Inhaltsverzeichnis

Development of
Algorithmic Constructions

12:12:38
Deutsch
29.Mar 2024

Polynom = x^2-126x+149

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) = 149 = 149
f(1) = 3 = 3
f(2) = 99 = 3*3*11
f(3) = 55 = 5*11
f(4) = 339 = 3*113
f(5) = 57 = 3*19
f(6) = 571 = 571
f(7) = 171 = 3*3*19
f(8) = 795 = 3*5*53
f(9) = 113 = 113
f(10) = 1011 = 3*337
f(11) = 279 = 3*3*31
f(12) = 1219 = 23*53
f(13) = 165 = 3*5*11
f(14) = 1419 = 3*11*43
f(15) = 379 = 379
f(16) = 1611 = 3*3*179
f(17) = 213 = 3*71
f(18) = 1795 = 5*359
f(19) = 471 = 3*157
f(20) = 1971 = 3*3*3*73
f(21) = 257 = 257
f(22) = 2139 = 3*23*31
f(23) = 555 = 3*5*37
f(24) = 2299 = 11*11*19
f(25) = 297 = 3*3*3*11
f(26) = 2451 = 3*19*43
f(27) = 631 = 631
f(28) = 2595 = 3*5*173
f(29) = 333 = 3*3*37
f(30) = 2731 = 2731
f(31) = 699 = 3*233
f(32) = 2859 = 3*953
f(33) = 365 = 5*73
f(34) = 2979 = 3*3*331
f(35) = 759 = 3*11*23
f(36) = 3091 = 11*281
f(37) = 393 = 3*131
f(38) = 3195 = 3*3*5*71
f(39) = 811 = 811
f(40) = 3291 = 3*1097
f(41) = 417 = 3*139
f(42) = 3379 = 31*109
f(43) = 855 = 3*3*5*19
f(44) = 3459 = 3*1153
f(45) = 437 = 19*23
f(46) = 3531 = 3*11*107
f(47) = 891 = 3*3*3*3*11
f(48) = 3595 = 5*719
f(49) = 453 = 3*151
f(50) = 3651 = 3*1217
f(51) = 919 = 919
f(52) = 3699 = 3*3*3*137
f(53) = 465 = 3*5*31
f(54) = 3739 = 3739
f(55) = 939 = 3*313
f(56) = 3771 = 3*3*419
f(57) = 473 = 11*43
f(58) = 3795 = 3*5*11*23
f(59) = 951 = 3*317
f(60) = 3811 = 37*103
f(61) = 477 = 3*3*53
f(62) = 3819 = 3*19*67
f(63) = 955 = 5*191
f(64) = 3819 = 3*19*67
f(65) = 477 = 3*3*53
f(66) = 3811 = 37*103
f(67) = 951 = 3*317
f(68) = 3795 = 3*5*11*23
f(69) = 473 = 11*43
f(70) = 3771 = 3*3*419
f(71) = 939 = 3*313
f(72) = 3739 = 3739
f(73) = 465 = 3*5*31
f(74) = 3699 = 3*3*3*137
f(75) = 919 = 919
f(76) = 3651 = 3*1217
f(77) = 453 = 3*151
f(78) = 3595 = 5*719
f(79) = 891 = 3*3*3*3*11
f(80) = 3531 = 3*11*107
f(81) = 437 = 19*23
f(82) = 3459 = 3*1153
f(83) = 855 = 3*3*5*19
f(84) = 3379 = 31*109
f(85) = 417 = 3*139
f(86) = 3291 = 3*1097
f(87) = 811 = 811
f(88) = 3195 = 3*3*5*71
f(89) = 393 = 3*131
f(90) = 3091 = 11*281
f(91) = 759 = 3*11*23
f(92) = 2979 = 3*3*331
f(93) = 365 = 5*73
f(94) = 2859 = 3*953
f(95) = 699 = 3*233
f(96) = 2731 = 2731
f(97) = 333 = 3*3*37
f(98) = 2595 = 3*5*173
f(99) = 631 = 631
f(100) = 2451 = 3*19*43

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^2-126x+149

f(0)=149
f(1)=3
f(2)=11
f(3)=5
f(4)=113
f(5)=19
f(6)=571
f(7)=1
f(8)=53
f(9)=1
f(10)=337
f(11)=31
f(12)=23
f(13)=1
f(14)=43
f(15)=379
f(16)=179
f(17)=71
f(18)=359
f(19)=157
f(20)=73
f(21)=257
f(22)=1
f(23)=37
f(24)=1
f(25)=1
f(26)=1
f(27)=631
f(28)=173
f(29)=1
f(30)=2731
f(31)=233
f(32)=953
f(33)=1
f(34)=331
f(35)=1
f(36)=281
f(37)=131
f(38)=1
f(39)=811
f(40)=1097
f(41)=139
f(42)=109
f(43)=1
f(44)=1153
f(45)=1
f(46)=107
f(47)=1
f(48)=719
f(49)=151
f(50)=1217
f(51)=919
f(52)=137
f(53)=1
f(54)=3739
f(55)=313
f(56)=419
f(57)=1
f(58)=1
f(59)=317
f(60)=103
f(61)=1
f(62)=67
f(63)=191
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)=1
f(91)=1
f(92)=1
f(93)=1
f(94)=1
f(95)=1
f(96)=1
f(97)=1
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-126x+149 could be written as f(y)= y^2-3820 with x=y+63

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-63
f'(x)>2x-127 with x > 62

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

149, 3, 11, 5, 113, 19, 571, 1, 53, 1, 337, 31, 23, 1, 43, 379, 179, 71, 359, 157, 73, 257, 1, 37, 1, 1, 1, 631, 173, 1, 2731, 233, 953, 1, 331, 1, 281, 131, 1, 811, 1097, 139, 109, 1, 1153, 1, 107, 1, 719, 151, 1217, 919, 137, 1, 3739, 313, 419, 1, 1, 317, 103, 1, 67, 191, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 223, 1, 941, 1, 1, 1, 503, 1, 1, 163, 1, 283, 269, 1, 2741, 1, 1, 809, 227, 1, 1, 1, 1367, 1, 1487, 1, 439, 1, 347, 1, 1, 241, 5981, 1, 709, 823, 1, 1, 1, 1, 2543, 1, 2687, 1, 8501, 727, 1, 1, 1, 1, 1, 421, 383, 1, 3607, 461, 263, 1, 787, 1, 373, 349, 12821, 1, 4447, 1, 1, 1, 1, 1223, 1, 1, 5167, 1, 16061, 1, 1, 4229, 1, 1, 17789, 1, 557, 467, 1, 1607, 1031, 829, 449, 1, 1, 881, 1951, 1, 1, 2803, 7583, 641, 1, 1, 1, 6101, 2749, 1, 2311, 1, 967, 3307, 1787, 1, 27509, 1, 409, 1429, 877, 1, 29669, 2503, 2027, 3847, 3461, 1, 1, 1, 1, 751, 1013, 1409, 6841, 1, 1, 4423, 11927, 1, 1, 1, 1, 859, 1, 1609, 1259, 1, 1, 1, 13567, 1, 41549, 1, 1, 10709, 14423, 607, 1193, 743, 1, 5683, 5101, 3863, 1, 1, 1, 12041, 853, 1, 463, 1, 16823, 6367, 1, 1, 4759, 1, 1, 2689, 6029, 2281, 55229, 4643, 1249, 1, 1733, 1, 797, 1, 19727, 1, 20063, 1, 12241, 1, 20743, 1, 1, 1063, 64301, 1, 1, 1, 1, 2789, 1, 1889, 1, 1, 1009, 1949, 70709, 2969, 4787, 18089, 8101, 3061, 74021, 1, 1, 9463, 25423, 1, 1, 1, 26183, 1, 857, 1, 7351, 617, 739, 1, 1, 6983, 84389, 3541, 1, 1, 28927, 1, 1, 1, 1, 1, 701, 1, 1291, 769, 1, 1, 1, 1, 19081, 1, 1, 12163, 32647, 1, 1, 1, 33503, 1, 1, 1423, 4483, 8647, 34807, 1, 1, 8867, 107069, 1, 1, 1, 36583, 1, 1, 1, 1973, 14143, 1997, 3181, 23041, 1, 3533, 1, 13109, 1, 3851, 10007, 13421, 15187, 8147, 10243, 11239, 1, 41687, 1, 1, 1, 5563, 10723, 8627, 16267, 1, 997, 1, 1109, 14869, 33641, 1, 5669, 27361, 3821, 46103, 17383, 1, 1, 1, 1, 47623, 1, 3209, 1, 145949, 12227, 1, 1, 4517, 12487, 1, 1, 1, 1663, 51263, 1, 155381, 1, 1, 1, 1, 1, 1, 6709, 17981, 1, 1, 1, 1543, 1, 1, 20947, 1, 1567, 8951, 7121, 1, 8629, 2141, 1, 15919, 1, 3929, 1, 1, 14947, 180221, 1, 60647, 1987, 61223, 1, 3371, 1, 62383, 1237, 1, 3163, 5153, 1, 7127, 1559, 1, 1, 5297, 1823, 65927, 4967, 937, 5569, 201389, 8429, 1, 4639, 1, 8581, 1, 3463, 23189, 26203, 70183, 17623, 1847, 2963, 1, 1, 72047, 1, 218021, 1, 73303, 1453, 1, 977, 1, 1, 1, 11329, 1039, 9521, 229469, 1, 15427, 1, 77783, 1, 21391, 1, 79087, 59561, 1399, 10009, 2539, 20183, 9007, 1, 81727, 1, 1, 1, 83063, 62549, 16747, 1, 253229, 21187, 85087, 1, 1, 1, 1163, 1, 1, 65609, 2833, 1, 265541, 1481, 2411, 1, 1, 1, 54361, 11369, 91303, 1297, 10223, 2309, 14639, 1, 1, 1, 1, 23623, 1, 3967, 1, 14389, 1319, 4027, 4099, 2213, 1777, 36787, 1427, 1, 297581, 1, 3701, 1, 2341, 1, 5531, 1, 4441, 1, 1, 1721, 310901, 13001, 104383, 78569, 1, 1, 28879, 1, 1871, 1, 5653, 26947, 1699, 4523, 21787, 1, 9973, 1, 10691, 1, 2099, 41863, 1, 1, 1277, 14149, 37861, 1, 1, 1, 345461, 9629, 1, 43627, 1, 1, 352589, 14741, 1, 1619, 39709, 1, 15643, 1, 2683, 45427, 6397, 1, 19319, 1, 11197, 92681, 123983, 1, 1, 1361, 1, 1, 1, 6343, 1, 1451, 42701, 1, 1, 1, 5333, 1, 1, 1, 131447, 1, 1, 1, 26627, 100169, 14887, 1, 1, 6763, 1459, 51043, 12413, 1, 4339, 5743, 1, 104021, 1, 1, 9767, 35107, 140863, 4817, 1, 35543, 2473, 17881, 1, 21589, 4657, 1, 11777, 1, 2657, 1, 1, 12289, 1, 3709, 2221, 1, 1, 1, 1, 3433, 4591, 56983, 152407, 7643, 3511, 1, 8117, 116009, 1, 2161, 1, 3557, 156967, 11807, 1, 1, 476429, 1, 1, 1, 160663, 1831, 44071, 1, 162527, 3217, 5273, 1, 98641, 1, 165343, 5407, 5039, 1, 501701, 41927, 6229, 63247, 33827, 42403, 3671, 1, 171047, 2339, 1, 7187, 1, 1, 1, 65407, 1, 1, 2053, 4409, 1, 1, 177823, 1, 1511, 1, 1, 1, 1, 1, 1, 1, 1, 2593, 1, 23029, 554189, 46307, 61909, 13967, 186727, 46807, 51199, 1, 37747, 1999, 189743, 7927, 30119, 1, 10093, 72103, 1, 1, 1, 1, 21647, 3407, 1, 4909, 2087, 5483, 2711, 2011, 3617, 1, 26083, 1, 201007, 1, 67349, 25321, 609269, 1, 1, 6977, 1, 51427, 1783, 1723, 207287, 155861, 3931, 1, 125641, 1693, 1, 7193, 1, 1, 637781, 26641, 1, 160649, 1, 1, 647429, 1, 19717, 1, 1, 1, 2557, 27449, 44027, 1, 1993, 1, 60631, 1, 1, 83983, 1, 56263, 1, 1, 1, 170441, 1, 1, 1, 57367, 1, 86467, 1, 57923, 36671, 29101, 1, 35089, 21317, 2671, 706829, 19681, 1, 1, 7673, 1, 716981, 1, 2203, 16411, 2437, 1, 145441, 1, 81181, 2953, 244687, 12263, 737501, 10267, 22453, 16879, 49627, 1, 747869, 62467, 10889, 1, 83869, 1, 1, 2879, 1, 190889, 3593, 1, 33427, 1, 257447, 96763, 258623, 21601, 1, 1, 260983, 196181, 2819, 6569, 1, 1, 1, 5233, 53147, 6053, 1, 1, 5059, 1, 269327, 1, 15313, 67783, 54347, 1, 1, 6217, 1, 6869, 4831, 1, 14557, 34649, 7247, 23201, 279023, 9533, 1, 1, 844421, 1, 282703, 2063, 1, 35569, 3023, 1931, 1, 1, 1, 3793, 1, 4021, 58027, 218069,

6. Sequence of the polynom (only primes)

149, 3, 11, 5, 113, 19, 571, 53, 337, 31, 23, 43, 379, 179, 71, 359, 157, 73, 257, 37, 631, 173, 2731, 233, 953, 331, 281, 131, 811, 1097, 139, 109, 1153, 107, 719, 151, 1217, 919, 137, 3739, 313, 419, 317, 103, 67, 191, 223, 941, 503, 163, 283, 269, 2741, 809, 227, 1367, 1487, 439, 347, 241, 5981, 709, 823, 2543, 2687, 8501, 727, 421, 383, 3607, 461, 263, 787, 373, 349, 12821, 4447, 1223, 5167, 16061, 4229, 17789, 557, 467, 1607, 1031, 829, 449, 881, 1951, 2803, 7583, 641, 6101, 2749, 2311, 967, 3307, 1787, 27509, 409, 1429, 877, 29669, 2503, 2027, 3847, 3461, 751, 1013, 1409, 6841, 4423, 11927, 859, 1609, 1259, 13567, 41549, 10709, 14423, 607, 1193, 743, 5683, 5101, 3863, 12041, 853, 463, 16823, 6367, 4759, 2689, 6029, 2281, 55229, 4643, 1249, 1733, 797, 19727, 20063, 12241, 20743, 1063, 64301, 2789, 1889, 1009, 1949, 70709, 2969, 4787, 18089, 8101, 3061, 74021, 9463, 25423, 26183, 857, 7351, 617, 739, 6983, 84389, 3541, 28927, 701, 1291, 769, 19081, 12163, 32647, 33503, 1423, 4483, 8647, 34807, 8867, 107069, 36583, 1973, 14143, 1997, 3181, 23041, 3533, 13109, 3851, 10007, 13421, 15187, 8147, 10243, 11239, 41687, 5563, 10723, 8627, 16267, 997, 1109, 14869, 33641, 5669, 27361, 3821, 46103, 17383, 47623, 3209, 145949, 12227, 4517, 12487, 1663, 51263, 155381, 6709, 17981, 1543, 20947, 1567, 8951, 7121, 8629, 2141, 15919, 3929, 14947, 180221, 60647, 1987, 61223, 3371, 62383, 1237, 3163, 5153, 7127, 1559, 5297, 1823, 65927, 4967, 937, 5569, 201389, 8429, 4639, 8581, 3463, 23189, 26203, 70183, 17623, 1847, 2963, 72047, 218021, 73303, 1453, 977, 11329, 1039, 9521, 229469, 15427, 77783, 21391, 79087, 59561, 1399, 10009, 2539, 20183, 9007, 81727, 83063, 62549, 16747, 253229, 21187, 85087, 1163, 65609, 2833, 265541, 1481, 2411, 54361, 11369, 91303, 1297, 10223, 2309, 14639, 23623, 3967, 14389, 1319, 4027, 4099, 2213, 1777, 36787, 1427, 297581, 3701, 2341, 5531, 4441, 1721, 310901, 13001, 104383, 78569, 28879, 1871, 5653, 26947, 1699, 4523, 21787, 9973, 10691, 2099, 41863, 1277, 14149, 37861, 345461, 9629, 43627, 352589, 14741, 1619, 39709, 15643, 2683, 45427, 6397, 19319, 11197, 92681, 123983, 1361, 6343, 1451, 42701, 5333, 131447, 26627, 100169, 14887, 6763, 1459, 51043, 12413, 4339, 5743, 104021, 9767, 35107, 140863, 4817, 35543, 2473, 17881, 21589, 4657, 11777, 2657, 12289, 3709, 2221, 3433, 4591, 56983, 152407, 7643, 3511, 8117, 116009, 2161, 3557, 156967, 11807, 476429, 160663, 1831, 44071, 162527, 3217, 5273, 98641, 165343, 5407, 5039, 501701, 41927, 6229, 63247, 33827, 42403, 3671, 171047, 2339, 7187, 65407, 2053, 4409, 177823, 1511, 2593, 23029, 554189, 46307, 61909, 13967, 186727, 46807, 51199, 37747, 1999, 189743, 7927, 30119, 10093, 72103, 21647, 3407, 4909, 2087, 5483, 2711, 2011, 3617, 26083, 201007, 67349, 25321, 609269, 6977, 51427, 1783, 1723, 207287, 155861, 3931, 125641, 1693, 7193, 637781, 26641, 160649, 647429, 19717, 2557, 27449, 44027, 1993, 60631, 83983, 56263, 170441, 57367, 86467, 57923, 36671, 29101, 35089, 21317, 2671, 706829, 19681, 7673, 716981, 2203, 16411, 2437, 145441, 81181, 2953, 244687, 12263, 737501, 10267, 22453, 16879, 49627, 747869, 62467, 10889, 83869, 2879, 190889, 3593, 33427, 257447, 96763, 258623, 21601, 260983, 196181, 2819, 6569, 5233, 53147, 6053, 5059, 269327, 15313, 67783, 54347, 6217, 6869, 4831, 14557, 34649, 7247, 23201, 279023, 9533, 844421, 282703, 2063, 35569, 3023, 1931, 3793, 4021, 58027, 218069,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-126x+149 and
the reducible primes which appear as divisor for the first time
p | x^2-126x+149 and p < x^2-126x+149

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)
1109270.9000000.2000000.9000000.0000000.0000000.000000
2100424380.4200000.0400000.4200004.6666672.0000005.428571
31.000514434710.5140000.0430000.51400012.23809510.75000012.394737
410.0006.0613425.7190.6061000.0342000.60610011.7918297.95348812.142250
5100.00063.2182.53060.6880.6321800.0253000.63218010.4302927.39766110.611646
61.000.000643.26820.832622.4360.6432680.0208320.64326810.1753938.23399210.256328
710.000.0006.506.979176.2446.330.7350.6506980.0176240.65069810.1155028.46025410.170901
8100.000.00065.621.3991.527.11264.094.2870.6562140.0152710.65621410.0847728.66476110.124305
91.000.000.000660.384.30113.489.324646.894.9770.6603840.0134890.66038410.0635518.83322510.092865
1010.000.000.0006.637.247.914120.720.8526.516.527.0620.6637250.0120720.66372510.0505848.94936310.073547


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)
123121.5000000.5000001.0000000.0000000.0000000.000000
245141.2500000.2500001.0000001.6666671.0000002.000000
388261.0000000.2500000.7500001.6000002.0000001.500000
416132110.8125000.1250000.6875001.6250001.0000001.833333
532233200.7187500.0937500.6250001.7692311.5000001.818182
664424380.6562500.0625000.5937501.8260871.3333331.900000
7128424380.3281250.0312500.2968751.0000001.0000001.000000
82569713840.3789060.0507810.3281252.3095243.2500002.210526
9512230232070.4492190.0449220.4042972.3711341.7692312.464286
101.024523434800.5107420.0419920.4687502.2739131.8695652.318841
112.0481.142831.0590.5576170.0405270.5170902.1835561.9302332.206250
124.0962.3891592.2300.5832520.0388180.5444342.0919441.9156632.105760
138.1924.9212924.6290.6007080.0356450.5650632.0598581.8364782.075785
1416.38410.0515249.5270.6134640.0319820.5814822.0424711.7945202.058112
1532.76820.41896319.4550.6231080.0293880.5937192.0314401.8377862.042091
1665.53641.2581.75439.5040.6295470.0267640.6027832.0206681.8213912.030532
17131.07283.0323.23779.7950.6334840.0246960.6087882.0125071.8454962.019922
18262.144167.0426.129160.9130.6372150.0233800.6138342.0117791.8934202.016580
19524.288335.66811.547324.1210.6402360.0220240.6182122.0094831.8839942.014262
201.048.576674.63221.746652.8860.6433790.0207390.6226412.0098191.8832602.014328
212.097.1521.354.48141.3261.313.1550.6458670.0197060.6261612.0077331.9003962.011308
224.194.3042.718.44278.4752.639.9670.6481270.0187100.6294172.0069991.8989262.010400
238.388.6085.454.198149.5405.304.6580.6501910.0178270.6323652.0063691.9055752.009365
2416.777.21610.940.305285.65010.654.6550.6520930.0170260.6350672.0058501.9101912.008547
2533.554.43221.938.451546.22521.392.2260.6538170.0162790.6375382.0052871.9122182.007782
2667.108.86443.981.9391.048.66942.933.2700.6553820.0156260.6397562.0047881.9198482.006957
27134.217.72888.154.1712.015.85986.138.3120.6568000.0150190.6417802.0043271.9223022.006330
28268.435.456176.663.9983.881.430172.782.5680.6581250.0144590.6436652.0040351.9254472.005873
29536.870.912353.989.3677.479.128346.510.2390.6593570.0139310.6454262.0037441.9269002.005470
301.073.741.824709.207.54714.432.379694.775.1680.6605010.0134410.6470602.0034711.9296872.005064
312.147.483.6481.420.721.74627.885.2211.392.836.5250.6615750.0129850.6485902.0032531.9321292.004730
324.294.967.2962.845.769.54353.935.5912.791.833.9520.6625820.0125580.6500242.0030451.9342002.004423
338.589.934.5925.699.642.943104.425.4325.595.217.5110.6635260.0121570.6513692.0028481.9361142.004137
3417.179.869.18411.414.502.370202.396.76011.212.105.6100.6644110.0117810.6526302.0026701.9381942.003873


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
121010010
241010010
382110110
4162110110
5323210210
6644310310
71284310310
82561331003100
95122332003200
101.0244334003400
112.0488338003800
124.0961593156031560
138.1922923289032890
1416.3845243521035210
1532.7689633960039600
1665.5361.75431.751031.7510
17131.0723.23733.234033.2340
18262.1446.12936.126036.1260
19524.28811.547311.5440311.5440
201.048.57621.746321.7430321.7430
212.097.15241.326341.3230341.3230
224.194.30478.475378.4720378.4720
238.388.608149.5403149.53703149.5370
2416.777.216285.6503285.64703285.6470
2533.554.432546.2253546.22203546.2220
2667.108.8641.048.66931.048.666031.048.6660
27134.217.7282.015.85932.015.856032.015.8560
28268.435.4563.881.43033.881.427033.881.4270
29536.870.9127.479.12837.479.125037.479.1250
301.073.741.82414.432.379314.432.3760314.432.3760
312.147.483.64827.885.221327.885.2180327.885.2180
324.294.967.29653.935.591353.935.5880353.935.5880
338.589.934.592104.425.4323104.425.42903104.425.4290
3417.179.869.184202.396.7603202.396.75703202.396.7570


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
122010200
244121201
386231311
41611642612
532209106635
664381720121259
7128381720121259
825684394416201731
951220710210440444578
101.02448025722210410399174
112.0481.059550508225229250355
124.0962.2301.1381.091491487548704
138.1924.6292.3402.2881.0691.0331.1151.412
1416.3849.5274.7694.7572.1842.1282.3152.900
1532.76819.4559.7719.6834.5064.3924.7675.790
1665.53639.50419.99619.5079.1889.1279.73211.457
17131.07279.79540.43439.36018.48618.62519.74022.944
18262.144160.91381.31779.59537.49237.89139.59945.931
19524.288324.121163.923160.19775.94476.46779.82491.886
201.048.576652.886329.854323.031153.834154.239161.151183.662
212.097.1521.313.155663.453649.701310.595311.015324.702366.843
224.194.3042.639.9671.332.5341.307.432626.444627.596652.387733.540
238.388.6085.304.6582.675.5332.629.1241.263.1351.264.1541.311.9781.465.391
2416.777.21610.654.6555.371.0895.283.5652.544.7072.545.7982.636.3312.927.819
2533.554.43221.392.22610.780.78010.611.4455.121.8025.121.8645.295.6775.852.883
2667.108.86442.933.27021.627.02321.306.24610.298.04410.299.23910.634.60111.701.386
27134.217.72886.138.31243.378.55642.759.75520.699.61020.702.24321.343.62123.392.838
28268.435.456172.782.56886.975.25685.807.31141.591.12141.600.52142.820.92546.770.001
29536.870.912346.510.239174.372.381172.137.85783.539.07683.563.90885.910.51793.496.738
301.073.741.824694.775.168349.551.178345.223.989167.744.921167.789.547172.299.771186.940.929
312.147.483.6481.392.836.525700.597.175692.239.349336.743.329336.815.304345.512.608373.765.284
324.294.967.2962.791.833.9521.403.989.7101.387.844.241675.821.607675.954.411692.727.403747.330.531
338.589.934.5925.595.217.5112.813.233.7482.781.983.7621.355.971.5291.356.186.2831.388.689.8651.494.369.834
3417.179.869.18411.212.105.6105.636.271.7615.575.833.8482.720.036.5412.720.479.3982.783.410.3202.988.179.351


8. Check for existing Integer Sequences by OEIS

Found in Database : 149, 3, 11, 5, 113, 19, 571, 1, 53, 1, 337, 31, 23, 1, 43, 379, 179, 71, 359, 157,
Found in Database : 149, 3, 11, 5, 113, 19, 571, 53, 337, 31, 23, 43, 379, 179, 71, 359, 157, 73, 257, 37, 631, 173, 2731, 233, 953, 331, 281, 131, 811,
Found in Database : 3, 5, 11, 19, 23, 31, 37, 43, 53, 67, 71, 73, 103, 107, 109, 113, 131, 137, 139, 149,