Inhaltsverzeichnis

Development of
Algorithmic Constructions

22:09:30
Deutsch
28.Mar 2024

Polynom = x^2+4x-31

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) = 31 = 31
f(1) = 13 = 13
f(2) = 19 = 19
f(3) = 5 = 5
f(4) = 1 = 1
f(5) = 7 = 7
f(6) = 29 = 29
f(7) = 23 = 23
f(8) = 65 = 5*13
f(9) = 43 = 43
f(10) = 109 = 109
f(11) = 67 = 67
f(12) = 161 = 7*23
f(13) = 95 = 5*19
f(14) = 221 = 13*17
f(15) = 127 = 127
f(16) = 289 = 17*17
f(17) = 163 = 163
f(18) = 365 = 5*73
f(19) = 203 = 7*29
f(20) = 449 = 449
f(21) = 247 = 13*19
f(22) = 541 = 541
f(23) = 295 = 5*59
f(24) = 641 = 641
f(25) = 347 = 347
f(26) = 749 = 7*107
f(27) = 403 = 13*31
f(28) = 865 = 5*173
f(29) = 463 = 463
f(30) = 989 = 23*43
f(31) = 527 = 17*31
f(32) = 1121 = 19*59
f(33) = 595 = 5*7*17
f(34) = 1261 = 13*97
f(35) = 667 = 23*29
f(36) = 1409 = 1409
f(37) = 743 = 743
f(38) = 1565 = 5*313
f(39) = 823 = 823
f(40) = 1729 = 7*13*19
f(41) = 907 = 907
f(42) = 1901 = 1901
f(43) = 995 = 5*199
f(44) = 2081 = 2081
f(45) = 1087 = 1087
f(46) = 2269 = 2269
f(47) = 1183 = 7*13*13
f(48) = 2465 = 5*17*29
f(49) = 1283 = 1283
f(50) = 2669 = 17*157
f(51) = 1387 = 19*73
f(52) = 2881 = 43*67
f(53) = 1495 = 5*13*23
f(54) = 3101 = 7*443
f(55) = 1607 = 1607
f(56) = 3329 = 3329
f(57) = 1723 = 1723
f(58) = 3565 = 5*23*31
f(59) = 1843 = 19*97
f(60) = 3809 = 13*293
f(61) = 1967 = 7*281
f(62) = 4061 = 31*131
f(63) = 2095 = 5*419
f(64) = 4321 = 29*149
f(65) = 2227 = 17*131
f(66) = 4589 = 13*353
f(67) = 2363 = 17*139
f(68) = 4865 = 5*7*139
f(69) = 2503 = 2503
f(70) = 5149 = 19*271
f(71) = 2647 = 2647
f(72) = 5441 = 5441
f(73) = 2795 = 5*13*43
f(74) = 5741 = 5741
f(75) = 2947 = 7*421
f(76) = 6049 = 23*263
f(77) = 3103 = 29*107
f(78) = 6365 = 5*19*67
f(79) = 3263 = 13*251
f(80) = 6689 = 6689
f(81) = 3427 = 23*149
f(82) = 7021 = 7*17*59
f(83) = 3595 = 5*719
f(84) = 7361 = 17*433
f(85) = 3767 = 3767
f(86) = 7709 = 13*593
f(87) = 3943 = 3943
f(88) = 8065 = 5*1613
f(89) = 4123 = 7*19*31
f(90) = 8429 = 8429
f(91) = 4307 = 59*73
f(92) = 8801 = 13*677
f(93) = 4495 = 5*29*31
f(94) = 9181 = 9181
f(95) = 4687 = 43*109
f(96) = 9569 = 7*1367
f(97) = 4883 = 19*257
f(98) = 9965 = 5*1993
f(99) = 5083 = 13*17*23
f(100) = 10369 = 10369

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+4x-31

f(0)=31
f(1)=13
f(2)=19
f(3)=5
f(4)=1
f(5)=7
f(6)=29
f(7)=23
f(8)=1
f(9)=43
f(10)=109
f(11)=67
f(12)=1
f(13)=1
f(14)=17
f(15)=127
f(16)=1
f(17)=163
f(18)=73
f(19)=1
f(20)=449
f(21)=1
f(22)=541
f(23)=59
f(24)=641
f(25)=347
f(26)=107
f(27)=1
f(28)=173
f(29)=463
f(30)=1
f(31)=1
f(32)=1
f(33)=1
f(34)=97
f(35)=1
f(36)=1409
f(37)=743
f(38)=313
f(39)=823
f(40)=1
f(41)=907
f(42)=1901
f(43)=199
f(44)=2081
f(45)=1087
f(46)=2269
f(47)=1
f(48)=1
f(49)=1283
f(50)=157
f(51)=1
f(52)=1
f(53)=1
f(54)=443
f(55)=1607
f(56)=3329
f(57)=1723
f(58)=1
f(59)=1
f(60)=293
f(61)=281
f(62)=131
f(63)=419
f(64)=149
f(65)=1
f(66)=353
f(67)=139
f(68)=1
f(69)=2503
f(70)=271
f(71)=2647
f(72)=5441
f(73)=1
f(74)=5741
f(75)=421
f(76)=263
f(77)=1
f(78)=1
f(79)=251
f(80)=6689
f(81)=1
f(82)=1
f(83)=719
f(84)=433
f(85)=3767
f(86)=593
f(87)=3943
f(88)=1613
f(89)=1
f(90)=8429
f(91)=1
f(92)=677
f(93)=1
f(94)=9181
f(95)=1
f(96)=1367
f(97)=257
f(98)=1993
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+4x-31 could be written as f(y)= y^2-35 with x=y-2

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+2
f'(x)>2x+3

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

31, 13, 19, 5, 1, 7, 29, 23, 1, 43, 109, 67, 1, 1, 17, 127, 1, 163, 73, 1, 449, 1, 541, 59, 641, 347, 107, 1, 173, 463, 1, 1, 1, 1, 97, 1, 1409, 743, 313, 823, 1, 907, 1901, 199, 2081, 1087, 2269, 1, 1, 1283, 157, 1, 1, 1, 443, 1607, 3329, 1723, 1, 1, 293, 281, 131, 419, 149, 1, 353, 139, 1, 2503, 271, 2647, 5441, 1, 5741, 421, 263, 1, 1, 251, 6689, 1, 1, 719, 433, 3767, 593, 3943, 1613, 1, 8429, 1, 677, 1, 9181, 1, 1367, 257, 1993, 1, 10369, 311, 10781, 1, 487, 439, 401, 5923, 1, 6143, 1787, 6367, 997, 1319, 13421, 6827, 1, 1009, 1, 1, 479, 7547, 1, 1559, 1, 619, 16349, 1, 3373, 8563, 17389, 1, 17921, 1, 18461, 1, 19009, 9643, 1, 9923, 20129, 1, 1, 2099, 1637, 1, 1151, 11083, 4493, 11383, 1, 1, 1, 2399, 1279, 397, 1, 971, 5113, 1, 26209, 13267, 26861, 2719, 1, 733, 4027, 839, 1, 859, 2273, 14947, 30241, 1, 30941, 15647, 31649, 1231, 6473, 16363, 1, 389, 1091, 1, 1, 17467, 1, 2549, 7213, 18223, 2833, 809, 1979, 1, 5483, 19387, 1, 1, 7993, 20183, 691, 1, 967, 1, 1, 21407, 1, 1, 1259, 1, 44909, 1193, 683, 1, 2027, 3361, 1, 773, 569, 1061, 2897, 24847, 1, 5059, 1187, 25747, 51949, 26203, 1, 1, 1, 27127, 54721, 5519, 55661, 1, 8087, 1, 1, 29023, 547, 1553, 1, 857, 1951, 709, 61469, 30983, 1, 1657, 9067, 1103, 3793, 1, 3853, 2539, 66529, 4789, 13513, 577, 1, 2659, 69661, 7019, 10103, 1549, 71789, 1, 1, 1, 1013, 1, 75041, 7559, 5857, 1237, 77249, 38903, 2239, 1, 2741, 3079, 80621, 1, 81761, 5881, 4877, 1, 1, 42323, 85229, 1, 12343, 8699, 6737, 44087, 3061, 44683, 947, 6469, 7013, 45887, 92381, 1, 93601, 1, 1, 3671, 19213, 1667, 1, 2129, 98561, 1, 1, 50227, 101089, 2677, 1, 51503, 1, 52147, 6173, 10559, 811, 1, 8273, 1, 21773, 2381, 1, 1289, 111521, 863, 701, 56767, 6011, 1, 797, 1, 116929, 1, 1, 1, 6299, 1, 1, 1, 3499, 61583, 1277, 1, 1, 1, 1, 1, 7537, 2801, 25913, 5011, 131009, 65867, 1, 1, 853, 5179, 135389, 2347, 883, 9829, 138349, 4091, 1, 827, 1, 3089, 20407, 1217, 2221, 1, 6343, 73327, 7759, 1, 148961, 1, 2551, 1129, 1789, 76423, 1291, 5939, 155201, 821, 156781, 78787, 1, 11369, 1, 2593, 161569, 4273, 163181, 1, 1811, 4871, 166429, 4919, 33613, 84443, 169709, 937, 1, 1, 173021, 86927, 2393, 1, 5039, 1, 9371, 3889, 1, 18059, 1, 1, 7963, 92003, 36973, 92863, 1, 4933, 26903, 18919, 6131, 95467, 3251, 7411, 38713, 1, 195329, 1, 197101, 1523, 8647, 1693, 1, 1039, 40493, 4421, 1, 102587, 206081, 2957, 1, 1, 1, 105323, 1, 106243, 1, 3457, 215261, 1663, 217121, 3517, 218989, 1, 1, 1, 7681, 111847, 9767, 1327, 32363, 6691, 17573, 6037, 46073, 115663, 3467, 16661, 1, 1, 1699, 1, 238109, 1, 1, 1, 1, 121507, 1, 24499, 245981, 1, 1, 1, 49993, 4327, 251969, 126487, 2791, 1, 4339, 1201, 1583, 1, 4001, 1097, 262109, 1, 9109, 1153, 266221, 1, 38327, 1, 2351, 1399, 272449, 1, 1, 1, 16273, 3229, 1, 4513, 1, 1931, 40427, 1303, 285121, 28619, 1163, 144167, 12583, 20749, 58313, 146323, 293729, 1, 295901, 1747, 1, 7873, 6983, 1, 60493, 151783, 1531, 21841, 1, 1621, 1, 1423, 311329, 5387, 1, 6841, 1429, 1481, 4357, 1, 1, 22961, 322589, 12451, 1511, 163003, 1, 164147, 2477, 2543, 2113, 9791, 334049, 9859, 67273, 24109, 26053, 1, 14827, 1801, 1, 5557, 1, 7541, 1, 1, 350429, 1, 20753, 1, 1229, 178187, 357569, 179383, 71993, 1, 51767, 181787, 1, 36599, 367201, 8009, 28433, 26489, 74413, 10979, 1, 1, 1, 1, 1, 6563, 1, 191563, 76873, 14831, 12479, 1459, 6599, 1, 2819, 1, 23197, 197803, 1, 10477, 399389, 1, 1, 1753, 404461, 1, 1, 204143, 6301, 205423, 1, 2131, 59243, 2447, 14389, 1, 1433, 6793, 4447, 30269, 3347, 1, 427681, 42899, 10007, 3221, 2689, 7487, 6701, 11497, 1, 219767, 25933, 6317, 1, 1, 446189, 11777, 1, 1381, 1, 17419, 7699, 1571, 19867, 1, 459649, 1, 1, 1, 465089, 1, 15091, 46919, 5171, 235967, 1, 237343, 95213, 238723, 36833, 34301, 1, 48299, 28493, 242887, 28657, 1, 13999, 245683, 492769, 247087, 1, 3823, 498401, 1879, 501229, 251323, 1, 1, 38993, 14951, 72823, 1, 512621, 8863, 2087, 1973, 2411, 1, 1, 261347, 7823, 1, 27739, 11489, 75707, 265703, 6269, 20551, 31517, 1, 538721, 7717, 541661, 14293, 41893, 273043, 1, 1, 1, 276007, 42577, 1, 1, 16411, 559469, 2357, 1, 1, 565469, 283487, 568481, 56999, 4297, 22039, 574529, 288023, 115513, 289543, 20021, 1, 1, 1, 34513, 294127, 25643, 295663, 1, 1, 595949, 1, 1697, 1, 602141, 1, 605249, 2389, 1, 17939, 1, 1, 87803, 1, 617761, 309667, 620909, 1, 9601, 1, 10631, 314407, 33179, 63199, 1, 317587, 5351, 319183, 7529, 5437, 33851, 24799, 1, 9257, 15107, 325607, 21059, 25171, 131213, 2207, 1, 17393, 28807, 3907, 51217, 1, 669089, 2083, 4637, 17737, 51973, 4639, 678941, 68059, 97463, 341947, 1, 26431, 1, 345263, 2143, 1709, 1, 1, 1, 1, 1, 11353, 1, 1777, 2371, 355307, 712301, 71399, 715681, 51241, 55313, 1, 1, 1, 1, 2777, 104183, 5623, 10037, 367207, 10987, 19417, 1, 4073, 25621, 372367, 1, 3253, 1, 1, 1, 377563, 1, 8821, 5987, 381047, 58757, 10937, 1, 384547, 1, 386303, 154873, 29851, 111127, 1, 1, 1, 784961, 30259, 34283, 2971, 1, 13687, 18503, 1, 4729, 1, 1, 1, 806369, 1, 1, 13093, 47857, 1, 5879, 81899, 1, 1, 43391, 4259, 1, 18041, 831709, 32059, 19427, 83719, 44159, 3533, 1, 1, 1, 424103, 850049, 1, 121963, 1, 65957, 22613, 27779, 1, 172973, 61909, 28019, 33479, 1, 1, 1, 438967, 125687, 33911, 176713, 1, 2003, 444607, 891101, 12757, 3623, 448387, 898669, 15527, 1, 1, 1, 26711, 2521, 91199, 1, 1, 917729, 1, 1, 1, 925409, 14957, 929261, 1, 133303, 467527, 55117, 1, 11069, 471403, 72673, 3559, 3607, 4133, 952541, 3203, 1, 1, 1, 3673, 964289, 483127, 1, 1, 972161, 4093, 976109, 489043, 6323, 1, 1, 4523, 1, 98999, 991981, 3049, 3331, 3929, 4651, 71569,

6. Sequence of the polynom (only primes)

31, 13, 19, 5, 7, 29, 23, 43, 109, 67, 17, 127, 163, 73, 449, 541, 59, 641, 347, 107, 173, 463, 97, 1409, 743, 313, 823, 907, 1901, 199, 2081, 1087, 2269, 1283, 157, 443, 1607, 3329, 1723, 293, 281, 131, 419, 149, 353, 139, 2503, 271, 2647, 5441, 5741, 421, 263, 251, 6689, 719, 433, 3767, 593, 3943, 1613, 8429, 677, 9181, 1367, 257, 1993, 10369, 311, 10781, 487, 439, 401, 5923, 6143, 1787, 6367, 997, 1319, 13421, 6827, 1009, 479, 7547, 1559, 619, 16349, 3373, 8563, 17389, 17921, 18461, 19009, 9643, 9923, 20129, 2099, 1637, 1151, 11083, 4493, 11383, 2399, 1279, 397, 971, 5113, 26209, 13267, 26861, 2719, 733, 4027, 839, 859, 2273, 14947, 30241, 30941, 15647, 31649, 1231, 6473, 16363, 389, 1091, 17467, 2549, 7213, 18223, 2833, 809, 1979, 5483, 19387, 7993, 20183, 691, 967, 21407, 1259, 44909, 1193, 683, 2027, 3361, 773, 569, 1061, 2897, 24847, 5059, 1187, 25747, 51949, 26203, 27127, 54721, 5519, 55661, 8087, 29023, 547, 1553, 857, 1951, 709, 61469, 30983, 1657, 9067, 1103, 3793, 3853, 2539, 66529, 4789, 13513, 577, 2659, 69661, 7019, 10103, 1549, 71789, 1013, 75041, 7559, 5857, 1237, 77249, 38903, 2239, 2741, 3079, 80621, 81761, 5881, 4877, 42323, 85229, 12343, 8699, 6737, 44087, 3061, 44683, 947, 6469, 7013, 45887, 92381, 93601, 3671, 19213, 1667, 2129, 98561, 50227, 101089, 2677, 51503, 52147, 6173, 10559, 811, 8273, 21773, 2381, 1289, 111521, 863, 701, 56767, 6011, 797, 116929, 6299, 3499, 61583, 1277, 7537, 2801, 25913, 5011, 131009, 65867, 853, 5179, 135389, 2347, 883, 9829, 138349, 4091, 827, 3089, 20407, 1217, 2221, 6343, 73327, 7759, 148961, 2551, 1129, 1789, 76423, 1291, 5939, 155201, 821, 156781, 78787, 11369, 2593, 161569, 4273, 163181, 1811, 4871, 166429, 4919, 33613, 84443, 169709, 937, 173021, 86927, 2393, 5039, 9371, 3889, 18059, 7963, 92003, 36973, 92863, 4933, 26903, 18919, 6131, 95467, 3251, 7411, 38713, 195329, 197101, 1523, 8647, 1693, 1039, 40493, 4421, 102587, 206081, 2957, 105323, 106243, 3457, 215261, 1663, 217121, 3517, 218989, 7681, 111847, 9767, 1327, 32363, 6691, 17573, 6037, 46073, 115663, 3467, 16661, 1699, 238109, 121507, 24499, 245981, 49993, 4327, 251969, 126487, 2791, 4339, 1201, 1583, 4001, 1097, 262109, 9109, 1153, 266221, 38327, 2351, 1399, 272449, 16273, 3229, 4513, 1931, 40427, 1303, 285121, 28619, 1163, 144167, 12583, 20749, 58313, 146323, 293729, 295901, 1747, 7873, 6983, 60493, 151783, 1531, 21841, 1621, 1423, 311329, 5387, 6841, 1429, 1481, 4357, 22961, 322589, 12451, 1511, 163003, 164147, 2477, 2543, 2113, 9791, 334049, 9859, 67273, 24109, 26053, 14827, 1801, 5557, 7541, 350429, 20753, 1229, 178187, 357569, 179383, 71993, 51767, 181787, 36599, 367201, 8009, 28433, 26489, 74413, 10979, 6563, 191563, 76873, 14831, 12479, 1459, 6599, 2819, 23197, 197803, 10477, 399389, 1753, 404461, 204143, 6301, 205423, 2131, 59243, 2447, 14389, 1433, 6793, 4447, 30269, 3347, 427681, 42899, 10007, 3221, 2689, 7487, 6701, 11497, 219767, 25933, 6317, 446189, 11777, 1381, 17419, 7699, 1571, 19867, 459649, 465089, 15091, 46919, 5171, 235967, 237343, 95213, 238723, 36833, 34301, 48299, 28493, 242887, 28657, 13999, 245683, 492769, 247087, 3823, 498401, 1879, 501229, 251323, 38993, 14951, 72823, 512621, 8863, 2087, 1973, 2411, 261347, 7823, 27739, 11489, 75707, 265703, 6269, 20551, 31517, 538721, 7717, 541661, 14293, 41893, 273043, 276007, 42577, 16411, 559469, 2357, 565469, 283487, 568481, 56999, 4297, 22039, 574529, 288023, 115513, 289543, 20021, 34513, 294127, 25643, 295663, 595949, 1697, 602141, 605249, 2389, 17939, 87803, 617761, 309667, 620909, 9601, 10631, 314407, 33179, 63199, 317587, 5351, 319183, 7529, 5437, 33851, 24799, 9257, 15107, 325607, 21059, 25171, 131213, 2207, 17393, 28807, 3907, 51217, 669089, 2083, 4637, 17737, 51973, 4639, 678941, 68059, 97463, 341947, 26431, 345263, 2143, 1709, 11353, 1777, 2371, 355307, 712301, 71399, 715681, 51241, 55313, 2777, 104183, 5623, 10037, 367207, 10987, 19417, 4073, 25621, 372367, 3253, 377563, 8821, 5987, 381047, 58757, 10937, 384547, 386303, 154873, 29851, 111127, 784961, 30259, 34283, 2971, 13687, 18503, 4729, 806369, 13093, 47857, 5879, 81899, 43391, 4259, 18041, 831709, 32059, 19427, 83719, 44159, 3533, 424103, 850049, 121963, 65957, 22613, 27779, 172973, 61909, 28019, 33479, 438967, 125687, 33911, 176713, 2003, 444607, 891101, 12757, 3623, 448387, 898669, 15527, 26711, 2521, 91199, 917729, 925409, 14957, 929261, 133303, 467527, 55117, 11069, 471403, 72673, 3559, 3607, 4133, 952541, 3203, 3673, 964289, 483127, 972161, 4093, 976109, 489043, 6323, 4523, 98999, 991981, 3049, 3331, 3929, 4651, 71569,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+4x-31 and
the reducible primes which appear as divisor for the first time
p | x^2+4x-31 and p < x^2+4x-31

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
21006718490.6700000.1800000.6700007.4444454.5000009.800000
31.0006911215700.6910000.1210000.69100010.3134336.72222211.632653
410.0006.9628496.1130.6962000.0849000.69620010.0752537.01652910.724562
5100.00069.6986.45163.2470.6969800.0645100.69698010.0112047.59835110.346312
61.000.000696.25452.382643.8720.6962540.0523820.6962549.9895848.11998210.180277
710.000.0006.955.700443.0516.512.6490.6955700.0443050.6955709.9901768.45807710.114820
8100.000.00069.507.4043.839.08665.668.3180.6950740.0383910.6950749.9928708.66511110.083197
91.000.000.000694.724.83033.857.833660.866.9970.6947250.0338580.6947259.9949758.81924210.063711
1010.000.000.0006.944.647.370303.022.5196.641.624.8510.6944650.0303020.6944659.9962568.94985010.049867


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
244221.0000000.5000000.5000001.3333331.0000002.000000
387340.8750000.3750000.5000001.7500001.5000002.000000
41611470.6875000.2500000.4375001.5714291.3333331.750000
532217140.6562500.2187500.4375001.9090911.7500002.000000
6644312310.6718750.1875000.4843752.0476191.7142862.214286
71288721660.6796880.1640620.5156252.0232561.7500002.129032
8256175371380.6835940.1445310.5390622.0114941.7619052.090909
9512353712820.6894530.1386720.5507812.0171431.9189192.043478
101.0247121225900.6953120.1191410.5761722.0169971.7183102.092199
112.0481.4362261.2100.7011720.1103520.5908202.0168541.8524592.050848
124.0962.8523962.4560.6962890.0966800.5996091.9860721.7522122.029752
138.1925.6827084.9740.6936040.0864260.6071781.9922861.7878792.025244
1416.38411.4191.30110.1180.6969600.0794070.6175542.0096801.8375712.034178
1532.76822.8052.38220.4230.6959530.0726930.6232601.9971101.8308992.018482
1665.53645.6734.42541.2480.6969150.0675200.6293952.0027631.8576832.019684
17131.07291.3168.21783.0990.6966860.0626910.6339951.9993431.8569492.014619
18262.144182.65915.478167.1810.6967890.0590440.6377452.0002961.8836562.011829
19524.288365.20028.934336.2660.6965640.0551870.6413761.9993541.8693632.011389
201.048.576730.02354.787675.2360.6962040.0522490.6439551.9989681.8935162.008041
212.097.1521.459.480103.9461.355.5340.6959340.0495650.6463691.9992251.8972752.007497
224.194.3042.918.411197.2992.721.1120.6958030.0470400.6487641.9996241.8980912.007410
238.388.6085.835.445376.0205.459.4250.6956390.0448250.6508141.9995281.9058382.006321
2416.777.21611.667.586718.49310.949.0930.6954420.0428260.6526171.9994341.9107842.005540
2533.554.43223.329.7281.374.23021.955.4980.6952800.0409550.6543251.9995331.9126562.005234
2667.108.86446.649.5302.637.16944.012.3610.6951320.0392970.6558351.9995741.9190162.004617
27134.217.72893.284.4825.065.21888.219.2640.6950240.0377390.6572851.9996881.9207032.004420
28268.435.456186.538.9169.743.547176.795.3690.6949120.0362980.6586141.9996781.9236182.004045
29536.870.912373.021.11718.771.758354.249.3590.6948060.0349650.6598411.9996961.9265842.003725
301.073.741.824745.947.79536.222.232709.725.5630.6947180.0337350.6609831.9997471.9296132.003463
312.147.483.6481.491.701.90569.984.6191.421.717.2860.6946280.0325890.6620391.9997401.9320902.003193
324.294.967.2962.983.088.046135.361.1402.847.726.9060.6945540.0315160.6630381.9997881.9341562.003019
338.589.934.5925.965.531.109262.108.2805.703.422.8290.6944790.0305130.6639661.9997841.9363632.002799
3417.179.869.18411.929.970.704508.040.55211.421.930.1520.6944160.0295720.6648441.9998171.9382852.002645


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
122200101
242200101
383210111
4164310121
5327432131
66412575151
71282171481111
8256371423161191
9512712447321371
101.0241223983581621
112.0482268014611111131
124.09639613526119611981
138.19270823647234113651
1416.3841.30142687563216671
1532.7682.3827951.5871.18411.1961
1665.5364.4251.4962.9292.20312.2201
17131.0728.2172.7515.4664.11014.1051
18262.14415.4785.15910.3197.70017.7761
19524.28828.9349.65419.28014.430114.5021
201.048.57654.78718.18536.60227.386127.3991
212.097.152103.94634.68969.25751.968151.9761
224.194.304197.29965.800131.49998.658198.6391
238.388.608376.020125.249250.771187.9241188.0941
2416.777.216718.493239.536478.957359.0111359.4801
2533.554.4321.374.230458.127916.103687.0411687.1871
2667.108.8642.637.169878.7621.758.4071.318.63111.318.5361
27134.217.7285.065.2181.688.7673.376.4512.532.47212.532.7441
28268.435.4569.743.5473.247.6856.495.8624.871.93914.871.6061
29536.870.91218.771.7586.257.30812.514.4509.386.39119.385.3651
301.073.741.82436.222.23212.073.08624.149.14618.112.360118.109.8701
312.147.483.64869.984.61923.326.10246.658.51734.993.438134.991.1791
324.294.967.296135.361.14045.114.60390.246.53767.680.617167.680.5211
338.589.934.592262.108.28087.370.926174.737.354131.053.2941131.054.9841
3417.179.869.184508.040.552169.345.911338.694.641254.015.2491254.025.3011


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
121100010
242110020
384220022
4167520223
53214861634
66431161541269
712866323411191125
8256138706825462245
951228214513753905683
101.024590301289115179118178
112.0481.210624586246366248350
124.0962.4561.2371.219508722498728
138.1924.9742.5832.3911.0461.4301.0601.438
1416.38410.1185.2764.8422.1482.8482.2132.909
1532.76820.42310.6559.7684.4275.6984.5025.796
1665.53641.24821.37519.8739.12811.4199.16211.539
17131.07283.09943.01640.08318.56422.89018.66522.980
18262.144167.18186.17881.00337.63645.84637.87045.829
19524.288336.266173.228163.03876.47291.42476.78091.590
201.048.576675.236347.098328.138154.792182.971154.404183.069
212.097.1521.355.534695.398660.136312.707365.794311.719365.314
224.194.3042.721.1121.394.3401.326.772629.602731.825629.173730.512
238.388.6085.459.4252.794.0612.665.3641.267.4331.462.0991.268.5781.461.315
2416.777.21610.949.0935.596.8335.352.2602.552.1312.922.5442.553.9802.920.438
2533.554.43221.955.49811.211.48310.744.0155.136.5095.840.9575.138.4775.839.555
2667.108.86444.012.36122.454.60621.557.75510.329.12511.678.98410.330.86011.673.392
27134.217.72888.219.26444.967.88043.251.38420.762.96723.351.79120.762.40723.342.099
28268.435.456176.795.36990.045.03686.750.33341.710.52546.690.02341.716.17946.678.642
29536.870.912354.249.359180.311.191173.938.16883.774.03493.352.72183.779.34193.343.263
301.073.741.824709.725.563360.997.327348.728.236168.201.087186.657.372168.214.670186.652.434
312.147.483.6481.421.717.286722.707.789699.009.497337.615.210373.233.916337.638.246373.229.914
324.294.967.2962.847.726.9061.446.778.0851.400.948.821677.512.964746.372.047677.518.728746.323.167
338.589.934.5925.703.422.8292.896.055.0502.807.367.7791.359.254.2261.492.469.5001.359.274.1381.492.424.965
3417.179.869.18411.421.930.1525.796.889.0975.625.041.0552.726.387.9472.984.584.0452.726.485.9732.984.472.187


8. Check for existing Integer Sequences by OEIS

Found in Database : 31, 13, 19, 5, 1, 7, 29, 23, 1, 43, 109, 67, 1, 1, 17, 127, 1, 163, 73, 1,
Found in Database : 31, 13, 19, 5, 7, 29, 23, 43, 109, 67, 17, 127, 163, 73, 449, 541, 59, 641, 347, 107, 173, 463, 97, 1409, 743, 313, 823,
Found in Database : 5, 7, 13, 17, 19, 23, 29, 31, 43, 59, 67, 73, 97, 107, 109, 127, 131, 139, 149,