Inhaltsverzeichnis

Development of
Algorithmic Constructions

14:39:08
Deutsch
29.Mar 2024

Polynom = x^2-52x+79

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) = 79 = 79
f(1) = 7 = 7
f(2) = 21 = 3*7
f(3) = 17 = 17
f(4) = 113 = 113
f(5) = 39 = 3*13
f(6) = 197 = 197
f(7) = 59 = 59
f(8) = 273 = 3*7*13
f(9) = 77 = 7*11
f(10) = 341 = 11*31
f(11) = 93 = 3*31
f(12) = 401 = 401
f(13) = 107 = 107
f(14) = 453 = 3*151
f(15) = 119 = 7*17
f(16) = 497 = 7*71
f(17) = 129 = 3*43
f(18) = 533 = 13*41
f(19) = 137 = 137
f(20) = 561 = 3*11*17
f(21) = 143 = 11*13
f(22) = 581 = 7*83
f(23) = 147 = 3*7*7
f(24) = 593 = 593
f(25) = 149 = 149
f(26) = 597 = 3*199
f(27) = 149 = 149
f(28) = 593 = 593
f(29) = 147 = 3*7*7
f(30) = 581 = 7*83
f(31) = 143 = 11*13
f(32) = 561 = 3*11*17
f(33) = 137 = 137
f(34) = 533 = 13*41
f(35) = 129 = 3*43
f(36) = 497 = 7*71
f(37) = 119 = 7*17
f(38) = 453 = 3*151
f(39) = 107 = 107
f(40) = 401 = 401
f(41) = 93 = 3*31
f(42) = 341 = 11*31
f(43) = 77 = 7*11
f(44) = 273 = 3*7*13
f(45) = 59 = 59
f(46) = 197 = 197
f(47) = 39 = 3*13
f(48) = 113 = 113
f(49) = 17 = 17
f(50) = 21 = 3*7
f(51) = 7 = 7
f(52) = 79 = 79
f(53) = 33 = 3*11
f(54) = 187 = 11*17
f(55) = 61 = 61
f(56) = 303 = 3*101
f(57) = 91 = 7*13
f(58) = 427 = 7*61
f(59) = 123 = 3*41
f(60) = 559 = 13*43
f(61) = 157 = 157
f(62) = 699 = 3*233
f(63) = 193 = 193
f(64) = 847 = 7*11*11
f(65) = 231 = 3*7*11
f(66) = 1003 = 17*59
f(67) = 271 = 271
f(68) = 1167 = 3*389
f(69) = 313 = 313
f(70) = 1339 = 13*103
f(71) = 357 = 3*7*17
f(72) = 1519 = 7*7*31
f(73) = 403 = 13*31
f(74) = 1707 = 3*569
f(75) = 451 = 11*41
f(76) = 1903 = 11*173
f(77) = 501 = 3*167
f(78) = 2107 = 7*7*43
f(79) = 553 = 7*79
f(80) = 2319 = 3*773
f(81) = 607 = 607
f(82) = 2539 = 2539
f(83) = 663 = 3*13*17
f(84) = 2767 = 2767
f(85) = 721 = 7*103
f(86) = 3003 = 3*7*11*13
f(87) = 781 = 11*71
f(88) = 3247 = 17*191
f(89) = 843 = 3*281
f(90) = 3499 = 3499
f(91) = 907 = 907
f(92) = 3759 = 3*7*179
f(93) = 973 = 7*139
f(94) = 4027 = 4027
f(95) = 1041 = 3*347
f(96) = 4303 = 13*331
f(97) = 1111 = 11*101
f(98) = 4587 = 3*11*139
f(99) = 1183 = 7*13*13
f(100) = 4879 = 7*17*41

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^2-52x+79

f(0)=79
f(1)=7
f(2)=3
f(3)=17
f(4)=113
f(5)=13
f(6)=197
f(7)=59
f(8)=1
f(9)=11
f(10)=31
f(11)=1
f(12)=401
f(13)=107
f(14)=151
f(15)=1
f(16)=71
f(17)=43
f(18)=41
f(19)=137
f(20)=1
f(21)=1
f(22)=83
f(23)=1
f(24)=593
f(25)=149
f(26)=199
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)=1
f(39)=1
f(40)=1
f(41)=1
f(42)=1
f(43)=1
f(44)=1
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)=61
f(56)=101
f(57)=1
f(58)=1
f(59)=1
f(60)=1
f(61)=157
f(62)=233
f(63)=193
f(64)=1
f(65)=1
f(66)=1
f(67)=271
f(68)=389
f(69)=313
f(70)=103
f(71)=1
f(72)=1
f(73)=1
f(74)=569
f(75)=1
f(76)=173
f(77)=167
f(78)=1
f(79)=1
f(80)=773
f(81)=607
f(82)=2539
f(83)=1
f(84)=2767
f(85)=1
f(86)=1
f(87)=1
f(88)=191
f(89)=281
f(90)=3499
f(91)=907
f(92)=179
f(93)=139
f(94)=4027
f(95)=347
f(96)=331
f(97)=1
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-52x+79 could be written as f(y)= y^2-597 with x=y+26

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-26
f'(x)>2x-53 with x > 24

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

79, 7, 3, 17, 113, 13, 197, 59, 1, 11, 31, 1, 401, 107, 151, 1, 71, 43, 41, 137, 1, 1, 83, 1, 593, 149, 199, 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, 61, 101, 1, 1, 1, 1, 157, 233, 193, 1, 1, 1, 271, 389, 313, 103, 1, 1, 1, 569, 1, 173, 167, 1, 1, 773, 607, 2539, 1, 2767, 1, 1, 1, 191, 281, 3499, 907, 179, 139, 4027, 347, 331, 1, 1, 1, 1, 419, 5179, 1, 1, 1, 829, 1, 557, 1, 2153, 1657, 523, 1, 1021, 1831, 1, 1, 7867, 1, 1, 1, 1, 2203, 9007, 1, 9403, 1, 467, 2503, 929, 1, 10639, 2713, 1, 1, 11503, 977, 919, 1, 4133, 1, 1, 1091, 13327, 3391, 1, 3511, 1, 1, 14779, 1, 463, 353, 15787, 1, 1, 1, 1, 4273, 17359, 1, 2557, 1, 1, 1, 613, 1607, 1151, 709, 1, 5107, 20719, 1, 1, 491, 1, 1, 317, 1901, 379, 5857, 1, 859, 1, 1, 2273, 487, 1, 1, 1, 1, 26959, 6823, 9209, 6991, 1, 1, 1, 7333, 761, 7507, 1787, 1, 4441, 1123, 10601, 1, 2957, 2741, 421, 1201, 1619, 661, 34747, 2927, 2731, 8971, 1, 1, 37039, 3119, 641, 1, 757, 1, 433, 1, 1297, 1, 1, 10357, 1, 503, 42667, 10771, 853, 1, 727, 1, 587, 1, 1181, 1, 1, 3947, 6829, 1723, 16229, 1, 1153, 1, 1, 1, 2447, 997, 1, 4397, 1, 1, 2579, 1951, 5009, 1, 1367, 1087, 19001, 2053, 1, 4871, 3467, 1, 19973, 1373, 1, 1, 601, 15601, 1613, 1, 1559, 1, 9277, 16363, 1999, 1511, 67003, 1, 9721, 1, 743, 1, 4127, 1, 5479, 1, 1, 18211, 73387, 1, 1049, 18757, 1, 2719, 1, 1, 643, 1, 26309, 1, 11437, 6719, 81199, 20443, 27449, 20731, 1, 1, 1, 21313, 1, 1, 1, 1, 12601, 1, 1753, 22501, 8237, 691, 1009, 3301, 31013, 1801, 881, 7907, 2221, 3433, 1, 2213, 1, 8219, 99259, 1, 4787, 3613, 1, 8537, 103087, 25933, 3163, 1, 1, 8861, 1289, 1583, 2777, 27241, 15661, 1, 1, 1, 1, 1, 113647, 1361, 2347, 28921, 1, 2251, 1, 1, 1, 1, 40169, 1, 877, 1, 123307, 1, 5939, 31357, 1, 1, 11597, 2467, 6143, 1, 130447, 1, 131899, 1, 44453, 4789, 1, 1, 136303, 34261, 3533, 1, 1, 1667, 140779, 863, 1103, 3251, 769, 1721, 1597, 36523, 48953, 1, 148399, 1, 1, 1, 4591, 3461, 153067, 12821, 2621, 1, 1, 2309, 1, 13217, 159403, 1, 1, 5779, 1187, 13619, 164239, 41263, 4253, 5953, 23929, 1, 15377, 3863, 1, 1, 1, 2063, 1, 1, 58601, 1, 177487, 1, 1, 45007, 60293, 45433, 182587, 15287, 1, 1, 1051, 46723, 1, 1429, 1, 1, 1301, 1117, 193003, 1, 194767, 1193, 1, 1, 1, 1277, 1, 1621, 1, 7243, 29101, 1, 205519, 51607, 1, 4733, 29881, 1, 16231, 52981, 70949, 4111, 4993, 1, 30937, 4943, 6619, 1, 12959, 1, 31741, 1, 1, 1, 17383, 18911, 20717, 1, 1, 1861, 1, 19391, 1, 58657, 1, 1, 237547, 1, 21773, 1019, 80489, 1237, 1, 20369, 245419, 61603, 82469, 1, 1, 1, 1, 63103, 4969, 63607, 1627, 1, 36781, 64621, 86501, 1, 1, 21881, 37657, 1, 5209, 1093, 267727, 22397, 4423, 1, 1, 6203, 273979, 1, 276079, 1, 1, 9973, 280303, 23447, 282427, 1, 8623, 1, 5851, 23981, 1307, 72481, 97001, 1, 1, 1, 26849, 6737, 99173, 74653, 2801, 3581, 1, 5827, 1, 76303, 23563, 1, 4007, 11059, 1459, 1, 313003, 26177, 315247, 11299, 1163, 79657, 1, 1, 7489, 1, 15443, 1, 19211, 1, 1, 82507, 1, 1, 1, 1, 1987, 1381, 2749, 1, 2861, 1, 342799, 1, 10459, 1, 26731, 1, 1, 1, 117413, 1, 354619, 1, 51001, 1, 10891, 5303, 361807, 1, 1, 1, 1, 2243, 369067, 1, 33773, 1, 17807, 13399, 376399, 31469, 1, 1, 7477, 1, 54829, 2917, 35117, 1, 129593, 1, 55897, 4673, 2833, 1, 2239, 99391, 2789, 1, 8191, 1, 1, 101293, 406447, 1, 1, 14653, 1, 1, 37649, 2663, 416719, 14929, 1, 1481, 2357, 35267, 24971, 106453, 1, 1, 3803, 2113, 1, 108421, 1747, 15583, 1, 1, 10739, 10037, 1, 3583, 1, 1, 448303, 8647, 150329, 113083, 26687, 5417, 5927, 1, 1, 2677, 461803, 2969, 66361, 2377, 11981, 1483, 27647, 3571, 3907, 1, 22643, 7013, 3491, 39971, 37003, 120607, 23039, 1, 1427, 3697, 1, 122701, 1, 1, 70717, 1, 497839, 9601, 9817, 11411, 1, 6011, 506347, 126943, 169733, 127657, 512059, 6113, 73561, 1, 1, 11801, 12109, 3347, 74797, 1, 175493, 2237, 17077, 1, 12983, 1733, 1, 10321, 538159, 1, 1, 1, 3701, 19483, 12721, 45707, 1, 1, 1, 19801, 1, 46451, 9473, 1, 1, 1697, 80701, 1, 3037, 142357, 190313, 143113, 573967, 1, 2659, 2371, 1, 1, 1, 1, 83737, 1, 1, 1, 592303, 1, 1, 21319, 28499, 1, 3217, 50261, 5987, 151561, 1, 3109, 3413, 1, 3677, 11839, 1, 1, 1, 51827, 20113, 1, 208889, 9239, 89977, 7517, 3181, 14423, 1, 1, 1, 1, 91801, 161053, 215273, 9521, 649039, 1, 1, 1, 12853, 12637, 16067, 1, 50923, 1, 1, 166723, 668527, 5077, 1, 5431, 1, 24169, 52183, 1, 1, 1, 5569, 1, 1, 5227, 1, 1, 2791, 174157, 14251, 1, 6947, 10343, 18077, 16061, 1, 1, 5981, 3023, 238373, 179203, 1, 60017, 7933, 1, 1, 1, 2689, 60869, 43067, 26209, 35027, 184321, 739003, 3631, 10457, 1, 3229, 26701, 1, 62591, 7309, 1, 252101, 27073, 108541, 3733, 69389, 17387, 19661, 192133, 6473, 1, 773803, 193891, 2293, 1, 25189, 1, 1, 196543, 262649, 15187, 46559, 66107, 8737, 1, 1, 1, 1, 6091, 805807, 28843, 38543, 1, 62539, 67901, 13841, 15739, 1, 1, 74897, 1, 20183, 207331, 1, 1, 3847, 1, 838459, 2531, 1, 19181, 120829, 10091, 10753, 1, 16729, 3623, 7583, 10223, 1, 1, 2381, 16657, 2459, 1, 17791, 1, 291833, 5101, 879247, 1, 80273, 1, 42227, 222163, 1, 2399, 1, 3673, 42767, 32143, 8429, 1, 1, 226903, 23321, 1, 130477, 5867, 53951, 229771, 1, 1, 12011, 1, 928699, 1, 2269, 17971, 4019, 11171, 10333, 2287, 1, 21503,

6. Sequence of the polynom (only primes)

79, 7, 3, 17, 113, 13, 197, 59, 11, 31, 401, 107, 151, 71, 43, 41, 137, 83, 593, 149, 199, 61, 101, 157, 233, 193, 271, 389, 313, 103, 569, 173, 167, 773, 607, 2539, 2767, 191, 281, 3499, 907, 179, 139, 4027, 347, 331, 419, 5179, 829, 557, 2153, 1657, 523, 1021, 1831, 7867, 2203, 9007, 9403, 467, 2503, 929, 10639, 2713, 11503, 977, 919, 4133, 1091, 13327, 3391, 3511, 14779, 463, 353, 15787, 4273, 17359, 2557, 613, 1607, 1151, 709, 5107, 20719, 491, 317, 1901, 379, 5857, 859, 2273, 487, 26959, 6823, 9209, 6991, 7333, 761, 7507, 1787, 4441, 1123, 10601, 2957, 2741, 421, 1201, 1619, 661, 34747, 2927, 2731, 8971, 37039, 3119, 641, 757, 433, 1297, 10357, 503, 42667, 10771, 853, 727, 587, 1181, 3947, 6829, 1723, 16229, 1153, 2447, 997, 4397, 2579, 1951, 5009, 1367, 1087, 19001, 2053, 4871, 3467, 19973, 1373, 601, 15601, 1613, 1559, 9277, 16363, 1999, 1511, 67003, 9721, 743, 4127, 5479, 18211, 73387, 1049, 18757, 2719, 643, 26309, 11437, 6719, 81199, 20443, 27449, 20731, 21313, 12601, 1753, 22501, 8237, 691, 1009, 3301, 31013, 1801, 881, 7907, 2221, 3433, 2213, 8219, 99259, 4787, 3613, 8537, 103087, 25933, 3163, 8861, 1289, 1583, 2777, 27241, 15661, 113647, 1361, 2347, 28921, 2251, 40169, 877, 123307, 5939, 31357, 11597, 2467, 6143, 130447, 131899, 44453, 4789, 136303, 34261, 3533, 1667, 140779, 863, 1103, 3251, 769, 1721, 1597, 36523, 48953, 148399, 4591, 3461, 153067, 12821, 2621, 2309, 13217, 159403, 5779, 1187, 13619, 164239, 41263, 4253, 5953, 23929, 15377, 3863, 2063, 58601, 177487, 45007, 60293, 45433, 182587, 15287, 1051, 46723, 1429, 1301, 1117, 193003, 194767, 1193, 1277, 1621, 7243, 29101, 205519, 51607, 4733, 29881, 16231, 52981, 70949, 4111, 4993, 30937, 4943, 6619, 12959, 31741, 17383, 18911, 20717, 1861, 19391, 58657, 237547, 21773, 1019, 80489, 1237, 20369, 245419, 61603, 82469, 63103, 4969, 63607, 1627, 36781, 64621, 86501, 21881, 37657, 5209, 1093, 267727, 22397, 4423, 6203, 273979, 276079, 9973, 280303, 23447, 282427, 8623, 5851, 23981, 1307, 72481, 97001, 26849, 6737, 99173, 74653, 2801, 3581, 5827, 76303, 23563, 4007, 11059, 1459, 313003, 26177, 315247, 11299, 1163, 79657, 7489, 15443, 19211, 82507, 1987, 1381, 2749, 2861, 342799, 10459, 26731, 117413, 354619, 51001, 10891, 5303, 361807, 2243, 369067, 33773, 17807, 13399, 376399, 31469, 7477, 54829, 2917, 35117, 129593, 55897, 4673, 2833, 2239, 99391, 2789, 8191, 101293, 406447, 14653, 37649, 2663, 416719, 14929, 1481, 2357, 35267, 24971, 106453, 3803, 2113, 108421, 1747, 15583, 10739, 10037, 3583, 448303, 8647, 150329, 113083, 26687, 5417, 5927, 2677, 461803, 2969, 66361, 2377, 11981, 1483, 27647, 3571, 3907, 22643, 7013, 3491, 39971, 37003, 120607, 23039, 1427, 3697, 122701, 70717, 497839, 9601, 9817, 11411, 6011, 506347, 126943, 169733, 127657, 512059, 6113, 73561, 11801, 12109, 3347, 74797, 175493, 2237, 17077, 12983, 1733, 10321, 538159, 3701, 19483, 12721, 45707, 19801, 46451, 9473, 1697, 80701, 3037, 142357, 190313, 143113, 573967, 2659, 2371, 83737, 592303, 21319, 28499, 3217, 50261, 5987, 151561, 3109, 3413, 3677, 11839, 51827, 20113, 208889, 9239, 89977, 7517, 3181, 14423, 91801, 161053, 215273, 9521, 649039, 12853, 12637, 16067, 50923, 166723, 668527, 5077, 5431, 24169, 52183, 5569, 5227, 2791, 174157, 14251, 6947, 10343, 18077, 16061, 5981, 3023, 238373, 179203, 60017, 7933, 2689, 60869, 43067, 26209, 35027, 184321, 739003, 3631, 10457, 3229, 26701, 62591, 7309, 252101, 27073, 108541, 3733, 69389, 17387, 19661, 192133, 6473, 773803, 193891, 2293, 25189, 196543, 262649, 15187, 46559, 66107, 8737, 6091, 805807, 28843, 38543, 62539, 67901, 13841, 15739, 74897, 20183, 207331, 3847, 838459, 2531, 19181, 120829, 10091, 10753, 16729, 3623, 7583, 10223, 2381, 16657, 2459, 17791, 291833, 5101, 879247, 80273, 42227, 222163, 2399, 3673, 42767, 32143, 8429, 226903, 23321, 130477, 5867, 53951, 229771, 12011, 928699, 2269, 17971, 4019, 11171, 10333, 2287, 21503,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-52x+79 and
the reducible primes which appear as divisor for the first time
p | x^2-52x+79 and p < x^2-52x+79

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)
1109450.9000000.4000000.9000000.0000000.0000000.000000
21004010300.4000000.1000000.4000004.4444452.5000006.000000
31.000590775130.5900000.0770000.59000014.7500007.70000017.100000
410.0006.4375565.8810.6437000.0556000.64370010.9101707.22077911.463938
5100.00065.6834.25961.4240.6568300.0425900.65683010.2039777.66007210.444482
61.000.000663.60534.819628.7860.6636050.0348190.66360510.1031478.17539310.236813
710.000.0006.677.031293.8686.383.1630.6677030.0293870.66770310.0617558.43987510.151567
8100.000.00067.088.6052.542.99664.545.6090.6708860.0254300.67088610.0476708.65353110.111854
91.000.000.000673.358.16022.451.950650.906.2100.6733580.0224520.67335810.0368488.82893710.084439
1010.000.000.0006.753.251.192200.969.8136.552.281.3790.6753250.0200970.67532510.0292128.95110710.066399


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
388441.0000000.5000000.5000001.6000001.3333332.000000
41613580.8125000.3125000.5000001.6250001.2500002.000000
532206140.6250000.1875000.4375001.5384621.2000001.750000
664236170.3593750.0937500.2656251.1500001.0000001.214286
71285414400.4218750.1093750.3125002.3478262.3333332.352941
8256127251020.4960940.0976560.3984382.3518521.7857142.550000
9512282452370.5507810.0878910.4628912.2204721.8000002.323529
101.024606775290.5917970.0751950.5166022.1489361.7111112.232068
112.0481.2661441.1220.6181640.0703120.5478522.0891091.8701302.120983
124.0962.5822642.3180.6303710.0644530.5659182.0394951.8333332.065954
138.1925.2484694.7790.6406250.0572510.5833742.0325331.7765152.061691
1416.38410.6268509.7760.6485600.0518800.5966802.0247711.8123672.045616
1532.76821.3551.56619.7890.6517030.0477910.6039122.0096931.8423532.024243
1665.53642.9232.90540.0180.6549530.0443270.6106262.0099741.8550452.022235
17131.07286.2365.40780.8290.6579280.0412520.6166762.0090861.8612742.019816
18262.144172.91110.192162.7190.6596030.0388790.6207242.0050911.8849642.013126
19524.288347.04319.252327.7910.6619320.0367200.6252122.0070611.8889322.014461
201.048.576695.93436.412659.5220.6636940.0347250.6289692.0053251.8913362.012020
212.097.1521.394.71468.8621.325.8520.6650510.0328360.6322152.0040891.8911902.010323
224.194.3042.794.459130.7612.663.6980.6662510.0311760.6350752.0036071.8988852.009046
238.388.6085.598.712249.5325.349.1800.6674180.0297470.6376722.0035051.9083062.008178
2416.777.21611.215.877475.85610.740.0210.6685180.0283630.6401552.0032961.9069942.007788
2533.554.43222.464.107910.84721.553.2600.6694830.0271450.6423372.0028851.9141232.006817
2667.108.86444.989.5151.746.46543.243.0500.6703960.0260240.6443722.0027291.9174082.006335
27134.217.72890.092.7203.356.66386.736.0570.6712430.0250090.6462342.0025271.9219752.005780
28268.435.456180.394.8386.458.770173.936.0680.6720230.0240610.6479622.0023241.9241642.005349
29536.870.912361.181.71412.447.418348.734.2960.6727530.0231850.6495682.0021731.9272122.004957
301.073.741.824723.083.92924.021.273699.062.6560.6734240.0223720.6510532.0019951.9298202.004571
312.147.483.6481.447.531.04646.410.2191.401.120.8270.6740590.0216110.6524482.0018851.9320472.004285
324.294.967.2962.897.612.12589.770.9692.807.841.1560.6746530.0209010.6537512.0017621.9342932.003997
338.589.934.5925.799.985.526173.831.1745.626.154.3520.6752070.0202370.6549712.0016431.9363852.003730
3417.179.869.18411.608.958.178336.933.50511.272.024.6730.6757300.0196120.6561182.0015491.9382802.003504


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
122100011
243111011
384121021
4165132021
5326143021
6646143021
712814943623
825625204310210
951245404320220
101.02477724334238
112.0481441394364275
124.096264259431152144
138.192469464432192245
1416.384850845434102435
1532.7681.5661.561437642797
1665.5362.9052.900431.43021.470
17131.0725.4075.402432.72122.681
18262.14410.19210.187435.15525.032
19524.28819.25219.247439.71129.536
201.048.57636.41236.4074318.271218.136
212.097.15268.86268.8574334.402234.455
224.194.304130.761130.7564365.447265.309
238.388.608249.532249.52743124.6612124.866
2416.777.216475.856475.85143238.0932237.758
2533.554.432910.847910.84243455.9312454.911
2667.108.8641.746.4651.746.46043873.8402872.620
27134.217.7283.356.6633.356.658431.678.57921.678.079
28268.435.4566.458.7706.458.765433.229.70823.229.057
29536.870.91212.447.41812.447.413436.222.90226.224.511
301.073.741.82424.021.27324.021.2684312.011.165212.010.103
312.147.483.64846.410.21946.410.2144323.202.891223.207.323
324.294.967.29689.770.96989.770.9644344.882.737244.888.227
338.589.934.592173.831.174173.831.1694386.913.552286.917.617
3417.179.869.184336.933.505336.933.50043168.465.9502168.467.550


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
121100001
242111001
384221111
4168441214
53214683425
66417895435
71284019211011910
8256102554725262625
951223712311459527155
101.024529278251132127160110
112.0481.122557565299254326243
124.0962.3181.1581.160602537653526
138.1924.7792.4172.3621.2341.1251.2971.123
1416.3849.7764.8884.8882.5572.3012.6072.311
1532.76819.7899.9609.8295.1594.6755.2744.681
1665.53640.01820.18319.83510.4559.46710.5609.536
17131.07280.82940.65740.17221.11619.25721.26819.188
18262.144162.71981.68981.03042.42638.92242.57738.794
19524.288327.791164.694163.09785.23878.73185.06478.758
201.048.576659.522331.602327.920170.896158.591171.019159.016
212.097.1521.325.852666.990658.862342.978320.284343.020319.570
224.194.3042.663.6981.338.9751.324.723687.950643.724688.341643.683
238.388.6085.349.1802.688.7132.660.4671.379.0981.294.3191.379.4231.296.340
2416.777.21610.740.0215.397.0365.342.9852.763.3082.602.9932.765.7672.607.953
2533.554.43221.553.26010.831.67110.721.5895.539.7635.232.8295.544.0425.236.626
2667.108.86443.243.05021.725.91021.517.14011.104.55010.513.61911.107.87110.517.010
27134.217.72886.736.05743.566.34243.169.71522.248.88521.113.74422.254.50821.118.920
28268.435.456173.936.06887.362.28586.573.78344.572.02642.388.42444.574.48342.401.135
29536.870.912348.734.296175.126.628173.607.66889.281.80085.079.11589.287.94085.085.441
301.073.741.824699.062.656351.033.301348.029.355178.809.234170.717.235178.822.102170.714.085
312.147.483.6481.401.120.827703.474.727697.646.100358.086.324342.453.246358.109.235342.472.022
324.294.967.2962.807.841.1561.409.539.3481.398.301.808717.065.207686.842.078717.068.988686.864.883
338.589.934.5925.626.154.3522.823.972.2572.802.182.0951.435.780.1601.377.311.6201.435.793.3491.377.269.223
3417.179.869.18411.272.024.6735.657.232.6425.614.792.0312.874.660.9102.761.353.6572.874.690.4722.761.319.634


8. Check for existing Integer Sequences by OEIS

Found in Database : 79, 7, 3, 17, 113, 13, 197, 59, 1, 11, 31, 1, 401, 107, 151, 1, 71, 43, 41, 137,
Found in Database : 79, 7, 3, 17, 113, 13, 197, 59, 11, 31, 401, 107, 151, 71, 43, 41, 137, 83, 593, 149, 199,
Found in Database : 3, 7, 11, 13, 17, 31, 41, 43, 59, 61, 71, 79, 83, 101, 103, 107, 113, 137, 139, 149,