Inhaltsverzeichnis

Development of
Algorithmic Constructions

06:52:57
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) = 3 = 3
f(2) = 51 = 3*17
f(3) = 7 = 7
f(4) = 63 = 3*3*7
f(5) = 9 = 3*3
f(6) = 83 = 83
f(7) = 3 = 3
f(8) = 111 = 3*37
f(9) = 1 = 1
f(10) = 147 = 3*7*7
f(11) = 21 = 3*7
f(12) = 191 = 191
f(13) = 27 = 3*3*3
f(14) = 243 = 3*3*3*3*3
f(15) = 17 = 17
f(16) = 303 = 3*101
f(17) = 21 = 3*7
f(18) = 371 = 7*53
f(19) = 51 = 3*17
f(20) = 447 = 3*149
f(21) = 61 = 61
f(22) = 531 = 3*3*59
f(23) = 9 = 3*3
f(24) = 623 = 7*89
f(25) = 21 = 3*7
f(26) = 723 = 3*241
f(27) = 97 = 97
f(28) = 831 = 3*277
f(29) = 111 = 3*37
f(30) = 947 = 947
f(31) = 63 = 3*3*7
f(32) = 1071 = 3*3*7*17
f(33) = 71 = 71
f(34) = 1203 = 3*401
f(35) = 159 = 3*53
f(36) = 1343 = 17*79
f(37) = 177 = 3*59
f(38) = 1491 = 3*7*71
f(39) = 49 = 7*7
f(40) = 1647 = 3*3*3*61
f(41) = 27 = 3*3*3
f(42) = 1811 = 1811
f(43) = 237 = 3*79
f(44) = 1983 = 3*661
f(45) = 259 = 7*37
f(46) = 2163 = 3*7*103
f(47) = 141 = 3*47
f(48) = 2351 = 2351
f(49) = 153 = 3*3*17
f(50) = 2547 = 3*3*283
f(51) = 331 = 331
f(52) = 2751 = 3*7*131
f(53) = 357 = 3*7*17
f(54) = 2963 = 2963
f(55) = 3 = 3
f(56) = 3183 = 3*1061
f(57) = 103 = 103
f(58) = 3411 = 3*3*379
f(59) = 441 = 3*3*7*7
f(60) = 3647 = 7*521
f(61) = 471 = 3*157
f(62) = 3891 = 3*1297
f(63) = 251 = 251
f(64) = 4143 = 3*1381
f(65) = 267 = 3*89
f(66) = 4403 = 7*17*37
f(67) = 567 = 3*3*3*3*7
f(68) = 4671 = 3*3*3*173
f(69) = 601 = 601
f(70) = 4947 = 3*17*97
f(71) = 159 = 3*53
f(72) = 5231 = 5231
f(73) = 21 = 3*7
f(74) = 5523 = 3*7*263
f(75) = 709 = 709
f(76) = 5823 = 3*3*647
f(77) = 747 = 3*3*83
f(78) = 6131 = 6131
f(79) = 393 = 3*131
f(80) = 6447 = 3*7*307
f(81) = 413 = 7*59
f(82) = 6771 = 3*37*61
f(83) = 867 = 3*17*17
f(84) = 7103 = 7103
f(85) = 909 = 3*3*101
f(86) = 7443 = 3*3*827
f(87) = 119 = 7*17
f(88) = 7791 = 3*7*7*53
f(89) = 249 = 3*83
f(90) = 8147 = 8147
f(91) = 1041 = 3*347
f(92) = 8511 = 3*2837
f(93) = 1087 = 1087
f(94) = 8883 = 3*3*3*7*47
f(95) = 567 = 3*3*3*3*7
f(96) = 9263 = 59*157
f(97) = 591 = 3*197
f(98) = 9651 = 3*3217
f(99) = 1231 = 1231
f(100) = 10047 = 3*17*197

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)=3
f(2)=17
f(3)=7
f(4)=1
f(5)=1
f(6)=83
f(7)=1
f(8)=37
f(9)=1
f(10)=1
f(11)=1
f(12)=191
f(13)=1
f(14)=1
f(15)=1
f(16)=101
f(17)=1
f(18)=53
f(19)=1
f(20)=149
f(21)=61
f(22)=59
f(23)=1
f(24)=89
f(25)=1
f(26)=241
f(27)=97
f(28)=277
f(29)=1
f(30)=947
f(31)=1
f(32)=1
f(33)=71
f(34)=401
f(35)=1
f(36)=79
f(37)=1
f(38)=1
f(39)=1
f(40)=1
f(41)=1
f(42)=1811
f(43)=1
f(44)=661
f(45)=1
f(46)=103
f(47)=1
f(48)=2351
f(49)=1
f(50)=283
f(51)=331
f(52)=131
f(53)=1
f(54)=2963
f(55)=1
f(56)=1061
f(57)=1
f(58)=379
f(59)=1
f(60)=521
f(61)=157
f(62)=1297
f(63)=251
f(64)=1381
f(65)=1
f(66)=1
f(67)=1
f(68)=173
f(69)=601
f(70)=1
f(71)=1
f(72)=5231
f(73)=1
f(74)=263
f(75)=709
f(76)=647
f(77)=1
f(78)=6131
f(79)=1
f(80)=307
f(81)=1
f(82)=1
f(83)=1
f(84)=7103
f(85)=1
f(86)=827
f(87)=1
f(88)=1
f(89)=1
f(90)=8147
f(91)=347
f(92)=2837
f(93)=1087
f(94)=1
f(95)=1
f(96)=1
f(97)=197
f(98)=3217
f(99)=1231

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, 3, 17, 7, 1, 1, 83, 1, 37, 1, 1, 1, 191, 1, 1, 1, 101, 1, 53, 1, 149, 61, 59, 1, 89, 1, 241, 97, 277, 1, 947, 1, 1, 71, 401, 1, 79, 1, 1, 1, 1, 1, 1811, 1, 661, 1, 103, 1, 2351, 1, 283, 331, 131, 1, 2963, 1, 1061, 1, 379, 1, 521, 157, 1297, 251, 1381, 1, 1, 1, 173, 601, 1, 1, 5231, 1, 263, 709, 647, 1, 6131, 1, 307, 1, 1, 1, 7103, 1, 827, 1, 1, 1, 8147, 347, 2837, 1087, 1, 1, 1, 197, 3217, 1231, 1, 1, 1493, 1, 1, 1, 3761, 479, 239, 1, 4049, 773, 1399, 1, 13043, 1, 643, 1, 4657, 1, 14447, 1, 1, 271, 1, 653, 15923, 337, 5477, 1, 269, 1, 17471, 739, 353, 571, 883, 1, 1123, 1, 1, 1, 6737, 1, 2969, 439, 7121, 2707, 1, 1, 3221, 1, 7717, 733, 1, 1, 659, 1, 397, 1583, 1, 541, 431, 1109, 1283, 487, 3067, 1, 1663, 1, 9649, 523, 1, 1249, 30323, 1, 383, 1, 1511, 1, 457, 1367, 11057, 1, 3767, 1, 1, 1459, 1, 1, 12049, 761, 5273, 1, 1, 4759, 12821, 1619, 39251, 1, 1907, 1, 1, 1, 683, 1753, 1, 1, 14437, 911, 1, 619, 4999, 811, 1, 1, 46703, 491, 1, 1, 769, 1, 49331, 1, 16741, 3167, 17041, 1, 7433, 1, 1, 1669, 17957, 1, 7829, 1, 1093, 7027, 6299, 1, 3391, 1, 2791, 1, 19861, 2503, 853, 1, 977, 1, 20849, 1, 617, 1, 21521, 1, 1, 1, 1129, 2797, 22549, 8521, 3271, 1, 719, 1, 7867, 1, 23957, 1, 613, 1531, 24677, 1, 1, 1051, 10889, 1, 1, 1217, 1, 823, 1693, 1, 1, 10159, 27281, 1, 2243, 1741, 4007, 1, 9479, 1193, 1, 907, 29221, 1, 4231, 3727, 1699, 1259, 10139, 5741, 1, 1, 93683, 3929, 1861, 11941, 1187, 1, 13913, 1021, 1, 12409, 33301, 1, 1, 1, 11383, 1, 1, 4349, 105023, 1, 1, 3343, 11959, 1, 108947, 4567, 1, 1, 37201, 2339, 2131, 1, 1, 2053, 5507, 1, 6883, 1, 39461, 1, 1901, 1, 121151, 5077, 40849, 7703, 1, 1, 17909, 1, 14087, 15937, 42737, 1, 18521, 1, 1181, 16477, 4909, 1, 1301, 1, 6451, 8513, 1, 5737, 1, 1933, 2221, 1, 1, 1481, 3863, 5987, 1, 2593, 1, 1019, 147503, 3089, 49681, 1, 1, 1, 152147, 1, 5693, 1, 1, 929, 22409, 6569, 52817, 1, 17783, 1117, 3299, 967, 54421, 20509, 1, 863, 1, 1, 1, 21121, 56597, 7109, 991, 1, 8243, 1553, 6473, 1, 1747, 1, 59377, 1, 8563, 941, 181523, 1, 20359, 23011, 8807, 1, 186671, 3907, 62801, 1, 1, 1, 1, 1, 3797, 6079, 1229, 8179, 1657, 1, 7369, 12491, 1097, 4201, 3433, 1, 1, 25657, 22907, 1, 1, 1, 1, 1, 70549, 1, 213491, 1489, 1, 1931, 10343, 1, 219071, 1, 1567, 3467, 1, 1, 13219, 9403, 75541, 28447, 4481, 1, 4703, 1607, 25819, 29167, 78101, 9803, 33749, 1, 79397, 1, 26683, 1, 2917, 1447, 1, 1, 82021, 5147, 248051, 1153, 1, 4483, 84017, 1, 3061, 2657, 85361, 4591, 1, 1, 1, 5441, 1, 16451, 12583, 1579, 1, 1, 29819, 8419, 90149, 1, 38933, 11399, 91541, 2027, 1, 1, 1, 1, 1319, 35251, 1, 11839, 1, 1, 4561, 9013, 96497, 12107, 291647, 12197, 1, 2633, 32887, 2063, 17539, 1, 1889, 1, 14407, 3163, 304751, 1, 11369, 1, 14723, 1847, 3499, 1, 104549, 1, 35099, 1, 1, 1, 106801, 1, 1823, 3373, 1, 1, 1, 41047, 109841, 1, 1, 1, 2273, 41911, 1, 1, 1223, 3541, 1, 1, 1, 1, 345791, 1, 1, 3119, 16691, 1, 352883, 14753, 1499, 2621, 1, 1, 1, 1, 120817, 1, 121621, 2179, 1, 1, 13693, 1, 7297, 1, 1, 2237, 7393, 11821, 1, 1, 7207, 2281, 18307, 1303, 1277, 8087, 2251, 2713, 6221, 1, 131477, 16487, 396947, 1, 133157, 1789, 1, 1867, 1, 1, 135697, 25523, 19507, 1, 412211, 5743, 2711, 52009, 139121, 1, 3529, 1, 2309, 52981, 1, 5923, 1, 1, 143461, 26981, 1, 18097, 435647, 1, 1, 6869, 1427, 1, 443603, 18539, 1, 7993, 1, 1, 6361, 9437, 4093, 1, 3109, 1, 27043, 1601, 51383, 7247, 1, 2777, 467903, 19553, 2659, 29501, 1, 1, 1, 1, 2617, 60037, 1, 1, 9887, 1, 54139, 3593, 1, 20477, 1, 1471, 23603, 1, 1, 1, 1, 20947, 24007, 1, 9941, 1, 509843, 1, 1, 9181, 24551, 10771, 518447, 10831, 173777, 1, 1, 1, 527123, 2753, 176677, 1, 3011, 3181, 2069, 1, 1, 33767, 2029, 11317, 77813, 3251, 182549, 1, 20393, 1, 553583, 1, 1559, 1, 1, 1, 33091, 3917, 1, 1, 2399, 23753, 571583, 23879, 191537, 1, 1, 2011, 580691, 1, 194581, 1, 27943, 1, 1, 1, 21961, 74311, 3257, 3557, 12227, 6257, 1, 1, 1, 8429, 5113, 3631, 203857, 38321, 1, 1, 617843, 1, 9857, 1, 208049, 3259, 6211, 6551, 4289, 11287, 7823, 1, 636851, 1, 213349, 1, 30631, 1, 10957, 9001, 1951, 20353, 31091, 1, 656147, 27407, 219797, 1, 1, 1, 1, 13907, 13121, 1, 224149, 28087, 2609, 1, 1, 1, 2879, 28499, 685631, 4091, 1, 2539, 76919, 1, 695603, 1709, 1, 12511, 1787, 7333, 1999, 1, 78779, 1, 33923, 29753, 1, 14947, 1, 45053, 3823, 1, 42703, 30319, 243121, 11423, 244261, 1, 105173, 1, 82183, 92671, 247697, 1, 106649, 1, 1, 93967, 83719, 1, 7349, 1, 1, 11909, 1, 1, 767423, 1, 4079, 6899, 6977, 1, 45763, 1, 1, 1, 1, 1, 4003, 8233, 3719, 99241, 5413, 1, 13103, 5563, 2411, 50291, 268817, 1, 1, 33827, 271217, 1, 30269, 1, 117269, 1, 274837, 103291, 276049, 17291, 10529, 1, 1, 104659, 16453, 1, 842771, 1, 2371, 1, 1, 11833, 853823, 1, 285841, 7673, 41011, 17981, 864947, 4013, 32173, 1, 41543, 1, 1, 1, 1, 110221, 2089, 1, 1, 1, 297061, 55817, 1, 37369, 128393, 1, 1, 3533, 1, 9461, 1, 1, 1, 114487, 1, 2129, 921647, 1, 44071, 16561, 309781, 38803, 15817, 1, 104119, 1, 1, 2311, 17827, 1, 316241, 59417, 15121, 1, 3637, 39937, 320149, 2039, 1, 1, 1, 1, 4001, 2591, 19141, 1, 1, 1, 328037, 61631, 1, 13751, 7573, 1, 47431, 31189,

6. Sequence of the polynom (only primes)

47, 3, 17, 7, 83, 37, 191, 101, 53, 149, 61, 59, 89, 241, 97, 277, 947, 71, 401, 79, 1811, 661, 103, 2351, 283, 331, 131, 2963, 1061, 379, 521, 157, 1297, 251, 1381, 173, 601, 5231, 263, 709, 647, 6131, 307, 7103, 827, 8147, 347, 2837, 1087, 197, 3217, 1231, 1493, 3761, 479, 239, 4049, 773, 1399, 13043, 643, 4657, 14447, 271, 653, 15923, 337, 5477, 269, 17471, 739, 353, 571, 883, 1123, 6737, 2969, 439, 7121, 2707, 3221, 7717, 733, 659, 397, 1583, 541, 431, 1109, 1283, 487, 3067, 1663, 9649, 523, 1249, 30323, 383, 1511, 457, 1367, 11057, 3767, 1459, 12049, 761, 5273, 4759, 12821, 1619, 39251, 1907, 683, 1753, 14437, 911, 619, 4999, 811, 46703, 491, 769, 49331, 16741, 3167, 17041, 7433, 1669, 17957, 7829, 1093, 7027, 6299, 3391, 2791, 19861, 2503, 853, 977, 20849, 617, 21521, 1129, 2797, 22549, 8521, 3271, 719, 7867, 23957, 613, 1531, 24677, 1051, 10889, 1217, 823, 1693, 10159, 27281, 2243, 1741, 4007, 9479, 1193, 907, 29221, 4231, 3727, 1699, 1259, 10139, 5741, 93683, 3929, 1861, 11941, 1187, 13913, 1021, 12409, 33301, 11383, 4349, 105023, 3343, 11959, 108947, 4567, 37201, 2339, 2131, 2053, 5507, 6883, 39461, 1901, 121151, 5077, 40849, 7703, 17909, 14087, 15937, 42737, 18521, 1181, 16477, 4909, 1301, 6451, 8513, 5737, 1933, 2221, 1481, 3863, 5987, 2593, 1019, 147503, 3089, 49681, 152147, 5693, 929, 22409, 6569, 52817, 17783, 1117, 3299, 967, 54421, 20509, 863, 21121, 56597, 7109, 991, 8243, 1553, 6473, 1747, 59377, 8563, 941, 181523, 20359, 23011, 8807, 186671, 3907, 62801, 3797, 6079, 1229, 8179, 1657, 7369, 12491, 1097, 4201, 3433, 25657, 22907, 70549, 213491, 1489, 1931, 10343, 219071, 1567, 3467, 13219, 9403, 75541, 28447, 4481, 4703, 1607, 25819, 29167, 78101, 9803, 33749, 79397, 26683, 2917, 1447, 82021, 5147, 248051, 1153, 4483, 84017, 3061, 2657, 85361, 4591, 5441, 16451, 12583, 1579, 29819, 8419, 90149, 38933, 11399, 91541, 2027, 1319, 35251, 11839, 4561, 9013, 96497, 12107, 291647, 12197, 2633, 32887, 2063, 17539, 1889, 14407, 3163, 304751, 11369, 14723, 1847, 3499, 104549, 35099, 106801, 1823, 3373, 41047, 109841, 2273, 41911, 1223, 3541, 345791, 3119, 16691, 352883, 14753, 1499, 2621, 120817, 121621, 2179, 13693, 7297, 2237, 7393, 11821, 7207, 2281, 18307, 1303, 1277, 8087, 2251, 2713, 6221, 131477, 16487, 396947, 133157, 1789, 1867, 135697, 25523, 19507, 412211, 5743, 2711, 52009, 139121, 3529, 2309, 52981, 5923, 143461, 26981, 18097, 435647, 6869, 1427, 443603, 18539, 7993, 6361, 9437, 4093, 3109, 27043, 1601, 51383, 7247, 2777, 467903, 19553, 2659, 29501, 2617, 60037, 9887, 54139, 3593, 20477, 1471, 23603, 20947, 24007, 9941, 509843, 9181, 24551, 10771, 518447, 10831, 173777, 527123, 2753, 176677, 3011, 3181, 2069, 33767, 2029, 11317, 77813, 3251, 182549, 20393, 553583, 1559, 33091, 3917, 2399, 23753, 571583, 23879, 191537, 2011, 580691, 194581, 27943, 21961, 74311, 3257, 3557, 12227, 6257, 8429, 5113, 3631, 203857, 38321, 617843, 9857, 208049, 3259, 6211, 6551, 4289, 11287, 7823, 636851, 213349, 30631, 10957, 9001, 1951, 20353, 31091, 656147, 27407, 219797, 13907, 13121, 224149, 28087, 2609, 2879, 28499, 685631, 4091, 2539, 76919, 695603, 1709, 12511, 1787, 7333, 1999, 78779, 33923, 29753, 14947, 45053, 3823, 42703, 30319, 243121, 11423, 244261, 105173, 82183, 92671, 247697, 106649, 93967, 83719, 7349, 11909, 767423, 4079, 6899, 6977, 45763, 4003, 8233, 3719, 99241, 5413, 13103, 5563, 2411, 50291, 268817, 33827, 271217, 30269, 117269, 274837, 103291, 276049, 17291, 10529, 104659, 16453, 842771, 2371, 11833, 853823, 285841, 7673, 41011, 17981, 864947, 4013, 32173, 41543, 110221, 2089, 297061, 55817, 37369, 128393, 3533, 9461, 114487, 2129, 921647, 44071, 16561, 309781, 38803, 15817, 104119, 2311, 17827, 316241, 59417, 15121, 3637, 39937, 320149, 2039, 4001, 2591, 19141, 328037, 61631, 13751, 7573, 47431, 31189,

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)
1106420.6000000.4000000.6000000.0000000.0000000.000000
21005222300.5200000.2200000.5200008.6666675.50000015.000000
31.0005931304630.5930000.1300000.59300011.4038465.90909115.433333
410.0006.2279215.3060.6227000.0921000.62270010.5008437.08461511.460043
5100.00064.0397.00557.0340.6403900.0700500.64039010.2840857.60586310.748963
61.000.000649.69756.271593.4260.6496970.0562710.64969710.1453338.03297610.404777
710.000.0006.560.873471.0186.089.8550.6560870.0471020.65608710.0983588.37052810.262197
8100.000.00066.076.7764.054.37062.022.4060.6607680.0405440.66076810.0713398.60767610.184546
91.000.000.000664.424.93935.597.184628.827.7550.6644250.0355970.66442510.0553478.77995510.138720
1010.000.000.0006.673.231.601317.177.5476.356.054.0540.6673230.0317180.66732310.0436198.91018610.107782


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
244311.0000000.7500000.2500001.3333331.5000001.000000
386420.7500000.5000000.2500001.5000001.3333332.000000
4168530.5000000.3125000.1875001.3333331.2500001.500000
53217890.5312500.2500000.2812502.1250001.6000003.000000
6643514210.5468750.2187500.3281252.0588241.7500002.333333
71286826420.5312500.2031250.3281251.9428571.8571432.000000
8256142391030.5546880.1523440.4023442.0882351.5000002.452381
9512301702310.5878910.1367190.4511722.1197181.7948722.242718
101.0246101344760.5957030.1308590.4648442.0265781.9142862.060606
112.0481.2422381.0040.6064450.1162110.4902342.0360661.7761192.109244
124.0962.5214242.0970.6154790.1035160.5119632.0297911.7815132.088645
138.1925.1077734.3340.6234130.0943600.5290532.0257831.8231132.066762
1416.38410.3271.4388.8890.6303100.0877690.5425422.0221261.8602852.050992
1532.76820.7842.58818.1960.6342770.0789790.5552982.0125881.7997222.047024
1665.53641.8024.78237.0200.6378480.0729680.5648802.0112591.8477592.034513
17131.07284.1258.93375.1920.6418230.0681530.5736692.0124641.8680472.031118
18262.144168.99116.643152.3480.6446500.0634880.5811612.0088081.8630922.026120
19524.288339.40331.219308.1840.6473600.0595460.5878142.0084091.8758042.022895
201.048.576681.39558.737622.6580.6498290.0560160.5938132.0076281.8814502.020410
212.097.1521.367.116110.9721.256.1440.6518920.0529160.5989762.0063491.8893032.017390
224.194.3042.742.576210.5002.532.0760.6538810.0501870.6036942.0061031.8968752.015753
238.388.6085.499.951400.1365.099.8150.6556450.0477000.6079452.0053961.9008842.014085
2416.777.21611.026.967762.37610.264.5910.6572580.0454410.6118172.0049211.9052922.012738
2533.554.43222.102.2071.456.64020.645.5670.6586970.0434110.6152862.0043781.9106582.011338
2667.108.86444.294.9302.787.76541.507.1650.6600460.0415410.6185052.0040951.9138332.010464
27134.217.72888.758.1665.346.34383.411.8230.6613000.0398330.6214672.0038001.9177882.009577
28268.435.456177.824.56210.271.634167.552.9280.6624480.0382650.6241832.0034731.9212452.008743
29536.870.912356.225.51819.762.292336.463.2260.6635220.0368100.6267122.0032411.9239682.008101
301.073.741.824713.526.99938.078.144675.448.8550.6645240.0354630.6290612.0030211.9268082.007497
312.147.483.6481.429.055.61273.461.2211.355.594.3910.6654560.0342080.6312482.0028051.9292232.006953
324.294.967.2962.861.858.040141.900.1592.719.957.8810.6663280.0330390.6332902.0026221.9316332.006469
338.589.934.5925.730.779.711274.426.7255.456.352.9860.6671510.0319470.6352032.0024681.9339422.006043
3417.179.869.18411.474.782.194531.342.61810.943.439.5760.6679200.0309280.6369922.0023071.9361912.005633
3534.359.738.36822.974.540.0001.029.811.70921.944.728.2910.6686470.0299710.6386762.0021771.9381312.005286
3668.719.476.73645.996.253.5441.997.843.40043.998.410.1440.6693340.0290720.6402612.0020531.9400092.004965


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
122010101
243110102
384120202
4165130203
5328341313
66414491715
712826817211310
8256391424217515
9512702841926926
101.024134488515482150
112.0482388914834923379
124.0964241482755815663147
138.192773268504103284112274
1416.3841.438507930205523200510
1532.7682.5889281.659367931345945
1665.5364.7821.7003.0816641.7516381.729
17131.0728.9333.1695.7631.2273.2491.1913.266
18262.14416.6435.82810.8142.2456.0592.2476.092
19524.28831.21910.91520.3034.18011.4624.19911.378
201.048.57658.73720.48438.2527.83421.6797.79921.425
212.097.152110.97238.57172.40014.71140.95114.62540.685
224.194.304210.50073.222137.27727.89277.62927.60977.370
238.388.608400.136138.843261.29252.652147.76152.572147.151
2416.777.216762.376263.628498.74799.902281.34699.650281.478
2533.554.4321.456.640503.026953.613190.044538.332190.083538.181
2667.108.8642.787.765961.7281.826.036363.0841.030.349363.7511.030.581
27134.217.7285.346.3431.842.3813.503.961695.9201.976.807696.2141.977.402
28268.435.45610.271.6343.535.1756.736.4581.334.6563.799.3061.336.0163.801.656
29536.870.91219.762.2926.793.00712.969.2842.564.1657.315.0762.565.8247.317.227
301.073.741.82438.078.14413.072.65025.005.4934.934.78314.103.3604.935.15414.104.847
312.147.483.64873.461.22125.197.47448.263.7469.512.01727.219.0849.510.44827.219.672
324.294.967.296141.900.15948.631.99793.268.16118.349.68752.600.52418.346.58952.603.359
338.589.934.592274.426.72593.966.853180.459.87135.446.132101.761.42735.438.726101.780.440
3417.179.869.184531.342.618181.798.301349.544.31668.559.819197.106.52568.551.641197.124.633
3534.359.738.3681.029.811.709352.072.531677.739.177132.747.959382.155.934132.730.431382.177.385
3668.719.476.7361.997.843.400682.491.7321.315.351.667257.266.757741.632.888257.278.045741.665.710


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
121011000
241011000
382111010
4163121020
5329363150
6642111106492
7128421824118158
8256103485530232921
951223111211965546745
101.024476232244134110129103
112.0481.004499505272225281226
124.0962.0971.0271.070574460579484
138.1924.3342.1272.2071.1879541.1761.017
1416.3848.8894.3994.4902.3882.0302.4132.058
1532.76818.1968.9819.2154.9114.2124.8124.261
1665.53637.02018.34218.6789.8458.6599.8458.671
17131.07275.19237.30637.88619.72817.82319.89017.751
18262.144152.34875.80976.53940.00736.17540.08836.078
19524.288308.184153.401154.78380.87773.25280.90373.152
201.048.576622.658310.218312.440162.613148.685162.655148.705
212.097.1521.256.144625.884630.260327.162300.643327.816300.523
224.194.3042.532.0761.262.1721.269.904658.182607.557659.440606.897
238.388.6085.099.8152.541.7532.558.0621.323.6941.225.5241.324.3541.226.243
2416.777.21610.264.5915.116.1155.148.4762.659.0322.472.0472.659.9582.473.554
2533.554.43220.645.56710.290.70210.354.8655.338.4754.984.1865.340.1894.982.717
2667.108.86441.507.16520.692.87120.814.29410.717.95210.036.32110.717.50910.035.383
27134.217.72883.411.82341.590.52141.821.30221.508.36520.196.04021.506.07520.201.343
28268.435.456167.552.92883.552.38184.000.54743.141.01140.627.54743.146.52540.637.845
29536.870.912336.463.226167.796.568168.666.65886.531.82581.693.37486.530.02481.708.003
301.073.741.824675.448.855336.896.587338.552.268173.521.949164.204.068173.516.192164.206.646
312.147.483.6481.355.594.391676.204.665679.389.726347.907.135329.902.553347.873.996329.910.707
324.294.967.2962.719.957.8811.356.922.2151.363.035.666697.413.101662.598.884697.332.191662.613.705
338.589.934.5925.456.352.9862.722.264.1052.734.088.8811.397.772.3211.330.412.3321.397.709.7311.330.458.602
3417.179.869.18410.943.439.5765.460.209.0305.483.230.5462.801.090.4952.670.632.6782.801.025.1632.670.691.240
3534.359.738.36821.944.728.29110.949.967.88410.994.760.4075.612.655.3845.359.698.6475.612.617.4785.359.756.782
3668.719.476.73643.998.410.14421.955.636.28222.042.773.86211.245.151.87210.754.041.64111.245.096.42810.754.120.203


8. Check for existing Integer Sequences by OEIS

Found in Database : 47, 3, 17, 7, 1, 1, 83, 1, 37, 1, 1, 1, 191, 1, 1, 1, 101, 1, 53, 1,
Found in Database : 47, 3, 17, 7, 83, 37, 191, 101, 53, 149, 61, 59, 89, 241, 97, 277, 947, 71, 401, 79,
Found in Database : 3, 7, 17, 37, 47, 53, 59, 61, 71, 79, 83, 89, 97, 101, 103, 131, 149,