Inhaltsverzeichnis

Development of
Algorithmic Constructions

16:43:02
Deutsch
28.Mar 2024

Polynom = x^2-94x+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) = 7 = 7
f(2) = 35 = 5*7
f(3) = 31 = 31
f(4) = 211 = 211
f(5) = 37 = 37
f(6) = 379 = 379
f(7) = 115 = 5*23
f(8) = 539 = 7*7*11
f(9) = 77 = 7*11
f(10) = 691 = 691
f(11) = 191 = 191
f(12) = 835 = 5*167
f(13) = 113 = 113
f(14) = 971 = 971
f(15) = 259 = 7*37
f(16) = 1099 = 7*157
f(17) = 145 = 5*29
f(18) = 1219 = 23*53
f(19) = 319 = 11*29
f(20) = 1331 = 11*11*11
f(21) = 173 = 173
f(22) = 1435 = 5*7*41
f(23) = 371 = 7*53
f(24) = 1531 = 1531
f(25) = 197 = 197
f(26) = 1619 = 1619
f(27) = 415 = 5*83
f(28) = 1699 = 1699
f(29) = 217 = 7*31
f(30) = 1771 = 7*11*23
f(31) = 451 = 11*41
f(32) = 1835 = 5*367
f(33) = 233 = 233
f(34) = 1891 = 31*61
f(35) = 479 = 479
f(36) = 1939 = 7*277
f(37) = 245 = 5*7*7
f(38) = 1979 = 1979
f(39) = 499 = 499
f(40) = 2011 = 2011
f(41) = 253 = 11*23
f(42) = 2035 = 5*11*37
f(43) = 511 = 7*73
f(44) = 2051 = 7*293
f(45) = 257 = 257
f(46) = 2059 = 29*71
f(47) = 515 = 5*103
f(48) = 2059 = 29*71
f(49) = 257 = 257
f(50) = 2051 = 7*293
f(51) = 511 = 7*73
f(52) = 2035 = 5*11*37
f(53) = 253 = 11*23
f(54) = 2011 = 2011
f(55) = 499 = 499
f(56) = 1979 = 1979
f(57) = 245 = 5*7*7
f(58) = 1939 = 7*277
f(59) = 479 = 479
f(60) = 1891 = 31*61
f(61) = 233 = 233
f(62) = 1835 = 5*367
f(63) = 451 = 11*41
f(64) = 1771 = 7*11*23
f(65) = 217 = 7*31
f(66) = 1699 = 1699
f(67) = 415 = 5*83
f(68) = 1619 = 1619
f(69) = 197 = 197
f(70) = 1531 = 1531
f(71) = 371 = 7*53
f(72) = 1435 = 5*7*41
f(73) = 173 = 173
f(74) = 1331 = 11*11*11
f(75) = 319 = 11*29
f(76) = 1219 = 23*53
f(77) = 145 = 5*29
f(78) = 1099 = 7*157
f(79) = 259 = 7*37
f(80) = 971 = 971
f(81) = 113 = 113
f(82) = 835 = 5*167
f(83) = 191 = 191
f(84) = 691 = 691
f(85) = 77 = 7*11
f(86) = 539 = 7*7*11
f(87) = 115 = 5*23
f(88) = 379 = 379
f(89) = 37 = 37
f(90) = 211 = 211
f(91) = 31 = 31
f(92) = 35 = 5*7
f(93) = 7 = 7
f(94) = 149 = 149
f(95) = 61 = 61
f(96) = 341 = 11*31
f(97) = 55 = 5*11
f(98) = 541 = 541
f(99) = 161 = 7*23
f(100) = 749 = 7*107

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^2-94x+149

f(0)=149
f(1)=7
f(2)=5
f(3)=31
f(4)=211
f(5)=37
f(6)=379
f(7)=23
f(8)=11
f(9)=1
f(10)=691
f(11)=191
f(12)=167
f(13)=113
f(14)=971
f(15)=1
f(16)=157
f(17)=29
f(18)=53
f(19)=1
f(20)=1
f(21)=173
f(22)=41
f(23)=1
f(24)=1531
f(25)=197
f(26)=1619
f(27)=83
f(28)=1699
f(29)=1
f(30)=1
f(31)=1
f(32)=367
f(33)=233
f(34)=61
f(35)=479
f(36)=277
f(37)=1
f(38)=1979
f(39)=499
f(40)=2011
f(41)=1
f(42)=1
f(43)=73
f(44)=293
f(45)=257
f(46)=71
f(47)=103
f(48)=1
f(49)=1
f(50)=1
f(51)=1
f(52)=1
f(53)=1
f(54)=1
f(55)=1
f(56)=1
f(57)=1
f(58)=1
f(59)=1
f(60)=1
f(61)=1
f(62)=1
f(63)=1
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)=541
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-94x+149 could be written as f(y)= y^2-2060 with x=y+47

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-47
f'(x)>2x-95 with x > 45

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, 7, 5, 31, 211, 37, 379, 23, 11, 1, 691, 191, 167, 113, 971, 1, 157, 29, 53, 1, 1, 173, 41, 1, 1531, 197, 1619, 83, 1699, 1, 1, 1, 367, 233, 61, 479, 277, 1, 1979, 499, 2011, 1, 1, 73, 293, 257, 71, 103, 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, 541, 1, 107, 1, 193, 269, 1, 163, 1, 1, 151, 223, 1, 509, 433, 1, 347, 641, 1, 1, 271, 1, 467, 1, 1, 929, 1, 503, 1, 1, 643, 1, 439, 1249, 1033, 1, 787, 1, 5861, 1, 6221, 1601, 599, 1, 199, 1789, 7349, 1, 7741, 397, 1163, 1, 1, 1, 1, 1, 229, 1, 1, 251, 331, 2621, 10709, 1367, 1, 1, 401, 1483, 12101, 617, 547, 1, 1867, 3329, 2713, 1, 1279, 3581, 2083, 1, 15101, 1, 15629, 1987, 1, 587, 1, 1, 421, 877, 1, 1, 1, 1, 3793, 1, 1, 1, 1831, 1, 2963, 5261, 577, 2707, 1, 5569, 461, 409, 2111, 1, 1, 3023, 24509, 887, 719, 3187, 1123, 1, 26501, 1, 353, 983, 1, 3527, 1, 7229, 29269, 1, 4283, 1, 2791, 1, 593, 7949, 919, 1, 32909, 1, 821, 1, 34421, 1, 457, 4447, 7193, 1, 36749, 4643, 1, 1, 1667, 1, 3559, 1, 7993, 1, 5827, 10301, 41621, 1051, 42461, 1, 1, 1, 1, 11149, 1217, 5683, 1, 1, 1, 5903, 653, 523, 883, 557, 1, 1783, 607, 1, 1, 12941, 1801, 941, 1, 1, 4919, 6823, 55061, 2777, 1, 1009, 56989, 14369, 11593, 7307, 1, 1, 8563, 1511, 1, 15361, 61949, 1, 1, 2267, 1049, 733, 1, 1, 2131, 1, 9587, 1, 13633, 1, 647, 1, 1, 1, 1741, 17981, 72469, 9127, 14713, 2647, 10667, 9403, 75781, 1, 6991, 1, 1, 1, 1, 9967, 80309, 1, 1, 1, 1, 1, 3643, 1, 16993, 1, 1, 1549, 1, 4397, 2393, 1013, 1, 1, 1, 11447, 92189, 23201, 907, 2351, 13523, 1, 8719, 1097, 19433, 1063, 98429, 1, 14243, 1, 100981, 12703, 3299, 2339, 1, 1861, 104869, 1, 106181, 2671, 1, 3863, 2221, 13687, 2003, 1, 111509, 1, 701, 811, 114221, 1, 115589, 709, 1, 1, 1, 29761, 4129, 3011, 1, 1, 1, 1, 24793, 1, 11399, 1433, 4091, 911, 1, 1, 859, 32609, 1, 16487, 18947, 1, 1, 3371, 135581, 1, 1213, 1, 1, 1201, 140069, 1, 1, 1, 20443, 1, 144629, 1, 1, 18367, 147709, 5303, 21323, 1, 13711, 1, 152389, 1, 1, 5527, 6763, 19543, 157141, 1, 14431, 1, 22907, 40289, 1117, 20347, 5641, 1787, 23603, 1, 166861, 1, 15319, 1, 34033, 1, 24547, 1, 173501, 1, 5651, 22003, 2297, 1, 1, 1, 2539, 45281, 181981, 1, 1, 46141, 185429, 1, 1, 1621, 26987, 3389, 5153, 1, 997, 1, 8443, 6967, 1, 2237, 6379, 937, 5393, 5011, 1, 1, 3833, 1109, 40993, 4679, 1709, 3709, 29803, 10477, 210461, 26423, 7321, 53309, 1, 1, 1, 4931, 217901, 5471, 1, 7883, 31667, 27827, 1, 1, 9803, 1, 2953, 1, 2143, 1, 7459, 58049, 46633, 1, 33587, 59021, 1, 1, 1, 2069, 1, 1, 48593, 1, 1, 1, 246941, 1, 1, 1, 250949, 62989, 50593, 1, 1, 1, 8291, 6451, 2141, 1, 7057, 1, 1, 1, 6469, 1, 267301, 13417, 1327, 1, 1, 1, 54713, 34327, 275669, 9883, 5669, 6971, 2477, 70241, 25639, 3217, 1, 1, 1499, 35923, 288461, 1, 290621, 5209, 1, 6679, 1, 1609, 1823, 74561, 1, 1, 1, 2609, 303749, 1, 5563, 1, 44027, 1, 5857, 1, 3767, 1, 44987, 11287, 1, 3617, 1, 80141, 1, 1153, 1493, 81281, 1, 40927, 65713, 2659, 4297, 1, 11489, 1, 1, 42083, 1, 12107, 9719, 42667, 9257, 1, 1, 1, 49603, 1, 349589, 1, 70393, 88289, 1, 1, 1, 1627, 4327, 1453, 361549, 3943, 10399, 6521, 1, 3169, 1931, 1, 33751, 1, 1, 46867, 3271, 94349, 378629, 1, 54443, 2731, 34871, 4373, 6329, 2617, 77713, 6961, 1, 98081, 393581, 9871, 9661, 1, 1, 1, 80233, 100609, 13921, 1, 14009, 1, 58403, 51263, 1, 1, 82793, 51907, 8501, 14923, 419141, 1, 421741, 105761, 5813, 1, 1, 107069, 429589, 1, 2753, 1, 1, 7789, 437509, 3539, 1, 1, 442829, 1, 2053, 11171, 12113, 2741, 19603, 56527, 12959, 1, 41479, 57203, 458981, 23017, 11261, 8269, 66347, 1, 1, 58567, 42719, 10711, 67523, 1693, 475421, 1, 478189, 1, 1, 1, 2383, 1, 4021, 1, 2459, 61343, 1, 17627, 98993, 62047, 1, 3373, 1, 1, 1, 1, 1, 63467, 101833, 127649, 1, 1, 1, 2347, 1, 1583, 1, 1, 2137, 1, 6343, 1, 529381, 1, 1, 1733, 535229, 1, 2909, 134909, 1, 9689, 77723, 27277, 547021, 1, 49999, 137869, 1, 9901, 1, 139361, 1523, 14011, 2089, 20123, 1, 1, 3917, 142369, 1, 71563, 82003, 4111, 1663, 2333, 14149, 13219, 1, 1, 1, 146941, 2293, 14771, 11177, 4013, 1, 1, 10883, 1, 601669, 75403, 604781, 1, 86843, 1, 1, 5281, 122833, 6997, 8017, 1, 620461, 15551, 623621, 1, 20219, 1, 1, 157889, 57559, 7213, 1, 1, 1, 1, 6007, 3929, 129193, 1, 8893, 2113, 1, 1, 1, 164321, 22721, 82567, 18919, 1, 665429, 83383, 1, 1, 671981, 1, 13781, 1, 1, 85027, 3533, 170881, 97883, 1, 62591, 172541, 691829, 3769, 139033, 1, 99787, 87523, 4057, 1, 1, 1, 14461, 25367, 1, 2411, 4801, 179261, 718741, 1, 1, 16451, 1, 90907, 4703, 4937, 4549, 13109, 735821, 36877, 1, 92623, 1, 2417, 21319, 93487, 12289, 8167, 25969, 1, 3727, 1, 4663, 1, 13883, 1, 1, 13729, 110083, 1, 24971, 1, 21017, 1, 2029, 1, 3719, 1, 788261, 19751, 1, 4049, 1, 1, 159793, 18199, 72959, 3467, 115163, 1, 809741, 1, 35363, 203789, 163393, 14621, 10657, 18691, 7703, 1, 5273, 6691, 118787, 1, 167033, 209249, 27059, 1, 1, 1, 2467, 1, 849869, 2999, 3221, 4649, 1, 1, 2699, 1, 1997, 216641, 10463, 15541, 24919, 218509, 38083, 1, 879661, 4007, 1, 15809, 28619, 222269, 2441, 1, 894749, 1, 5581, 22511, 82031, 20551,

6. Sequence of the polynom (only primes)

149, 7, 5, 31, 211, 37, 379, 23, 11, 691, 191, 167, 113, 971, 157, 29, 53, 173, 41, 1531, 197, 1619, 83, 1699, 367, 233, 61, 479, 277, 1979, 499, 2011, 73, 293, 257, 71, 103, 541, 107, 193, 269, 163, 151, 223, 509, 433, 347, 641, 271, 467, 929, 503, 643, 439, 1249, 1033, 787, 5861, 6221, 1601, 599, 199, 1789, 7349, 7741, 397, 1163, 229, 251, 331, 2621, 10709, 1367, 401, 1483, 12101, 617, 547, 1867, 3329, 2713, 1279, 3581, 2083, 15101, 15629, 1987, 587, 421, 877, 3793, 1831, 2963, 5261, 577, 2707, 5569, 461, 409, 2111, 3023, 24509, 887, 719, 3187, 1123, 26501, 353, 983, 3527, 7229, 29269, 4283, 2791, 593, 7949, 919, 32909, 821, 34421, 457, 4447, 7193, 36749, 4643, 1667, 3559, 7993, 5827, 10301, 41621, 1051, 42461, 11149, 1217, 5683, 5903, 653, 523, 883, 557, 1783, 607, 12941, 1801, 941, 4919, 6823, 55061, 2777, 1009, 56989, 14369, 11593, 7307, 8563, 1511, 15361, 61949, 2267, 1049, 733, 2131, 9587, 13633, 647, 1741, 17981, 72469, 9127, 14713, 2647, 10667, 9403, 75781, 6991, 9967, 80309, 3643, 16993, 1549, 4397, 2393, 1013, 11447, 92189, 23201, 907, 2351, 13523, 8719, 1097, 19433, 1063, 98429, 14243, 100981, 12703, 3299, 2339, 1861, 104869, 106181, 2671, 3863, 2221, 13687, 2003, 111509, 701, 811, 114221, 115589, 709, 29761, 4129, 3011, 24793, 11399, 1433, 4091, 911, 859, 32609, 16487, 18947, 3371, 135581, 1213, 1201, 140069, 20443, 144629, 18367, 147709, 5303, 21323, 13711, 152389, 5527, 6763, 19543, 157141, 14431, 22907, 40289, 1117, 20347, 5641, 1787, 23603, 166861, 15319, 34033, 24547, 173501, 5651, 22003, 2297, 2539, 45281, 181981, 46141, 185429, 1621, 26987, 3389, 5153, 997, 8443, 6967, 2237, 6379, 937, 5393, 5011, 3833, 1109, 40993, 4679, 1709, 3709, 29803, 10477, 210461, 26423, 7321, 53309, 4931, 217901, 5471, 7883, 31667, 27827, 9803, 2953, 2143, 7459, 58049, 46633, 33587, 59021, 2069, 48593, 246941, 250949, 62989, 50593, 8291, 6451, 2141, 7057, 6469, 267301, 13417, 1327, 54713, 34327, 275669, 9883, 5669, 6971, 2477, 70241, 25639, 3217, 1499, 35923, 288461, 290621, 5209, 6679, 1609, 1823, 74561, 2609, 303749, 5563, 44027, 5857, 3767, 44987, 11287, 3617, 80141, 1153, 1493, 81281, 40927, 65713, 2659, 4297, 11489, 42083, 12107, 9719, 42667, 9257, 49603, 349589, 70393, 88289, 1627, 4327, 1453, 361549, 3943, 10399, 6521, 3169, 1931, 33751, 46867, 3271, 94349, 378629, 54443, 2731, 34871, 4373, 6329, 2617, 77713, 6961, 98081, 393581, 9871, 9661, 80233, 100609, 13921, 14009, 58403, 51263, 82793, 51907, 8501, 14923, 419141, 421741, 105761, 5813, 107069, 429589, 2753, 7789, 437509, 3539, 442829, 2053, 11171, 12113, 2741, 19603, 56527, 12959, 41479, 57203, 458981, 23017, 11261, 8269, 66347, 58567, 42719, 10711, 67523, 1693, 475421, 478189, 2383, 4021, 2459, 61343, 17627, 98993, 62047, 3373, 63467, 101833, 127649, 2347, 1583, 2137, 6343, 529381, 1733, 535229, 2909, 134909, 9689, 77723, 27277, 547021, 49999, 137869, 9901, 139361, 1523, 14011, 2089, 20123, 3917, 142369, 71563, 82003, 4111, 1663, 2333, 14149, 13219, 146941, 2293, 14771, 11177, 4013, 10883, 601669, 75403, 604781, 86843, 5281, 122833, 6997, 8017, 620461, 15551, 623621, 20219, 157889, 57559, 7213, 6007, 3929, 129193, 8893, 2113, 164321, 22721, 82567, 18919, 665429, 83383, 671981, 13781, 85027, 3533, 170881, 97883, 62591, 172541, 691829, 3769, 139033, 99787, 87523, 4057, 14461, 25367, 2411, 4801, 179261, 718741, 16451, 90907, 4703, 4937, 4549, 13109, 735821, 36877, 92623, 2417, 21319, 93487, 12289, 8167, 25969, 3727, 4663, 13883, 13729, 110083, 24971, 21017, 2029, 3719, 788261, 19751, 4049, 159793, 18199, 72959, 3467, 115163, 809741, 35363, 203789, 163393, 14621, 10657, 18691, 7703, 5273, 6691, 118787, 167033, 209249, 27059, 2467, 849869, 2999, 3221, 4649, 2699, 1997, 216641, 10463, 15541, 24919, 218509, 38083, 879661, 4007, 15809, 28619, 222269, 2441, 894749, 5581, 22511, 82031, 20551,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-94x+149 and
the reducible primes which appear as divisor for the first time
p | x^2-94x+149 and p < x^2-94x+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)
1109540.9000000.5000000.9000000.0000000.0000000.000000
21003212200.3200000.1200000.3200003.5555562.4000005.000000
31.000572904820.5720000.0900000.57200017.8750007.50000024.100000
410.0006.4077395.6680.6407000.0739000.64070011.2010498.21111111.759336
5100.00065.7685.66060.1080.6576800.0566000.65768010.2650227.65899810.604799
61.000.000663.99646.372617.6240.6639960.0463720.66399610.0960358.19293310.275238
710.000.0006.683.479392.7926.290.6870.6683480.0392790.66834810.0655418.47045610.185302
8100.000.00067.150.8183.408.33163.742.4870.6715080.0340830.67150810.0472858.67719110.132834
91.000.000.000673.927.82430.080.735643.847.0890.6739280.0300810.67392810.0360338.82565010.100753
1010.000.000.0006.758.534.457269.185.8256.489.348.6320.6758530.0269190.67585310.0285748.94877810.079021


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
41614680.8750000.3750000.5000001.7500001.5000002.000000
532229130.6875000.2812500.4062501.5714291.5000001.625000
6643111200.4843750.1718750.3125001.4090911.2222221.538462
71284312310.3359380.0937500.2421881.3870971.0909091.550000
825611527880.4492190.1054690.3437502.6744192.2500002.838710
9512268532150.5234380.1035160.4199222.3304351.9629632.443182
101.024586914950.5722660.0888670.4833982.1865671.7169812.302325
112.0481.2371671.0700.6040040.0815430.5224612.1109211.8351652.161616
124.0962.5443142.2300.6210940.0766600.5444342.0565881.8802392.084112
138.1925.2066204.5860.6354980.0756840.5598142.0463841.9745222.056502
1416.38410.5601.1519.4090.6445310.0702510.5742802.0284291.8564522.051679
1532.76821.3602.07419.2860.6518550.0632930.5885622.0227271.8019112.049740
1665.53643.0283.85739.1710.6565550.0588530.5977022.0144201.8596912.031059
17131.07286.3657.18479.1810.6589130.0548100.6041032.0071811.8625872.021419
18262.144173.25813.520159.7380.6609270.0515750.6093522.0061141.8819602.017378
19524.288347.42925.691321.7380.6626680.0490020.6136672.0052701.9002222.014161
201.048.576696.41448.402648.0120.6641520.0461600.6179922.0044791.8840062.014098
212.097.1521.395.92991.8841.304.0450.6656310.0438140.6218172.0044531.8983512.012378
224.194.3042.797.291174.8172.622.4740.6669260.0416800.6252472.0038921.9025842.011030
238.388.6085.604.504333.5055.270.9990.6681090.0397570.6283522.0035471.9077382.009934
2416.777.21611.227.194637.34010.589.8540.6691930.0379880.6312042.0032451.9110362.009079
2533.554.43222.486.3741.220.37621.265.9980.6701460.0363700.6337762.0028491.9147962.008148
2667.108.86445.032.6102.341.80642.690.8040.6710380.0348960.6361432.0026621.9189222.007468
27134.217.72890.176.0054.499.38985.676.6160.6718640.0335230.6383412.0024601.9213332.006910
28268.435.456180.557.6108.655.722171.901.8880.6726300.0322450.6403842.0022801.9237552.006404
29536.870.912361.494.39616.678.527344.815.8690.6733360.0310660.6422702.0021001.9268792.005888
301.073.741.824723.697.20432.182.685691.514.5190.6739960.0299720.6440232.0019601.9295882.005460
312.147.483.6481.448.724.30462.167.7961.386.556.5080.6746150.0289490.6456662.0018381.9317162.005101
324.294.967.2962.899.929.903120.255.8102.779.674.0930.6751930.0279990.6471932.0017131.9343752.004732
338.589.934.5925.804.557.611232.845.5255.571.712.0860.6757390.0271070.6486332.0016201.9362522.004448
3417.179.869.18411.617.924.809451.324.34711.166.600.4620.6762520.0262710.6499822.0015181.9383002.004160


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
122020110
243120210
384220310
4166330510
5329540810
664116501010
7128127501020
825627918010170
9512531934010430
101.024913160010810
112.048167591080101570
124.0963141102040103040
138.1926202144060106100
1416.3841.1513867650101.1410
1532.7682.0746871.3870102.0640
1665.5363.8571.2982.5590103.8470
17131.0727.1842.4074.7770107.1740
18262.14413.5204.4999.02101013.5100
19524.28825.6918.59117.10001025.6810
201.048.57648.40216.17932.22301048.3920
212.097.15291.88430.69961.18501091.8740
224.194.304174.81758.364116.453010174.8070
238.388.608333.505110.911222.594010333.4950
2416.777.216637.340212.265425.075010637.3300
2533.554.4321.220.376406.735813.6410101.220.3660
2667.108.8642.341.806779.6661.562.1400102.341.7960
27134.217.7284.499.3891.498.6663.000.7230104.499.3790
28268.435.4568.655.7222.883.7345.771.9880108.655.7120
29536.870.91216.678.5275.558.57711.119.95001016.678.5170
301.073.741.82432.182.68510.724.23521.458.45001032.182.6750
312.147.483.64862.167.79620.717.46541.450.33101062.167.7860
324.294.967.296120.255.81040.079.59680.176.214010120.255.8000
338.589.934.592232.845.52577.610.266155.235.259010232.845.5150
3417.179.869.184451.324.347150.430.418300.893.929010451.324.3370


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
242200002
384310013
4168441025
53213581156
664208123278
712831121965911
825688414719222126
951221510810750614658
101.024495246249133137105120
112.0481.070542528291292236251
124.0962.2301.1431.087603609476542
138.1924.5862.3472.2391.2071.2549951.130
1416.3849.4094.7714.6382.4662.5282.0832.332
1532.76819.2869.7129.5745.0265.2324.2844.744
1665.53639.17119.72219.44910.29810.5478.7559.571
17131.07279.18139.77039.41120.61121.16217.94319.465
18262.144159.73880.25179.48741.60342.30436.48539.346
19524.288321.738161.819159.91983.26684.79073.96379.719
201.048.576648.012326.395321.617167.360170.392149.692160.568
212.097.1521.304.045656.017648.028336.515341.430303.205322.895
224.194.3042.622.4741.319.4621.303.012676.015685.159611.400649.900
238.388.6085.270.9992.650.4302.620.5691.355.5001.375.5141.233.3291.306.656
2416.777.21610.589.8545.321.4625.268.3922.718.7312.757.7772.486.2562.627.090
2533.554.43221.265.99810.685.57910.580.4195.454.3695.525.3485.011.7145.274.567
2667.108.86442.690.80421.447.60321.243.20110.939.61711.068.46410.086.87910.595.844
27134.217.72885.676.61643.030.79242.645.82421.926.37622.174.87120.301.71621.273.653
28268.435.456171.901.88886.324.94885.576.94043.948.59244.430.13840.835.29442.687.864
29536.870.912344.815.869173.132.847171.683.02288.073.44388.999.30382.099.82585.643.298
301.073.741.824691.514.519347.164.821344.349.698176.470.290178.264.266164.978.867171.801.096
312.147.483.6481.386.556.508695.983.040690.573.468353.574.553357.020.719331.396.280344.564.956
324.294.967.2962.779.674.0931.395.048.5611.384.625.532708.321.131714.952.442665.513.955690.886.565
338.589.934.5925.571.712.0862.795.962.1762.775.749.9101.418.832.9351.431.644.5621.336.099.0751.385.135.514
3417.179.869.18411.166.600.4625.602.929.0075.563.671.4552.841.767.3432.866.588.8142.681.732.0222.776.512.283


8. Check for existing Integer Sequences by OEIS

Found in Database : 149, 7, 5, 31, 211, 37, 379, 23, 11, 1, 691, 191, 167, 113, 971, 1, 157, 29, 53, 1,
Found in Database : 149, 7, 5, 31, 211, 37, 379, 23, 11, 691, 191, 167, 113, 971, 157, 29, 53, 173, 41, 1531, 197, 1619, 83, 1699, 367, 233, 61, 479, 277, 1979, 499,
Found in Database : 5, 7, 11, 23, 29, 31, 37, 41, 53, 61, 71, 73, 83, 103, 107, 113, 149,