Inhaltsverzeichnis

Development of
Algorithmic Constructions

06:05:36
Deutsch
29.Mar 2024

Polynom = x^2-36x+41

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) = 41 = 41
f(1) = 3 = 3
f(2) = 27 = 3*3*3
f(3) = 29 = 29
f(4) = 87 = 3*29
f(5) = 57 = 3*19
f(6) = 139 = 139
f(7) = 81 = 3*3*3*3
f(8) = 183 = 3*61
f(9) = 101 = 101
f(10) = 219 = 3*73
f(11) = 117 = 3*3*13
f(12) = 247 = 13*19
f(13) = 129 = 3*43
f(14) = 267 = 3*89
f(15) = 137 = 137
f(16) = 279 = 3*3*31
f(17) = 141 = 3*47
f(18) = 283 = 283
f(19) = 141 = 3*47
f(20) = 279 = 3*3*31
f(21) = 137 = 137
f(22) = 267 = 3*89
f(23) = 129 = 3*43
f(24) = 247 = 13*19
f(25) = 117 = 3*3*13
f(26) = 219 = 3*73
f(27) = 101 = 101
f(28) = 183 = 3*61
f(29) = 81 = 3*3*3*3
f(30) = 139 = 139
f(31) = 57 = 3*19
f(32) = 87 = 3*29
f(33) = 29 = 29
f(34) = 27 = 3*3*3
f(35) = 3 = 3
f(36) = 41 = 41
f(37) = 39 = 3*13
f(38) = 117 = 3*3*13
f(39) = 79 = 79
f(40) = 201 = 3*67
f(41) = 123 = 3*41
f(42) = 293 = 293
f(43) = 171 = 3*3*19
f(44) = 393 = 3*131
f(45) = 223 = 223
f(46) = 501 = 3*167
f(47) = 279 = 3*3*31
f(48) = 617 = 617
f(49) = 339 = 3*113
f(50) = 741 = 3*13*19
f(51) = 403 = 13*31
f(52) = 873 = 3*3*97
f(53) = 471 = 3*157
f(54) = 1013 = 1013
f(55) = 543 = 3*181
f(56) = 1161 = 3*3*3*43
f(57) = 619 = 619
f(58) = 1317 = 3*439
f(59) = 699 = 3*233
f(60) = 1481 = 1481
f(61) = 783 = 3*3*3*29
f(62) = 1653 = 3*19*29
f(63) = 871 = 13*67
f(64) = 1833 = 3*13*47
f(65) = 963 = 3*3*107
f(66) = 2021 = 43*47
f(67) = 1059 = 3*353
f(68) = 2217 = 3*739
f(69) = 1159 = 19*61
f(70) = 2421 = 3*3*269
f(71) = 1263 = 3*421
f(72) = 2633 = 2633
f(73) = 1371 = 3*457
f(74) = 2853 = 3*3*317
f(75) = 1483 = 1483
f(76) = 3081 = 3*13*79
f(77) = 1599 = 3*13*41
f(78) = 3317 = 31*107
f(79) = 1719 = 3*3*191
f(80) = 3561 = 3*1187
f(81) = 1843 = 19*97
f(82) = 3813 = 3*31*41
f(83) = 1971 = 3*3*3*73
f(84) = 4073 = 4073
f(85) = 2103 = 3*701
f(86) = 4341 = 3*1447
f(87) = 2239 = 2239
f(88) = 4617 = 3*3*3*3*3*19
f(89) = 2379 = 3*13*61
f(90) = 4901 = 13*13*29
f(91) = 2523 = 3*29*29
f(92) = 5193 = 3*3*577
f(93) = 2671 = 2671
f(94) = 5493 = 3*1831
f(95) = 2823 = 3*941
f(96) = 5801 = 5801
f(97) = 2979 = 3*3*331
f(98) = 6117 = 3*2039
f(99) = 3139 = 43*73
f(100) = 6441 = 3*19*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^2-36x+41

f(0)=41
f(1)=3
f(2)=1
f(3)=29
f(4)=1
f(5)=19
f(6)=139
f(7)=1
f(8)=61
f(9)=101
f(10)=73
f(11)=13
f(12)=1
f(13)=43
f(14)=89
f(15)=137
f(16)=31
f(17)=47
f(18)=283
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)=1
f(39)=79
f(40)=67
f(41)=1
f(42)=293
f(43)=1
f(44)=131
f(45)=223
f(46)=167
f(47)=1
f(48)=617
f(49)=113
f(50)=1
f(51)=1
f(52)=97
f(53)=157
f(54)=1013
f(55)=181
f(56)=1
f(57)=619
f(58)=439
f(59)=233
f(60)=1481
f(61)=1
f(62)=1
f(63)=1
f(64)=1
f(65)=107
f(66)=1
f(67)=353
f(68)=739
f(69)=1
f(70)=269
f(71)=421
f(72)=2633
f(73)=457
f(74)=317
f(75)=1483
f(76)=1
f(77)=1
f(78)=1
f(79)=191
f(80)=1187
f(81)=1
f(82)=1
f(83)=1
f(84)=4073
f(85)=701
f(86)=1447
f(87)=2239
f(88)=1
f(89)=1
f(90)=1
f(91)=1
f(92)=577
f(93)=2671
f(94)=1831
f(95)=941
f(96)=5801
f(97)=331
f(98)=2039
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-36x+41 could be written as f(y)= y^2-283 with x=y+18

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
f'(x)>2x-37 with x > 17

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

41, 3, 1, 29, 1, 19, 139, 1, 61, 101, 73, 13, 1, 43, 89, 137, 31, 47, 283, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 79, 67, 1, 293, 1, 131, 223, 167, 1, 617, 113, 1, 1, 97, 157, 1013, 181, 1, 619, 439, 233, 1481, 1, 1, 1, 1, 107, 1, 353, 739, 1, 269, 421, 2633, 457, 317, 1483, 1, 1, 1, 191, 1187, 1, 1, 1, 4073, 701, 1447, 2239, 1, 1, 1, 1, 577, 2671, 1831, 941, 5801, 331, 2039, 1, 1, 367, 521, 1, 2371, 3643, 829, 1, 7817, 1, 1, 1, 2851, 1, 8933, 1, 239, 4759, 1, 1, 349, 1721, 3511, 1, 1217, 1861, 599, 1933, 1, 463, 1, 2081, 12713, 719, 4391, 6703, 4547, 257, 743, 2393, 1, 571, 1, 2557, 503, 1, 1789, 8179, 1, 1, 281, 967, 1, 8971, 467, 1, 1, 1, 1, 1, 1, 3361, 1, 3457, 1, 10663, 7207, 1, 1709, 1, 7607, 373, 1, 1319, 359, 1, 433, 431, 1, 4273, 1997, 337, 2957, 709, 9091, 1, 27941, 1571, 9539, 14479, 9767, 1, 1, 389, 787, 1, 1163, 1, 32117, 5413, 1, 16603, 1, 5657, 34313, 1, 1, 1, 11939, 2011, 1, 1, 12451, 1, 1, 6421, 38921, 6553, 1471, 1543, 1039, 1, 41333, 773, 14051, 21283, 14327, 2411, 1019, 1, 14887, 22543, 1, 1, 587, 1, 1, 769, 1, 8093, 2579, 1, 16631, 25171, 16931, 1, 1, 8693, 17539, 26539, 661, 9001, 1, 9157, 1, 27943, 18787, 9473, 4409, 1, 19427, 29383, 19751, 3319, 1, 1, 20407, 30859, 1, 10453, 1, 1, 1, 32371, 21751, 1, 839, 1237, 1181, 1, 22787, 1, 1693, 11657, 1, 2731, 7949, 1, 72617, 1, 1, 37123, 1, 1, 2617, 1, 25667, 1, 2003, 1, 79241, 1, 1, 40471, 3019, 13681, 631, 13873, 1, 2221, 2179, 1097, 853, 1, 677, 43963, 1553, 4951, 1229, 15053, 30307, 45763, 1, 1, 1, 15661, 1, 47599, 1, 16073, 1, 1, 32771, 811, 33191, 1, 7757, 1301, 34039, 1, 11489, 17341, 104693, 1, 11777, 53323, 761, 947, 2311, 1, 2819, 1907, 1279, 691, 5927, 1, 37987, 1, 4271, 19333, 116681, 1, 1009, 4567, 971, 20021, 6359, 1, 40739, 1499, 1, 6907, 1289, 1103, 1, 1, 1093, 21433, 1453, 21673, 1, 2267, 44071, 22157, 133673, 1, 1, 67939, 1, 1, 138101, 1, 1, 1493, 1, 1, 142601, 23893, 1, 1, 1129, 1877, 11321, 8219, 2609, 1823, 50087, 2797, 151817, 877, 1, 4057, 1913, 1, 12041, 2017, 17569, 1, 53239, 863, 1, 9007, 54311, 1, 54851, 9187, 1213, 2141, 1, 84319, 991, 1, 1249, 28657, 1, 1423, 58147, 1, 6073, 1, 1, 6871, 1, 1, 181193, 30341, 1487, 1, 1, 1627, 6011, 1, 20897, 1, 1, 1, 191561, 10691, 64439, 1091, 65027, 1, 196853, 32957, 907, 1489, 1, 1, 1, 1, 22669, 1297, 68611, 34457, 207653, 11587, 69827, 105199, 70439, 1, 1, 1151, 71671, 107971, 24097, 1171, 5087, 1, 8171, 2357, 74167, 1, 1, 1, 75431, 113623, 1, 1, 230117, 2027, 77347, 1031, 25997, 39157, 1, 3037, 1, 119419, 79939, 1, 241781, 13487, 2621, 1549, 1, 1523, 1, 41453, 1, 9643, 9323, 1, 253733, 42457, 1, 1, 2963, 43133, 6337, 1, 1, 10111, 1, 1, 5659, 44501, 1901, 1, 30013, 1051, 272201, 45541, 10159, 4441, 1, 3557, 278501, 1, 1, 140839, 1193, 15767, 2521, 47657, 3299, 4967, 1, 1, 22409, 48733, 32609, 147283, 1, 1, 297833, 16607, 1, 150559, 1, 1, 23417, 3917, 2377, 153871, 1, 2719, 4643, 1, 1123, 157219, 1, 1, 1, 1, 1, 160603, 1, 17971, 324617, 54293, 108967, 164023, 1, 55057, 1, 55441, 1, 1, 1, 1, 3163, 1, 113591, 170971, 114371, 1, 345461, 57773, 1, 1, 1, 1, 12157, 1, 1, 1, 119107, 59753, 5897, 20051, 6353, 1, 1, 1, 366953, 1427, 123127, 185299, 1, 1, 7963, 62581, 2203, 1871, 1303, 4877, 1, 1, 2099, 10141, 4157, 1, 13417, 1, 130531, 2207, 43789, 1, 30509, 5101, 14783, 1, 1, 67157, 404213, 2503, 3307, 204019, 1, 22811, 1847, 5297, 10627, 2143, 46337, 1, 6263, 1, 1, 1, 1, 1, 427433, 23819, 11027, 1, 1, 1, 435317, 1, 145987, 1399, 16319, 1, 443273, 74101, 49549, 17203, 11503, 3947, 451301, 25147, 3691, 227671, 152231, 25447, 24179, 1873, 4969, 1, 1, 1, 15083, 78157, 17419, 1, 1, 1, 1, 8837, 2381, 5581, 1, 2063, 1, 4259, 1607, 244159, 54413, 81853, 4603, 1, 1, 248371, 3533, 1, 1, 1, 5791, 1, 168887, 1, 509513, 1, 1, 256903, 6361, 86113, 12637, 6661, 1, 261223, 174631, 87557, 12251, 29347, 9293, 265579, 177539, 29671, 6779, 89501, 13807, 1, 1, 90481, 7457, 3137, 1, 1, 1, 91961, 553253, 10273, 5981, 1, 1, 31151, 3581, 1999, 188407, 6029, 3323, 1, 13933, 95461, 63809, 4297, 1, 1, 580361, 32327, 194471, 292471, 10289, 10889, 1, 1, 197539, 297079, 22063, 1, 46061, 100057, 66877, 9733, 201667, 2351, 608117, 33871, 3041, 5023, 204791, 1801, 47501, 7937, 206887, 1, 69313, 104233, 32999, 104761, 23339, 4327, 1973, 1, 1, 1, 1, 320659, 5227, 1, 1, 1, 216451, 1949, 2339, 1, 6761, 1, 1, 25411, 220771, 1, 665573, 1, 222947, 2447, 1, 4159, 1, 5939, 226231, 1, 1, 3929, 685301, 114493, 4027, 345139, 230647, 115601, 1, 1, 232871, 1, 1, 1, 705317, 117833, 12433, 8663, 1, 2531, 5147, 2543, 1, 1, 240739, 9281, 55817, 13469, 243011, 1, 8419, 40787, 3079, 1, 246439, 19501, 82529, 1, 1, 1, 2687, 375703, 1, 125813, 24407, 42131, 253367, 3371, 1, 14173, 767093, 9857, 19759, 9419, 1, 129313, 777641, 129901, 1, 13499, 2939, 6899, 788261, 43891, 1, 1, 5641, 1, 1, 1, 267511, 12973, 2083, 1, 8017, 4363, 3347, 1, 20947, 2237, 2273, 1, 6389, 1, 275939, 46091, 831461, 4789, 1, 418471, 1, 1, 1, 1, 2293, 1, 283267, 141941, 853493, 1, 285731, 5437, 9257, 1, 66509, 144413, 9337, 435103, 32299, 1, 9029, 7699, 97729, 9377, 294439, 1, 1, 1, 15629, 1, 298211, 49807, 898421, 150053, 300739, 10513, 100669, 1, 909833, 11689, 1, 6271, 1, 4943, 13751, 1, 2251, 1, 309671, 51719, 1, 155801, 24019, 1, 104513, 5417, 32569, 1, 1, 1, 317431, 1, 1, 53231, 320039, 37003,

6. Sequence of the polynom (only primes)

41, 3, 29, 19, 139, 61, 101, 73, 13, 43, 89, 137, 31, 47, 283, 79, 67, 293, 131, 223, 167, 617, 113, 97, 157, 1013, 181, 619, 439, 233, 1481, 107, 353, 739, 269, 421, 2633, 457, 317, 1483, 191, 1187, 4073, 701, 1447, 2239, 577, 2671, 1831, 941, 5801, 331, 2039, 367, 521, 2371, 3643, 829, 7817, 2851, 8933, 239, 4759, 349, 1721, 3511, 1217, 1861, 599, 1933, 463, 2081, 12713, 719, 4391, 6703, 4547, 257, 743, 2393, 571, 2557, 503, 1789, 8179, 281, 967, 8971, 467, 3361, 3457, 10663, 7207, 1709, 7607, 373, 1319, 359, 433, 431, 4273, 1997, 337, 2957, 709, 9091, 27941, 1571, 9539, 14479, 9767, 389, 787, 1163, 32117, 5413, 16603, 5657, 34313, 11939, 2011, 12451, 6421, 38921, 6553, 1471, 1543, 1039, 41333, 773, 14051, 21283, 14327, 2411, 1019, 14887, 22543, 587, 769, 8093, 2579, 16631, 25171, 16931, 8693, 17539, 26539, 661, 9001, 9157, 27943, 18787, 9473, 4409, 19427, 29383, 19751, 3319, 20407, 30859, 10453, 32371, 21751, 839, 1237, 1181, 22787, 1693, 11657, 2731, 7949, 72617, 37123, 2617, 25667, 2003, 79241, 40471, 3019, 13681, 631, 13873, 2221, 2179, 1097, 853, 677, 43963, 1553, 4951, 1229, 15053, 30307, 45763, 15661, 47599, 16073, 32771, 811, 33191, 7757, 1301, 34039, 11489, 17341, 104693, 11777, 53323, 761, 947, 2311, 2819, 1907, 1279, 691, 5927, 37987, 4271, 19333, 116681, 1009, 4567, 971, 20021, 6359, 40739, 1499, 6907, 1289, 1103, 1093, 21433, 1453, 21673, 2267, 44071, 22157, 133673, 67939, 138101, 1493, 142601, 23893, 1129, 1877, 11321, 8219, 2609, 1823, 50087, 2797, 151817, 877, 4057, 1913, 12041, 2017, 17569, 53239, 863, 9007, 54311, 54851, 9187, 1213, 2141, 84319, 991, 1249, 28657, 1423, 58147, 6073, 6871, 181193, 30341, 1487, 1627, 6011, 20897, 191561, 10691, 64439, 1091, 65027, 196853, 32957, 907, 1489, 22669, 1297, 68611, 34457, 207653, 11587, 69827, 105199, 70439, 1151, 71671, 107971, 24097, 1171, 5087, 8171, 2357, 74167, 75431, 113623, 230117, 2027, 77347, 1031, 25997, 39157, 3037, 119419, 79939, 241781, 13487, 2621, 1549, 1523, 41453, 9643, 9323, 253733, 42457, 2963, 43133, 6337, 10111, 5659, 44501, 1901, 30013, 1051, 272201, 45541, 10159, 4441, 3557, 278501, 140839, 1193, 15767, 2521, 47657, 3299, 4967, 22409, 48733, 32609, 147283, 297833, 16607, 150559, 23417, 3917, 2377, 153871, 2719, 4643, 1123, 157219, 160603, 17971, 324617, 54293, 108967, 164023, 55057, 55441, 3163, 113591, 170971, 114371, 345461, 57773, 12157, 119107, 59753, 5897, 20051, 6353, 366953, 1427, 123127, 185299, 7963, 62581, 2203, 1871, 1303, 4877, 2099, 10141, 4157, 13417, 130531, 2207, 43789, 30509, 5101, 14783, 67157, 404213, 2503, 3307, 204019, 22811, 1847, 5297, 10627, 2143, 46337, 6263, 427433, 23819, 11027, 435317, 145987, 1399, 16319, 443273, 74101, 49549, 17203, 11503, 3947, 451301, 25147, 3691, 227671, 152231, 25447, 24179, 1873, 4969, 15083, 78157, 17419, 8837, 2381, 5581, 2063, 4259, 1607, 244159, 54413, 81853, 4603, 248371, 3533, 5791, 168887, 509513, 256903, 6361, 86113, 12637, 6661, 261223, 174631, 87557, 12251, 29347, 9293, 265579, 177539, 29671, 6779, 89501, 13807, 90481, 7457, 3137, 91961, 553253, 10273, 5981, 31151, 3581, 1999, 188407, 6029, 3323, 13933, 95461, 63809, 4297, 580361, 32327, 194471, 292471, 10289, 10889, 197539, 297079, 22063, 46061, 100057, 66877, 9733, 201667, 2351, 608117, 33871, 3041, 5023, 204791, 1801, 47501, 7937, 206887, 69313, 104233, 32999, 104761, 23339, 4327, 1973, 320659, 5227, 216451, 1949, 2339, 6761, 25411, 220771, 665573, 222947, 2447, 4159, 5939, 226231, 3929, 685301, 114493, 4027, 345139, 230647, 115601, 232871, 705317, 117833, 12433, 8663, 2531, 5147, 2543, 240739, 9281, 55817, 13469, 243011, 8419, 40787, 3079, 246439, 19501, 82529, 2687, 375703, 125813, 24407, 42131, 253367, 3371, 14173, 767093, 9857, 19759, 9419, 129313, 777641, 129901, 13499, 2939, 6899, 788261, 43891, 5641, 267511, 12973, 2083, 8017, 4363, 3347, 20947, 2237, 2273, 6389, 275939, 46091, 831461, 4789, 418471, 2293, 283267, 141941, 853493, 285731, 5437, 9257, 66509, 144413, 9337, 435103, 32299, 9029, 7699, 97729, 9377, 294439, 15629, 298211, 49807, 898421, 150053, 300739, 10513, 100669, 909833, 11689, 6271, 4943, 13751, 2251, 309671, 51719, 155801, 24019, 104513, 5417, 32569, 317431, 53231, 320039, 37003,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-36x+41 and
the reducible primes which appear as divisor for the first time
p | x^2-36x+41 and p < x^2-36x+41

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)
1108530.8000000.5000000.3000000.0000000.0000000.000000
21004820280.4800000.2000000.2800006.0000004.0000009.333333
31.0006281195090.6280000.1190000.50900013.0833335.95000018.178572
410.0006.6788945.7840.6678000.0894000.57840010.6337587.51260511.363458
5100.00067.6336.73560.8980.6763300.0673500.60898010.1277337.53355710.528700
61.000.000678.69055.120623.5700.6786900.0551200.62357010.0348948.18411310.239581
710.000.0006.808.498464.8756.343.6230.6808500.0464870.63436210.0318238.43387110.173073
8100.000.00068.233.0904.019.40564.213.6850.6823310.0401940.64213710.0217538.64620610.122557
91.000.000.000683.447.12435.419.330648.027.7940.6834470.0354190.64802810.0163598.81208210.091740
1010.000.000.0006.843.593.241316.654.3916.526.938.8500.6843590.0316650.65269410.0133488.94015810.072004


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)
122201.0000001.0000000.0000000.0000000.0000000.000000
243300.7500000.7500000.0000001.5000001.500000-nan
386420.7500000.5000000.2500002.0000001.333333inf
41611650.6875000.3750000.3125001.8333331.5000002.500000
53213760.4062500.2187500.1875001.1818181.1666671.200000
6642714130.4218750.2187500.2031252.0769232.0000002.166667
71286524410.5078120.1875000.3203122.4074081.7142863.153846
8256144411030.5625000.1601560.4023442.2153851.7083332.512195
9512311702410.6074220.1367190.4707032.1597221.7073172.339806
101.0246441245200.6289060.1210940.5078122.0707401.7714292.157676
112.0481.3242261.0980.6464840.1103520.5361332.0559011.8225812.111538
124.0962.7034172.2860.6599120.1018070.5581052.0415411.8451332.081967
138.1925.4757544.7210.6683350.0920410.5762942.0255271.8081542.065179
1416.38410.9781.3759.6030.6700440.0839230.5861212.0051141.8236072.034103
1532.76822.0662.52919.5370.6734010.0771790.5962222.0100201.8392732.034468
1665.53644.2654.59839.6670.6754300.0701600.6052702.0060271.8181102.030353
17131.07288.6768.61680.0600.6765440.0657350.6108092.0032981.8738582.018302
18262.144177.51716.079161.4380.6771740.0613370.6158372.0018611.8661792.016463
19524.288355.52630.395325.1310.6781120.0579740.6201382.0027721.8903542.013968
201.048.576711.82857.544654.2840.6788520.0548780.6239742.0021831.8932062.012370
212.097.1521.424.898109.0441.315.8540.6794440.0519960.6274482.0017451.8949672.011136
224.194.3042.852.659207.3502.645.3090.6801270.0494360.6306912.0020091.9015262.010336
238.388.6085.710.239394.6475.315.5920.6807140.0470460.6336682.0017251.9032892.009441
2416.777.21611.428.796753.39010.675.4060.6812090.0449060.6363042.0014571.9090222.008319
2533.554.43222.873.8751.440.65221.433.2230.6816950.0429350.6387602.0014251.9122262.007720
2667.108.86445.776.2962.760.59843.015.6980.6821200.0411360.6409842.0012481.9162142.006963
27134.217.72891.599.7015.303.46486.296.2370.6824710.0395140.6429572.0010291.9211292.006157
28268.435.456183.293.50110.200.855173.092.6460.6828220.0380010.6448202.0010271.9234322.005796
29536.870.912366.771.32019.646.916347.124.4040.6831650.0365950.6465702.0010061.9260072.005426
301.073.741.824733.879.31237.893.804695.985.5080.6834780.0352910.6481872.0009181.9287412.005003
312.147.483.6481.468.378.74273.178.1611.395.200.5810.6837670.0340760.6496912.0008451.9311382.004640
324.294.967.2962.937.944.145141.501.8372.796.442.3080.6840430.0329460.6510972.0008081.9336622.004330
338.589.934.5925.878.119.329273.919.6725.604.199.6570.6843030.0318880.6524152.0007591.9358032.004046
3417.179.869.18411.760.453.486530.783.17411.229.670.3120.6845480.0308960.6536532.0007171.9377332.003796
3534.359.738.36823.528.890.2931.029.554.78222.499.335.5110.6847810.0299640.6548172.0006791.9396902.003561
3668.719.476.73647.072.859.1041.998.861.06545.073.998.0390.6850000.0290870.6559132.0006411.9414812.003348


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
122011100
243021110
384121210
4166142220
5327242320
66414584442
71282410138655
82564121191112810
951270363319211416
101.024124636029333131
112.04822611411158595356
124.09641721120510810897104
138.192754391362190205172187
1416.3841.375711663328359335353
1532.7682.5291.2921.236624643612650
1665.5364.5982.3582.2391.1331.1701.1061.189
17131.0728.6164.3474.2682.1562.1852.1122.163
18262.14416.0798.1157.9634.0014.0893.9624.027
19524.28830.39515.33915.0557.5777.7307.4787.610
201.048.57657.54429.10228.44114.24814.60314.19314.500
212.097.152109.04455.27753.76626.97727.62326.78927.655
224.194.304207.350104.961102.38851.40352.41150.98552.551
238.388.608394.647199.487195.15997.68999.65497.47099.834
2416.777.216753.390381.029372.360186.144190.557186.216190.473
2533.554.4321.440.652728.174712.477356.308363.916356.169364.259
2667.108.8642.760.5981.394.9531.365.644682.914697.424682.730697.530
27134.217.7285.303.4642.678.6862.624.7771.312.3691.339.1071.312.4081.339.580
28268.435.45610.200.8555.148.7195.052.1352.526.2322.573.7082.525.9032.575.012
29536.870.91219.646.9169.910.9599.735.9564.868.4574.955.5224.867.4994.955.438
301.073.741.82437.893.80419.114.24018.779.5639.389.0209.558.4399.390.5439.555.802
312.147.483.64873.178.16136.906.78736.271.37318.135.97518.457.97218.135.39818.448.816
324.294.967.296141.501.83771.343.62470.158.21235.073.78835.676.17335.084.42435.667.452
338.589.934.592273.919.672138.063.788135.855.88367.918.06669.037.69367.937.81769.026.096
3417.179.869.184530.783.174267.461.157263.322.016131.654.464133.733.222131.667.552133.727.936
3534.359.738.3681.029.554.782518.661.615510.893.166255.432.695259.328.430255.460.471259.333.186
3668.719.476.7361.998.861.0651.006.761.708992.099.356496.034.907503.370.996496.064.449503.390.713


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
382200110
4165412210
5326422211
66413764333
71284122191081211
8256103515220272729
951224111612552636165
101.024520250270116141128135
112.0481.098548550272265274287
124.0962.2861.1521.134547568581590
138.1924.7212.3932.3281.1641.1601.2001.197
1416.3849.6034.8164.7872.3552.4182.3982.432
1532.76819.5379.7779.7604.8134.9304.9324.862
1665.53639.66719.87119.7969.8409.86910.0079.951
17131.07280.06040.13839.92219.86420.06320.12920.004
18262.144161.43880.80980.62940.40540.38540.48440.164
19524.288325.131162.575162.55681.05481.45081.40581.222
201.048.576654.284327.055327.229163.250163.603164.174163.257
212.097.1521.315.854658.031657.823328.433328.869329.562328.990
224.194.3042.645.3091.323.3011.322.008661.471660.664661.746661.428
238.388.6085.315.5922.658.0962.657.4961.329.1781.327.5431.330.7061.328.165
2416.777.21610.675.4065.337.6015.337.8052.669.6722.666.9732.671.0712.667.690
2533.554.43221.433.22310.715.79810.717.4255.359.5935.355.3515.362.4005.355.879
2667.108.86443.015.69821.511.48021.504.21810.760.33310.745.84610.761.96610.747.553
27134.217.72886.296.23743.150.06543.146.17221.586.63721.555.81021.590.34821.563.442
28268.435.456173.092.64686.551.49386.541.15343.298.91343.242.33943.299.92743.251.467
29536.870.912347.124.404173.580.623173.543.78186.823.92886.724.88286.832.89286.742.702
301.073.741.824695.985.508348.038.319347.947.189174.087.774173.903.607174.084.063173.910.064
312.147.483.6481.395.200.581697.695.259697.505.322348.968.238348.638.923348.942.670348.650.750
324.294.967.2962.796.442.3081.398.390.8571.398.051.451699.430.056698.804.614699.408.614698.799.024
338.589.934.5925.604.199.6572.802.433.4092.801.766.2481.401.638.0931.400.485.1781.401.603.5811.400.472.805
3417.179.869.18411.229.670.3125.615.487.2505.614.183.0622.808.476.2542.806.337.4762.808.519.4502.806.337.132
3534.359.738.36822.499.335.51111.250.896.93811.248.438.5735.626.884.4145.622.796.5925.626.854.6795.622.799.826
3668.719.476.73645.073.998.03922.539.245.92822.534.752.11111.272.354.04611.264.706.89111.272.311.56411.264.625.538


8. Check for existing Integer Sequences by OEIS

Found in Database : 41, 3, 1, 29, 1, 19, 139, 1, 61, 101, 73, 13, 1, 43, 89, 137, 31, 47, 283, 1,
Found in Database : 41, 3, 29, 19, 139, 61, 101, 73, 13, 43, 89, 137, 31, 47, 283, 79,
Found in Database : 3, 13, 19, 29, 31, 41, 43, 47, 61, 67, 73, 79, 89, 97, 101, 107, 113, 131, 137, 139,