Inhaltsverzeichnis

Development of
Algorithmic Constructions

12:10:44
Deutsch
29.Mar 2024

Polynom = x^2+3x-23

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) = 23 = 23
f(1) = 19 = 19
f(2) = 13 = 13
f(3) = 5 = 5
f(4) = 5 = 5
f(5) = 17 = 17
f(6) = 31 = 31
f(7) = 47 = 47
f(8) = 65 = 5*13
f(9) = 85 = 5*17
f(10) = 107 = 107
f(11) = 131 = 131
f(12) = 157 = 157
f(13) = 185 = 5*37
f(14) = 215 = 5*43
f(15) = 247 = 13*19
f(16) = 281 = 281
f(17) = 317 = 317
f(18) = 355 = 5*71
f(19) = 395 = 5*79
f(20) = 437 = 19*23
f(21) = 481 = 13*37
f(22) = 527 = 17*31
f(23) = 575 = 5*5*23
f(24) = 625 = 5*5*5*5
f(25) = 677 = 677
f(26) = 731 = 17*43
f(27) = 787 = 787
f(28) = 845 = 5*13*13
f(29) = 905 = 5*181
f(30) = 967 = 967
f(31) = 1031 = 1031
f(32) = 1097 = 1097
f(33) = 1165 = 5*233
f(34) = 1235 = 5*13*19
f(35) = 1307 = 1307
f(36) = 1381 = 1381
f(37) = 1457 = 31*47
f(38) = 1535 = 5*307
f(39) = 1615 = 5*17*19
f(40) = 1697 = 1697
f(41) = 1781 = 13*137
f(42) = 1867 = 1867
f(43) = 1955 = 5*17*23
f(44) = 2045 = 5*409
f(45) = 2137 = 2137
f(46) = 2231 = 23*97
f(47) = 2327 = 13*179
f(48) = 2425 = 5*5*97
f(49) = 2525 = 5*5*101
f(50) = 2627 = 37*71
f(51) = 2731 = 2731
f(52) = 2837 = 2837
f(53) = 2945 = 5*19*31
f(54) = 3055 = 5*13*47
f(55) = 3167 = 3167
f(56) = 3281 = 17*193
f(57) = 3397 = 43*79
f(58) = 3515 = 5*19*37
f(59) = 3635 = 5*727
f(60) = 3757 = 13*17*17
f(61) = 3881 = 3881
f(62) = 4007 = 4007
f(63) = 4135 = 5*827
f(64) = 4265 = 5*853
f(65) = 4397 = 4397
f(66) = 4531 = 23*197
f(67) = 4667 = 13*359
f(68) = 4805 = 5*31*31
f(69) = 4945 = 5*23*43
f(70) = 5087 = 5087
f(71) = 5231 = 5231
f(72) = 5377 = 19*283
f(73) = 5525 = 5*5*13*17
f(74) = 5675 = 5*5*227
f(75) = 5827 = 5827
f(76) = 5981 = 5981
f(77) = 6137 = 17*19*19
f(78) = 6295 = 5*1259
f(79) = 6455 = 5*1291
f(80) = 6617 = 13*509
f(81) = 6781 = 6781
f(82) = 6947 = 6947
f(83) = 7115 = 5*1423
f(84) = 7285 = 5*31*47
f(85) = 7457 = 7457
f(86) = 7631 = 13*587
f(87) = 7807 = 37*211
f(88) = 7985 = 5*1597
f(89) = 8165 = 5*23*71
f(90) = 8347 = 17*491
f(91) = 8531 = 19*449
f(92) = 8717 = 23*379
f(93) = 8905 = 5*13*137
f(94) = 9095 = 5*17*107
f(95) = 9287 = 37*251
f(96) = 9481 = 19*499
f(97) = 9677 = 9677
f(98) = 9875 = 5*5*5*79
f(99) = 10075 = 5*5*13*31
f(100) = 10277 = 43*239

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+3x-23

f(0)=23
f(1)=19
f(2)=13
f(3)=5
f(4)=1
f(5)=17
f(6)=31
f(7)=47
f(8)=1
f(9)=1
f(10)=107
f(11)=131
f(12)=157
f(13)=37
f(14)=43
f(15)=1
f(16)=281
f(17)=317
f(18)=71
f(19)=79
f(20)=1
f(21)=1
f(22)=1
f(23)=1
f(24)=1
f(25)=677
f(26)=1
f(27)=787
f(28)=1
f(29)=181
f(30)=967
f(31)=1031
f(32)=1097
f(33)=233
f(34)=1
f(35)=1307
f(36)=1381
f(37)=1
f(38)=307
f(39)=1
f(40)=1697
f(41)=137
f(42)=1867
f(43)=1
f(44)=409
f(45)=2137
f(46)=97
f(47)=179
f(48)=1
f(49)=101
f(50)=1
f(51)=2731
f(52)=2837
f(53)=1
f(54)=1
f(55)=3167
f(56)=193
f(57)=1
f(58)=1
f(59)=727
f(60)=1
f(61)=3881
f(62)=4007
f(63)=827
f(64)=853
f(65)=4397
f(66)=197
f(67)=359
f(68)=1
f(69)=1
f(70)=5087
f(71)=5231
f(72)=283
f(73)=1
f(74)=227
f(75)=5827
f(76)=5981
f(77)=1
f(78)=1259
f(79)=1291
f(80)=509
f(81)=6781
f(82)=6947
f(83)=1423
f(84)=1
f(85)=7457
f(86)=587
f(87)=211
f(88)=1597
f(89)=1
f(90)=491
f(91)=449
f(92)=379
f(93)=1
f(94)=1
f(95)=251
f(96)=499
f(97)=9677
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+3x-23 could be written as f(y)= y^2-25.25 with x=y-1.5

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+1.5
f'(x)>2x+2

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

23, 19, 13, 5, 1, 17, 31, 47, 1, 1, 107, 131, 157, 37, 43, 1, 281, 317, 71, 79, 1, 1, 1, 1, 1, 677, 1, 787, 1, 181, 967, 1031, 1097, 233, 1, 1307, 1381, 1, 307, 1, 1697, 137, 1867, 1, 409, 2137, 97, 179, 1, 101, 1, 2731, 2837, 1, 1, 3167, 193, 1, 1, 727, 1, 3881, 4007, 827, 853, 4397, 197, 359, 1, 1, 5087, 5231, 283, 1, 227, 5827, 5981, 1, 1259, 1291, 509, 6781, 6947, 1423, 1, 7457, 587, 211, 1597, 1, 491, 449, 379, 1, 1, 251, 499, 9677, 1, 1, 239, 223, 10687, 2179, 2221, 11317, 887, 691, 2393, 2437, 653, 743, 1, 2617, 2663, 1, 13781, 1, 2851, 1, 14737, 1, 15227, 619, 1, 1229, 16231, 16487, 1, 1, 557, 373, 1, 3613, 1, 809, 1, 19157, 1, 3943, 19997, 1193, 1, 1, 4229, 1, 701, 22027, 1, 1, 1, 1787, 23537, 1, 4831, 569, 24781, 25097, 1, 5147, 367, 1, 1571, 5407, 421, 27697, 28031, 1493, 5741, 1, 29387, 2287, 1583, 1217, 1231, 1831, 31481, 1, 1, 383, 32917, 1447, 33647, 6803, 1, 34757, 1, 35507, 7177, 7253, 2819, 1949, 1, 7561, 7639, 821, 2293, 1, 1, 1607, 40577, 1, 41387, 643, 1, 2243, 1163, 1889, 1, 521, 1, 45131, 45557, 541, 9283, 593, 3637, 47717, 9631, 9719, 49037, 49481, 49927, 1, 1, 1091, 1, 2269, 10529, 1, 1, 761, 54497, 10993, 11087, 1511, 4337, 56857, 11467, 1, 601, 1367, 1, 1, 12049, 60737, 61231, 3631, 1, 1, 2749, 631, 64237, 563, 1, 5059, 839, 66797, 13463, 13567, 4021, 68881, 1, 1, 829, 70997, 1, 3793, 1117, 14629, 2377, 74231, 1, 1, 607, 5879, 3347, 4561, 15619, 15731, 2141, 1, 80347, 16183, 1, 2647, 4349, 83207, 1289, 1, 84947, 85531, 86117, 17341, 1, 87887, 3847, 1, 1, 1, 4783, 1, 1297, 18539, 18661, 4943, 94531, 1, 1, 1, 97007, 5743, 98257, 19777, 1531, 1, 3251, 2741, 20411, 1, 7949, 103981, 4549, 4211, 1, 1, 683, 1, 1277, 21841, 1, 1033, 1, 1721, 1, 1433, 2423, 1181, 1213, 1, 8969, 117281, 1, 1249, 23869, 1, 1, 3917, 977, 1, 123527, 124231, 1237, 1933, 1, 1, 127781, 6763, 1, 1999, 130657, 131381, 1, 857, 26713, 5839, 1, 991, 1187, 27449, 3209, 138731, 10729, 1, 5639, 1, 7499, 1, 929, 1, 145517, 7699, 147047, 1, 29717, 11489, 150131, 150907, 1319, 30493, 153247, 1, 11909, 31121, 1009, 8273, 9293, 158777, 1, 1283, 1, 3767, 162787, 32719, 1, 1, 166031, 166847, 33533, 1087, 169307, 1, 170957, 1, 1, 1621, 174281, 10301, 2707, 1861, 4801, 178481, 179327, 7207, 1, 5867, 1, 183587, 997, 37061, 1, 14387, 187897, 1987, 1, 190507, 1973, 1, 1, 38803, 6287, 195781, 1019, 39511, 1, 199337, 200231, 11831, 8081, 8117, 15679, 12043, 1, 1, 41491, 208367, 1, 1, 42223, 42407, 1, 213881, 214807, 3319, 2549, 5881, 218531, 219467, 2593, 44269, 17099, 1, 224177, 1801, 9043, 227027, 1, 4871, 45979, 46171, 10079, 13693, 233747, 1, 47137, 13921, 237631, 1, 47917, 3701, 12713, 1, 243517, 1, 1327, 12973, 19037, 248477, 1, 1, 251477, 252481, 1, 2213, 1, 256517, 11197, 5501, 51913, 1, 1997, 1451, 263657, 52937, 2797, 1, 267781, 268817, 1741, 3187, 1409, 272981, 1, 11003, 1, 277177, 12097, 1, 1, 2447, 6569, 1, 16741, 1, 57347, 1, 16993, 289957, 58207, 1579, 1, 22637, 6869, 1, 59509, 2791, 299731, 1, 1, 1, 304127, 305231, 1, 3617, 1, 309667, 310781, 311897, 62603, 62827, 315257, 24337, 1, 63727, 2063, 1439, 1, 24859, 1753, 65089, 19211, 1, 1, 1, 1, 14449, 9013, 334637, 1, 67391, 1, 1, 340447, 4019, 1, 1, 4861, 1567, 69497, 1, 18413, 351031, 352217, 5437, 70919, 1, 356981, 1427, 1, 14423, 1637, 7723, 364187, 73079, 1, 9941, 28387, 370247, 74293, 3923, 4733, 12101, 1, 1, 1, 380047, 8867, 22501, 1, 5923, 386237, 1, 388727, 1, 15649, 392477, 1, 394987, 1, 1, 4111, 400031, 30869, 80513, 4751, 2099, 406381, 407657, 1, 6311, 2621, 1, 1, 83071, 3623, 2473, 419231, 22133, 1, 1, 11471, 1, 1, 85669, 85931, 1, 432281, 1, 6691, 2029, 437557, 23099, 440207, 1, 3851, 1, 1, 19429, 5273, 89909, 450887, 1, 26681, 1, 18251, 1, 458981, 460337, 7103, 2503, 24443, 465781, 467147, 93703, 7229, 1459, 472631, 1, 3067, 1, 20789, 36887, 2657, 96461, 96739, 10321, 486481, 37529, 19571, 1, 492077, 493481, 1, 99259, 1, 499117, 29443, 501947, 100673, 100957, 1693, 507631, 10831, 1, 2381, 513347, 13913, 39709, 5449, 1, 2467, 521981, 523427, 1, 1, 5441, 3863, 5471, 1499, 106721, 1, 536531, 2309, 4691, 108187, 17497, 1, 28703, 109367, 109663, 32341, 551281, 1, 8527, 111149, 12959, 558731, 560227, 22469, 1733, 18217, 566231, 567737, 1, 1, 2521, 1, 1, 115363, 1, 8167, 1, 44839, 1, 117193, 587497, 19001, 590567, 1, 9133, 1, 1, 4567, 23993, 1, 2441, 604481, 606037, 1, 5297, 32143, 19751, 2053, 123083, 123397, 1723, 1, 36571, 1, 2659, 16931, 36943, 629617, 126241, 6661, 48799, 635981, 1, 1, 1, 14939, 49537, 28069, 129439, 1, 28279, 13873, 653647, 1, 1663, 658507, 660131, 21347, 6983, 1, 666647, 1, 2803, 7069, 134639, 674837, 3061, 6991, 27191, 1, 1747, 684731, 1, 1, 137941, 1, 1, 36563, 139273, 10739, 18911, 14923, 37003, 8291, 141283, 54469, 1, 41851, 1, 142969, 4003, 1, 1, 1, 28933, 1913, 1, 728437, 1, 1, 43151, 38699, 7297, 147743, 1, 57089, 743881, 1823, 1, 4049, 750797, 1, 754267, 151201, 151549, 1, 1, 1, 1, 1, 40433, 45293, 2069, 154699, 11927, 2531, 1, 1, 3329, 156817, 785857, 1, 789407, 2003, 1, 1, 796531, 61409, 9413, 1, 803687, 3457, 35099, 32363, 1, 17291, 22013, 11497, 1, 163981, 2039, 1, 825347, 8707, 4481, 48871, 832631, 64189, 8803, 167623, 6131, 4273, 1, 13007, 1, 36919, 1, 852827, 2011, 34261, 66029, 10889, 1, 1, 4027, 18461, 1, 45863, 174653, 1, 1, 28351, 1, 1, 176903, 2267, 12511, 4973, 7757, 1, 2339, 1, 19141, 36061, 36137, 8461, 3673, 29327, 182209, 1, 914867, 916781, 4157, 1, 1, 21499, 54493, 928307, 8089, 14341, 1, 40697, 1, 1, 188369, 3821, 945731, 22039, 1, 2239, 25771, 955481, 1, 11287, 192271, 1, 1, 4261, 1, 10223, 2609, 42397, 977107, 195817, 1, 75619, 57943, 1, 1, 4217, 58411, 76537, 1, 1, 40039,

6. Sequence of the polynom (only primes)

23, 19, 13, 5, 17, 31, 47, 107, 131, 157, 37, 43, 281, 317, 71, 79, 677, 787, 181, 967, 1031, 1097, 233, 1307, 1381, 307, 1697, 137, 1867, 409, 2137, 97, 179, 101, 2731, 2837, 3167, 193, 727, 3881, 4007, 827, 853, 4397, 197, 359, 5087, 5231, 283, 227, 5827, 5981, 1259, 1291, 509, 6781, 6947, 1423, 7457, 587, 211, 1597, 491, 449, 379, 251, 499, 9677, 239, 223, 10687, 2179, 2221, 11317, 887, 691, 2393, 2437, 653, 743, 2617, 2663, 13781, 2851, 14737, 15227, 619, 1229, 16231, 16487, 557, 373, 3613, 809, 19157, 3943, 19997, 1193, 4229, 701, 22027, 1787, 23537, 4831, 569, 24781, 25097, 5147, 367, 1571, 5407, 421, 27697, 28031, 1493, 5741, 29387, 2287, 1583, 1217, 1231, 1831, 31481, 383, 32917, 1447, 33647, 6803, 34757, 35507, 7177, 7253, 2819, 1949, 7561, 7639, 821, 2293, 1607, 40577, 41387, 643, 2243, 1163, 1889, 521, 45131, 45557, 541, 9283, 593, 3637, 47717, 9631, 9719, 49037, 49481, 49927, 1091, 2269, 10529, 761, 54497, 10993, 11087, 1511, 4337, 56857, 11467, 601, 1367, 12049, 60737, 61231, 3631, 2749, 631, 64237, 563, 5059, 839, 66797, 13463, 13567, 4021, 68881, 829, 70997, 3793, 1117, 14629, 2377, 74231, 607, 5879, 3347, 4561, 15619, 15731, 2141, 80347, 16183, 2647, 4349, 83207, 1289, 84947, 85531, 86117, 17341, 87887, 3847, 4783, 1297, 18539, 18661, 4943, 94531, 97007, 5743, 98257, 19777, 1531, 3251, 2741, 20411, 7949, 103981, 4549, 4211, 683, 1277, 21841, 1033, 1721, 1433, 2423, 1181, 1213, 8969, 117281, 1249, 23869, 3917, 977, 123527, 124231, 1237, 1933, 127781, 6763, 1999, 130657, 131381, 857, 26713, 5839, 991, 1187, 27449, 3209, 138731, 10729, 5639, 7499, 929, 145517, 7699, 147047, 29717, 11489, 150131, 150907, 1319, 30493, 153247, 11909, 31121, 1009, 8273, 9293, 158777, 1283, 3767, 162787, 32719, 166031, 166847, 33533, 1087, 169307, 170957, 1621, 174281, 10301, 2707, 1861, 4801, 178481, 179327, 7207, 5867, 183587, 997, 37061, 14387, 187897, 1987, 190507, 1973, 38803, 6287, 195781, 1019, 39511, 199337, 200231, 11831, 8081, 8117, 15679, 12043, 41491, 208367, 42223, 42407, 213881, 214807, 3319, 2549, 5881, 218531, 219467, 2593, 44269, 17099, 224177, 1801, 9043, 227027, 4871, 45979, 46171, 10079, 13693, 233747, 47137, 13921, 237631, 47917, 3701, 12713, 243517, 1327, 12973, 19037, 248477, 251477, 252481, 2213, 256517, 11197, 5501, 51913, 1997, 1451, 263657, 52937, 2797, 267781, 268817, 1741, 3187, 1409, 272981, 11003, 277177, 12097, 2447, 6569, 16741, 57347, 16993, 289957, 58207, 1579, 22637, 6869, 59509, 2791, 299731, 304127, 305231, 3617, 309667, 310781, 311897, 62603, 62827, 315257, 24337, 63727, 2063, 1439, 24859, 1753, 65089, 19211, 14449, 9013, 334637, 67391, 340447, 4019, 4861, 1567, 69497, 18413, 351031, 352217, 5437, 70919, 356981, 1427, 14423, 1637, 7723, 364187, 73079, 9941, 28387, 370247, 74293, 3923, 4733, 12101, 380047, 8867, 22501, 5923, 386237, 388727, 15649, 392477, 394987, 4111, 400031, 30869, 80513, 4751, 2099, 406381, 407657, 6311, 2621, 83071, 3623, 2473, 419231, 22133, 11471, 85669, 85931, 432281, 6691, 2029, 437557, 23099, 440207, 3851, 19429, 5273, 89909, 450887, 26681, 18251, 458981, 460337, 7103, 2503, 24443, 465781, 467147, 93703, 7229, 1459, 472631, 3067, 20789, 36887, 2657, 96461, 96739, 10321, 486481, 37529, 19571, 492077, 493481, 99259, 499117, 29443, 501947, 100673, 100957, 1693, 507631, 10831, 2381, 513347, 13913, 39709, 5449, 2467, 521981, 523427, 5441, 3863, 5471, 1499, 106721, 536531, 2309, 4691, 108187, 17497, 28703, 109367, 109663, 32341, 551281, 8527, 111149, 12959, 558731, 560227, 22469, 1733, 18217, 566231, 567737, 2521, 115363, 8167, 44839, 117193, 587497, 19001, 590567, 9133, 4567, 23993, 2441, 604481, 606037, 5297, 32143, 19751, 2053, 123083, 123397, 1723, 36571, 2659, 16931, 36943, 629617, 126241, 6661, 48799, 635981, 14939, 49537, 28069, 129439, 28279, 13873, 653647, 1663, 658507, 660131, 21347, 6983, 666647, 2803, 7069, 134639, 674837, 3061, 6991, 27191, 1747, 684731, 137941, 36563, 139273, 10739, 18911, 14923, 37003, 8291, 141283, 54469, 41851, 142969, 4003, 28933, 1913, 728437, 43151, 38699, 7297, 147743, 57089, 743881, 1823, 4049, 750797, 754267, 151201, 151549, 40433, 45293, 2069, 154699, 11927, 2531, 3329, 156817, 785857, 789407, 2003, 796531, 61409, 9413, 803687, 3457, 35099, 32363, 17291, 22013, 11497, 163981, 2039, 825347, 8707, 4481, 48871, 832631, 64189, 8803, 167623, 6131, 4273, 13007, 36919, 852827, 2011, 34261, 66029, 10889, 4027, 18461, 45863, 174653, 28351, 176903, 2267, 12511, 4973, 7757, 2339, 19141, 36061, 36137, 8461, 3673, 29327, 182209, 914867, 916781, 4157, 21499, 54493, 928307, 8089, 14341, 40697, 188369, 3821, 945731, 22039, 2239, 25771, 955481, 11287, 192271, 4261, 10223, 2609, 42397, 977107, 195817, 75619, 57943, 4217, 58411, 76537, 40039,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+3x-23 and
the reducible primes which appear as divisor for the first time
p | x^2+3x-23 and p < x^2+3x-23

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)
1107700.7000000.7000000.0000000.0000000.0000000.000000
21006836320.6800000.3600000.3200009.7142865.142857inf
31.0007092075020.7090000.2070000.50200010.4264715.75000015.687500
410.0007.0871.4935.5940.7087000.1493000.5594009.9957697.21256011.143426
5100.00070.69811.51659.1820.7069800.1151600.5918209.9757307.71332910.579550
61.000.000704.62294.228610.3940.7046220.0942280.6103949.9666478.18235510.313846
710.000.0007.027.775794.3786.233.3970.7027780.0794380.6233409.9738238.43038210.212088
8100.000.00070.135.4776.882.08663.253.3910.7013550.0688210.6325349.9797568.66349010.147499
91.000.000.000700.304.70560.717.232639.587.4730.7003050.0607170.6395879.9850278.82250410.111512
1010.000.000.0006.994.787.892543.358.0956.451.429.7970.6994790.0543360.6451439.9882068.94899310.086862


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)
123301.5000001.5000000.0000000.0000000.0000000.000000
243300.7500000.7500000.0000001.0000001.000000-nan
386600.7500000.7500000.0000002.0000002.000000-nan
416121020.7500000.6250000.1250002.0000001.666667inf
532211650.6562500.5000000.1562501.7500001.6000002.500000
6644227150.6562500.4218750.2343752.0000001.6875003.000000
71288943460.6953120.3359380.3593752.1190481.5925933.066667
8256180701100.7031250.2734380.4296882.0224721.6279072.391304
95123661262400.7148440.2460940.4687502.0333331.8000002.181818
101.0247272115160.7099610.2060550.5039061.9863391.6746032.150000
112.0481.4643841.0800.7148440.1875000.5273442.0137551.8199052.093023
124.0962.9176862.2310.7121580.1674800.5446781.9924861.7864582.065741
138.1925.8111.2444.5670.7093510.1518550.5574951.9921151.8134112.047064
1416.38411.6002.2859.3150.7080080.1394650.5685421.9962141.8368172.039632
1532.76823.2174.23818.9790.7085270.1293330.5791932.0014661.8547052.037467
1665.53646.3357.89738.4380.7070160.1204990.5865171.9957361.8633792.025291
17131.07292.63014.74777.8830.7067110.1125110.5942001.9991371.8674182.026198
18262.144185.16927.732157.4370.7063640.1057890.6005741.9990181.8805182.021455
19524.288369.78452.065317.7190.7053070.0993060.6060011.9970081.8774342.018071
201.048.576738.85698.413640.4430.7046280.0938540.6107741.9980751.8901952.015753
212.097.1521.476.375186.1541.290.2210.7039900.0887650.6152251.9981901.8915592.014576
224.194.3042.950.088353.9232.596.1650.7033560.0843820.6189741.9981971.9012382.012186
238.388.6085.896.321674.1165.222.2050.7028960.0803610.6225351.9986931.9046972.011507
2416.777.21611.784.1251.288.59510.495.5300.7023890.0768060.6255821.9985561.9115332.009789
2533.554.43223.553.4412.466.83721.086.6040.7019470.0735170.6284301.9987431.9143622.009103
2667.108.86447.082.4374.727.37342.355.0640.7015830.0704430.6311401.9989621.9163702.008624
27134.217.72894.115.1959.081.13785.034.0580.7012130.0676600.6335531.9989451.9209692.007648
28268.435.456188.139.96717.469.025170.670.9420.7008760.0650770.6357991.9990391.9236602.007089
29536.870.912376.111.77433.666.448342.445.3260.7005630.0627090.6378541.9991061.9272082.006465
301.073.741.824751.914.14564.959.380686.954.7650.7002750.0604980.6397761.9991771.9294992.006028
312.147.483.6481.503.255.144125.501.5571.377.753.5870.7000080.0584410.6415671.9992381.9320012.005596
324.294.967.2963.005.432.931242.739.1152.762.693.8160.6997570.0565170.6432401.9992831.9341522.005216
338.589.934.5926.008.878.563470.003.8855.538.874.6780.6995260.0547160.6448101.9993391.9362512.004882
3417.179.869.18412.014.108.476910.981.60611.103.126.8700.6993130.0530260.6462871.9993931.9382432.004582
3534.359.738.36824.021.323.3501.767.388.62022.253.934.7300.6991130.0514380.6476751.9994261.9400932.004294
3668.719.476.73648.029.684.2083.432.041.67344.597.642.5350.6989240.0499430.6489811.9994601.9418722.004034


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
123210111
243210111
386331113
41610462323
532166103445
6642710176777
71284316278101312
825670244617152216
9512126428430303531
101.0242117213948545851
112.04838413025483101100100
124.096686231455164176179167
138.1921.244427817314316315299
1416.3842.2857681.517590571569555
1532.7684.2381.4272.8111.0851.0581.0481.047
1665.5367.8972.6385.2592.0021.9811.9571.957
17131.07214.7474.9309.8173.6853.6773.7263.659
18262.14427.7329.22418.5086.9426.9036.9586.929
19524.28852.06517.40134.66413.05012.96013.03413.021
201.048.57698.41332.94165.47224.66124.48324.70924.560
212.097.152186.15462.046124.10846.62346.46446.70646.361
224.194.304353.923118.076235.84788.46688.41788.87188.169
238.388.608674.116224.786449.330168.529168.426168.955168.206
2416.777.2161.288.595429.602858.993322.292322.065322.669321.569
2533.554.4322.466.837822.7541.644.083617.152616.836616.896615.953
2667.108.8644.727.3731.576.0373.151.3361.182.8261.181.8361.181.3551.181.356
27134.217.7289.081.1373.027.7506.053.3872.271.7172.270.7642.269.1022.269.554
28268.435.45617.469.0255.823.06411.645.9614.367.8864.370.2754.365.4734.365.391
29536.870.91233.666.44811.222.90222.443.5468.417.1378.417.6238.415.1768.416.512
301.073.741.82464.959.38021.651.32443.308.05616.240.15416.238.81416.241.01716.239.395
312.147.483.648125.501.55741.829.68883.671.86931.378.07231.374.19331.378.25931.371.033
324.294.967.296242.739.11580.911.309161.827.80660.687.95860.681.98860.690.01260.679.157
338.589.934.592470.003.885156.663.259313.340.626117.496.596117.500.336117.506.970117.499.983
3417.179.869.184910.981.606303.665.073607.316.533227.746.802227.739.630227.753.152227.742.022
3534.359.738.3681.767.388.620589.129.3551.178.259.265441.843.240441.834.049441.866.898441.844.433
3668.719.476.7363.432.041.6731.143.999.3012.288.042.372858.006.499858.011.074858.024.526857.999.574


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
380000000
4162200110
5325410122
664151055433
71284625218181010
8256110545623312630
951224012211855586760
101.024516266250122127138129
112.0481.080563517260286284250
124.0962.2311.1561.075554569568540
138.1924.5672.3532.2141.0951.1851.1451.142
1416.3849.3154.7924.5232.2952.3352.3282.357
1532.76818.9799.7029.2774.6774.7784.7234.801
1665.53638.43819.67018.7689.4989.6539.6639.624
17131.07277.88339.85238.03119.31619.50919.67219.386
18262.144157.43780.41477.02339.18039.48339.53839.236
19524.288317.719161.984155.73579.33579.53379.61079.241
201.048.576640.443326.179314.264160.041160.098160.277160.027
212.097.1521.290.221657.021633.200322.900322.713322.503322.105
224.194.3042.596.1651.321.0761.275.089649.175648.995649.024648.971
238.388.6085.222.2052.654.5132.567.6921.306.3531.304.9531.305.1031.305.796
2416.777.21610.495.5305.333.7795.161.7512.623.4272.623.3722.624.2282.624.503
2533.554.43221.086.60410.704.94910.381.6555.271.3445.271.5835.270.7225.272.955
2667.108.86442.355.06421.485.96020.869.10410.589.35910.587.84010.585.41310.592.452
27134.217.72885.034.05843.104.93641.929.12221.259.29621.260.96321.252.90621.260.893
28268.435.456170.670.94286.463.43084.207.51242.667.03042.672.47042.660.91542.670.527
29536.870.912342.445.326173.396.923169.048.40385.612.49785.617.09885.599.90885.615.823
301.073.741.824686.954.765347.681.567339.273.198171.737.120171.753.542171.725.508171.738.595
312.147.483.6481.377.753.587696.973.355680.780.232344.449.292344.460.640344.412.430344.431.225
324.294.967.2962.762.693.8161.396.984.7561.365.709.060690.684.003690.693.603690.645.477690.670.733
338.589.934.5925.538.874.6782.799.653.2752.739.221.4031.384.731.2791.384.738.4311.384.681.8691.384.723.099
3417.179.869.18411.103.126.8705.610.041.7495.493.085.1212.775.766.2222.775.867.1962.775.710.8362.775.782.616
3534.359.738.36822.253.934.73011.240.344.22411.013.590.5065.563.500.3905.563.560.2415.563.410.5525.563.463.547
3668.719.476.73644.597.642.53522.518.833.91022.078.808.62511.149.519.25211.149.423.21011.149.350.47711.149.349.596


8. Check for existing Integer Sequences by OEIS

Found in Database : 23, 19, 13, 5, 1, 17, 31, 47, 1, 1, 107, 131, 157, 37, 43, 1, 281, 317, 71, 79,
Found in Database : 23, 19, 13, 5, 17, 31, 47, 107, 131, 157, 37, 43, 281, 317, 71, 79, 677, 787, 181, 967, 1031, 1097, 233, 1307, 1381, 307,
Found in Database : 5, 13, 17, 19, 23, 31, 37, 43, 47, 71, 79, 97, 101, 107, 131, 137,