, 197089, 8999, 99881, 200657, 11909, 5981, 1301, 5399, 206069, 2797, 18899, 9491, 209717, 2377, 213397, 1061, 215249, 108089, 1091, 1171, 3547, 2579, 1153, 224629, 1063, 1093, 10427, 230353, 115657, 10099, 6917, 2207, 2237, 10867, 6343, 121501, 243989, 122489, 1103, 247957, 124477, 14821, 126481, 253969, 255989, 3851, 129517, 13687, 11867, 23827, 131561, 6143, 132589, 14011, 2843, 5101, 135697, 24767, 274529, 276629, 138841, 12119, 12211, 4547, 282977, 12911, 25919, 144161, 289397, 5501, 146317, 293717, 147401, 13499, 2347, 1481, 15803, 1693, 2003, 6599, 17921, 13999, 28099, 3607, 156217, 313553, 3659, 159589, 1979, 9521, 162997, 1181, 14323, 2467, 14423, 15131, 30367, 167597, 336353, 4561, 2273, 3187, 5519, 10133, 2857, 15767, 348053, 174617, 18443, 3637, 176989, 2053, 362357, 9631, 367189, 4153, 1901, 187861, 376949, 12239, 190321, 10321, 3181, 7393, 196541, 2203, 1423, 20887, 2971, 36307, 18211, 401909, 201589, 404449, 11933, 204137, 11069, 205417, 6151, 37699, 9043, 7873, 4157, 11083, 422453, 211877, 425057, 213181, 2287, 430289, 215801, 217117, 435553, 218437, 23063, 219761, 440849, 1753, 2293, 223757, 448853, 13241, 26561, 1783, 454229, 227789, 41539, 14827, 4349, 12203, 465077, 233221, 7669, 234589, 2531, 11071, 12941, 240101, 481589, 241489, 484369, 2729, 1627, 492757, 495569, 1543, 501217, 45823, 252737, 22039, 254161, 255589, 512609, 8291, 515477, 47123, 23627, 30661, 15373, 4127, 13831, 1493, 2969, 2663, 532853, 6247, 538709, 11743, 3323, 11807, 28663, 16061, 274517, 50047, 2281, 29131, 4549, 2309, 278981, 18047, 2777, 1669, 7621, 25771, 51679, 284989, 33617, 24979, 1933, 25111, 2083, 15319, 4243, 1709, 592853, 9587, 13859, 17573, 27299, 301841, 605237, 9787, 9973, 13259, 32183, 13327, 614609, 5813, 28151, 311237, 36709, 18401, 627217, 314401, 16631, 633569, 28871, 6791, 5039, 16883, 20747, 8713, 6113, 3491, 21163, 328837, 34871, 7723, 7481, 333721, 669077, 3217, 336997, 39937, 682229, 14867, 685537, 14939, 5693, 31387, 10331, 695509, 698849, 13249, 18523, 19069, 5303, 712289, 356989, 15227, 358681, 719057, 2851, 33071, 729269, 365489, 732689, 367201, 21701, 739553, 3821, 33851, 374089, 749909, 377557, 756853, 19963, 24527, 16567, 69439, 45137, 770837, 386297, 5197, 388057, 777877, 781409, 71359, 393361, 788497, 395137, 41687, 7489, 2753, 4903, 12919, 36571, 35059, 5827, 9439, 13337, 13151, 817169, 820789, 7829, 831697, 416761, 22031, 839009, 420421, 76607, 846353, 18439, 2713, 2129, 427789, 39227, 78623, 433357, 868577, 435221, 23677, 438961, 879797, 26153, 894869, 23599, 20899, 450277, 2459, 82387, 23899, 910069, 455989, 913889, 14771, 9461, 459817, 84127, 20327, 9277, 469457, 940853, 2617, 86239, 475289, 3541, 2111, 960353, 481157, 9941, 2699, 88019, 5431, 8017, 42611, 984017, 493001, 2647, 2909, 9377, 995957, 498977,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2x-47 and
the reducible primes which appear as divisor for the first time
p | x^2x-47 and p < x^2x-47

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)
1108800.8000000.8000000.8000000.0000000.0000000.000000
21006948210.6900000.4800000.6900008.6250006.000000inf
31.0007063183880.7060000.3180000.70600010.2318846.62500018.476191
410.0007.1592.2234.9360.7159000.2223000.71590010.1402266.99056612.721649
5100.00071.04017.33653.7040.7104000.1733600.7104009.9231747.79847010.880065
61.000.000707.058141.016566.0420.7070580.1410160.7070589.9529568.13428710.540034
710.000.0007.049.4921.191.9965.857.4960.7049490.1192000.7049499.9701758.45291310.348165
8100.000.00070.324.87710.315.66060.009.2170.7032490.1031570.7032499.9758798.65410610.244858
91.000.000.000701.992.63790.912.540611.080.0970.7019930.0909130.7019939.9821388.81306210.183105
1010.000.000.0007.010.059.102812.747.8856.197.311.2170.7010060.0812750.7010069.9859448.93988710.141568


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)
123301.5000001.5000000.0000000.0000000.0000000.000000
245501.2500001.2500000.0000001.6666671.666667-nan
387700.8750000.8750000.0000001.4000001.400000-nan
416131300.8125000.8125000.0000001.8571431.857143-nan
532232210.7187500.6875000.0312501.7692311.692308inf
664443770.6875000.5781250.1093751.9130431.6818187.000000
71289061290.7031250.4765620.2265622.0454551.6486494.142857
8256181110710.7070310.4296880.2773442.0111111.8032792.448276
95123621851770.7070310.3613280.3457032.0000001.6818182.492958
101.0247233253980.7060550.3173830.3886721.9972381.7567572.248588
112.0481.4615738880.7133790.2797850.4335942.0207471.7630772.231156
124.0962.9231.0411.8820.7136230.2541500.4594732.0006841.8167542.119369
138.1925.8801.8694.0110.7177730.2281490.4896242.0116321.7953892.131243
1416.38411.7103.4308.2800.7147220.2093510.5053711.9914971.8352062.064323
1532.76823.3556.38016.9750.7127380.1947020.5180361.9944491.8600582.050121
1665.53646.59311.89234.7010.7109530.1814580.5294951.9949901.8639502.044241
17131.07293.06622.12370.9430.7100370.1687850.5412521.9974241.8603262.044408
18262.144185.81541.571144.2440.7088280.1585810.5502471.9965941.8790852.033238
19524.288371.17378.064293.1090.7079560.1488950.5590611.9975411.8778482.032036
201.048.576741.374147.348594.0260.7070290.1405220.5665071.9973811.8875282.026639
212.097.1521.481.520279.6831.201.8370.7064440.1333630.5730811.9983441.8981122.023206
224.194.3042.959.886531.4412.428.4450.7056920.1267050.5789861.9978711.9001552.020611
238.388.6085.914.7601.012.1444.902.6160.7050940.1206570.5844371.9983071.9045282.018829
2416.777.21611.819.7161.932.5829.887.1340.7045100.1151910.5893191.9983421.9093942.016706
2533.554.43223.622.5063.698.35519.924.1510.7040060.1102200.5937861.9985681.9136862.015160
2667.108.86447.212.7977.089.09540.123.7020.7035250.1056360.5978901.9986361.9168242.013822
27134.217.72894.364.07413.609.33380.754.7410.7030670.1013970.6016701.9986971.9197562.012644
28268.435.456188.621.69826.177.621162.444.0770.7026710.0975190.6051511.9988721.9235052.011573
29536.870.912377.044.44350.417.784326.626.6590.7023000.0939100.6083901.9989451.9259882.010702
301.073.741.824753.722.16297.261.479656.460.6830.7019580.0905820.6113771.9990271.9291112.009820
312.147.483.6481.506.751.752187.831.8641.318.919.8880.7016360.0874660.6141701.9990811.9312052.009138
324.294.967.2963.012.237.361363.195.0822.649.042.2790.7013410.0845630.6167781.9991601.9336182.008494
338.589.934.5926.022.089.441703.068.8785.319.020.5630.7010630.0818480.6192151.9992081.9357892.007903
3417.179.869.18412.039.735.4881.362.370.04810.677.365.4400.7008050.0793000.6215041.9992621.9377482.007393
3534.359.738.36824.071.154.0882.642.526.25821.428.627.8300.7005630.0769080.6236551.9993091.9396542.006921
3668.719.476.73648.126.644.7015.130.296.23342.996.348.4680.7003350.0746560.6256791.9993491.9414362.006491


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
123120102
245320203
387341303
41613673343
5322212107393
664371621163153
7128612932273283
82561105159523523
95121859095883913
101.02432515117415731623
112.04857328229127732903
124.0961.04151153050935263
138.1921.86993393692039433
1416.3843.4301.7311.6991.68531.7393
1532.7686.3803.2073.1733.14433.2303
1665.53611.8925.9595.9335.82936.0573
17131.07222.12311.08611.03710.921311.1963
18262.14441.57120.83120.74020.670320.8953
19524.28878.06439.26338.80139.002339.0563
201.048.576147.34874.06773.28173.698373.6443
212.097.152279.683140.425139.258139.6473140.0303
224.194.304531.441266.712264.729265.2303266.2053
238.388.6081.012.144507.864504.280505.6763506.4623
2416.777.2161.932.582970.065962.517966.7833965.7933
2533.554.4323.698.3551.856.8711.841.4841.849.50431.848.8453
2667.108.8647.089.0953.558.2243.530.8713.545.10933.543.9803
27134.217.72813.609.3336.829.9456.779.3886.805.25736.804.0703
28268.435.45626.177.62113.134.02713.043.59413.087.399313.090.2163
29536.870.91250.417.78425.289.46025.128.32425.209.532325.208.2463
301.073.741.82497.261.47948.774.04848.487.43148.634.097348.627.3763
312.147.483.648187.831.86494.187.44093.644.42493.919.262393.912.5963
324.294.967.296363.195.082182.106.181181.088.901181.599.1433181.595.9333
338.589.934.592703.068.878352.474.395350.594.483351.524.5093351.544.3633
3417.179.869.1841.362.370.048682.948.834679.421.214681.183.1743681.186.8683
3534.359.738.3682.642.526.2581.324.585.5311.317.940.7271.321.250.53831.321.275.7143
3668.719.476.7365.130.296.2332.571.386.3212.558.909.9122.565.125.44932.565.170.7783


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
380000000
4160000000
5321100100
6647430214
7128291316211412
82567137348261126
9512177888922613460
101.0243981972015713577129
112.048888446442167290169262
124.0961.882942940361567380574
138.1924.0112.0251.9868181.1858461.162
1416.3848.2804.1504.1301.7312.3831.7922.374
1532.76816.9758.4648.5113.6764.7973.7074.795
1665.53634.70117.35817.3437.6249.6927.6949.691
17131.07270.94335.45935.48415.99519.53315.88019.535
18262.144144.24472.23372.01132.68839.52132.56139.474
19524.288293.109146.343146.76666.93079.60366.91379.663
201.048.576594.026296.859297.167136.683160.126136.562160.655
212.097.1521.201.837600.883600.954278.457322.604278.094322.682
224.194.3042.428.4451.213.9501.214.495564.048649.925564.481649.991
238.388.6084.902.6162.451.6342.450.9821.144.1301.306.3031.144.7161.307.467
2416.777.2169.887.1344.943.6294.943.5052.317.9142.624.0462.319.1132.626.061
2533.554.43219.924.1519.960.7709.963.3814.689.3575.272.3664.689.3235.273.105
2667.108.86440.123.70220.056.57520.067.1279.472.33710.586.7809.475.21010.589.375
27134.217.72880.754.74140.369.03040.385.71119.119.45221.250.98019.121.51821.262.791
28268.435.456162.444.07781.209.24581.234.83238.566.64442.648.98438.564.56742.663.882
29536.870.912326.626.659163.284.874163.341.78577.727.50785.580.75577.723.33585.595.062
301.073.741.824656.460.683328.196.192328.264.491156.544.953171.682.610156.554.821171.678.299
312.147.483.6481.318.919.888659.405.641659.514.247315.160.517344.291.807315.163.315344.304.249
324.294.967.2962.649.042.2791.324.436.2341.324.606.045634.168.413690.372.816634.143.315690.357.735
338.589.934.5925.319.020.5632.659.356.0782.659.664.4851.275.456.0741.384.051.6201.275.467.1441.384.045.725
3417.179.869.18410.677.365.4405.338.404.5305.338.960.9102.564.345.6162.774.342.6212.564.392.0752.774.285.128
3534.359.738.36821.428.627.83010.713.780.95910.714.846.8715.153.893.9555.560.406.4175.153.935.6095.560.391.849
3668.719.476.73642.996.348.46821.497.041.19221.499.307.27610.355.118.65311.143.073.64410.355.092.45911.143.063.712


8. Check for existing Integer Sequences by OEIS

Found in Database : 47, 23, 43, 19, 31, 11, 1, 1, 17, 1, 53, 37, 97, 61, 149, 89, 1, 1, 277, 157,
Found in Database : 47, 23, 43, 19, 31, 11, 17, 53, 37, 97, 61, 149, 89, 277, 157, 353, 197, 241, 67, 397, 853, 457, 977, 521, 1109, 1249, 661, 127,
Found in Database : 11, 17, 19, 23, 31, 37, 43, 47, 53, 61, 67, 89, 97, 101, 107, 127, 139, 149,