Inhaltsverzeichnis

Development of
Algorithmic Constructions

15:30:58
Deutsch
29.Mar 2024

Polynom = x^2+1x-5

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) = 5 = 5
f(1) = 3 = 3
f(2) = 1 = 1
f(3) = 7 = 7
f(4) = 15 = 3*5
f(5) = 25 = 5*5
f(6) = 37 = 37
f(7) = 51 = 3*17
f(8) = 67 = 67
f(9) = 85 = 5*17
f(10) = 105 = 3*5*7
f(11) = 127 = 127
f(12) = 151 = 151
f(13) = 177 = 3*59
f(14) = 205 = 5*41
f(15) = 235 = 5*47
f(16) = 267 = 3*89
f(17) = 301 = 7*43
f(18) = 337 = 337
f(19) = 375 = 3*5*5*5
f(20) = 415 = 5*83
f(21) = 457 = 457
f(22) = 501 = 3*167
f(23) = 547 = 547
f(24) = 595 = 5*7*17
f(25) = 645 = 3*5*43
f(26) = 697 = 17*41
f(27) = 751 = 751
f(28) = 807 = 3*269
f(29) = 865 = 5*173
f(30) = 925 = 5*5*37
f(31) = 987 = 3*7*47
f(32) = 1051 = 1051
f(33) = 1117 = 1117
f(34) = 1185 = 3*5*79
f(35) = 1255 = 5*251
f(36) = 1327 = 1327
f(37) = 1401 = 3*467
f(38) = 1477 = 7*211
f(39) = 1555 = 5*311
f(40) = 1635 = 3*5*109
f(41) = 1717 = 17*101
f(42) = 1801 = 1801
f(43) = 1887 = 3*17*37
f(44) = 1975 = 5*5*79
f(45) = 2065 = 5*7*59
f(46) = 2157 = 3*719
f(47) = 2251 = 2251
f(48) = 2347 = 2347
f(49) = 2445 = 3*5*163
f(50) = 2545 = 5*509
f(51) = 2647 = 2647
f(52) = 2751 = 3*7*131
f(53) = 2857 = 2857
f(54) = 2965 = 5*593
f(55) = 3075 = 3*5*5*41
f(56) = 3187 = 3187
f(57) = 3301 = 3301
f(58) = 3417 = 3*17*67
f(59) = 3535 = 5*7*101
f(60) = 3655 = 5*17*43
f(61) = 3777 = 3*1259
f(62) = 3901 = 47*83
f(63) = 4027 = 4027
f(64) = 4155 = 3*5*277
f(65) = 4285 = 5*857
f(66) = 4417 = 7*631
f(67) = 4551 = 3*37*41
f(68) = 4687 = 43*109
f(69) = 4825 = 5*5*193
f(70) = 4965 = 3*5*331
f(71) = 5107 = 5107
f(72) = 5251 = 59*89
f(73) = 5397 = 3*7*257
f(74) = 5545 = 5*1109
f(75) = 5695 = 5*17*67
f(76) = 5847 = 3*1949
f(77) = 6001 = 17*353
f(78) = 6157 = 47*131
f(79) = 6315 = 3*5*421
f(80) = 6475 = 5*5*7*37
f(81) = 6637 = 6637
f(82) = 6801 = 3*2267
f(83) = 6967 = 6967
f(84) = 7135 = 5*1427
f(85) = 7305 = 3*5*487
f(86) = 7477 = 7477
f(87) = 7651 = 7*1093
f(88) = 7827 = 3*2609
f(89) = 8005 = 5*1601
f(90) = 8185 = 5*1637
f(91) = 8367 = 3*2789
f(92) = 8551 = 17*503
f(93) = 8737 = 8737
f(94) = 8925 = 3*5*5*7*17
f(95) = 9115 = 5*1823
f(96) = 9307 = 41*227
f(97) = 9501 = 3*3167
f(98) = 9697 = 9697
f(99) = 9895 = 5*1979
f(100) = 10095 = 3*5*673

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+1x-5

f(0)=5
f(1)=3
f(2)=1
f(3)=7
f(4)=1
f(5)=1
f(6)=37
f(7)=17
f(8)=67
f(9)=1
f(10)=1
f(11)=127
f(12)=151
f(13)=59
f(14)=41
f(15)=47
f(16)=89
f(17)=43
f(18)=337
f(19)=1
f(20)=83
f(21)=457
f(22)=167
f(23)=547
f(24)=1
f(25)=1
f(26)=1
f(27)=751
f(28)=269
f(29)=173
f(30)=1
f(31)=1
f(32)=1051
f(33)=1117
f(34)=79
f(35)=251
f(36)=1327
f(37)=467
f(38)=211
f(39)=311
f(40)=109
f(41)=101
f(42)=1801
f(43)=1
f(44)=1
f(45)=1
f(46)=719
f(47)=2251
f(48)=2347
f(49)=163
f(50)=509
f(51)=2647
f(52)=131
f(53)=2857
f(54)=593
f(55)=1
f(56)=3187
f(57)=3301
f(58)=1
f(59)=1
f(60)=1
f(61)=1259
f(62)=1
f(63)=4027
f(64)=277
f(65)=857
f(66)=631
f(67)=1
f(68)=1
f(69)=193
f(70)=331
f(71)=5107
f(72)=1
f(73)=257
f(74)=1109
f(75)=1
f(76)=1949
f(77)=353
f(78)=1
f(79)=421
f(80)=1
f(81)=6637
f(82)=2267
f(83)=6967
f(84)=1427
f(85)=487
f(86)=7477
f(87)=1093
f(88)=2609
f(89)=1601
f(90)=1637
f(91)=2789
f(92)=503
f(93)=8737
f(94)=1
f(95)=1823
f(96)=227
f(97)=3167
f(98)=9697
f(99)=1979

b) Substitution of the polynom
The polynom f(x)=x^2+1x-5 could be written as f(y)= y^2-5.25 with x=y-0.5

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+0.5
f'(x)>2x

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

5, 3, 1, 7, 1, 1, 37, 17, 67, 1, 1, 127, 151, 59, 41, 47, 89, 43, 337, 1, 83, 457, 167, 547, 1, 1, 1, 751, 269, 173, 1, 1, 1051, 1117, 79, 251, 1327, 467, 211, 311, 109, 101, 1801, 1, 1, 1, 719, 2251, 2347, 163, 509, 2647, 131, 2857, 593, 1, 3187, 3301, 1, 1, 1, 1259, 1, 4027, 277, 857, 631, 1, 1, 193, 331, 5107, 1, 257, 1109, 1, 1949, 353, 1, 421, 1, 6637, 2267, 6967, 1427, 487, 7477, 1093, 2609, 1601, 1637, 2789, 503, 8737, 1, 1823, 227, 3167, 9697, 1979, 673, 1471, 10501, 1, 1, 1, 3779, 11551, 1, 1, 2441, 1, 4217, 1, 2621, 1, 13567, 373, 4679, 571, 2903, 4919, 2143, 1, 1033, 1, 941, 5417, 971, 479, 1, 293, 17551, 5939, 3617, 3671, 887, 461, 1, 1297, 3947, 541, 1, 1, 1, 1, 499, 21751, 7349, 1, 647, 7649, 23251, 23557, 1, 967, 521, 1181, 25117, 5087, 1, 1, 1553, 1, 773, 5477, 9239, 28051, 28387, 383, 5813, 4201, 1, 30097, 6089, 2053, 31147, 1, 1, 379, 1303, 10979, 33301, 1, 2269, 983, 419, 11717, 35527, 1, 1, 991, 1, 12479, 1, 7643, 757, 907, 39397, 1, 8039, 40597, 1, 881, 8363, 563, 6091, 43051, 14489, 1, 8861, 877, 1, 1, 3067, 1, 46867, 15767, 47737, 1, 463, 49057, 839, 16649, 10079, 10169, 17099, 7393, 1, 3511, 1, 1307, 1, 54517, 1571, 3697, 55927, 56401, 18959, 11471, 1, 2777, 1, 587, 797, 709, 60757, 1201, 8821, 1, 1, 63247, 1723, 21419, 12953, 1, 21929, 66301, 613, 1, 1, 677, 1, 69427, 823, 1, 1511, 71551, 24029, 1, 14633, 1, 1811, 74797, 5023, 1, 1, 25667, 77557, 919, 1049, 1, 79801, 1, 16187, 16301, 1, 82651, 83227, 1, 2411, 84967, 28517, 86137, 3469, 5821, 5171, 1, 1747, 17939, 18059, 739, 2473, 92107, 883, 3733, 93937, 31517, 1613, 19157, 6427, 1, 97651, 1, 1, 1171, 1, 100801, 1, 1361, 20543, 103357, 34667, 1, 21059, 1009, 1, 107251, 35969, 21713, 1, 36629, 929, 111217, 7459, 22511, 113227, 37967, 114577, 1, 1, 1, 2861, 1, 1, 23873, 5717, 7103, 121447, 1, 1, 123547, 1, 17851, 1, 1, 761, 1, 42839, 25847, 1, 1013, 1301, 3571, 1, 26717, 7901, 1, 135787, 1, 9151, 138007, 1559, 46499, 4007, 1, 1277, 142501, 143257, 9601, 5791, 1223, 48767, 1, 29567, 9907, 149377, 150151, 7187, 30341, 30497, 1087, 1213, 2311, 1, 1, 157207, 52667, 9341, 1, 1, 1, 23143, 54269, 1, 6577, 55079, 3533, 166867, 1597, 1, 169327, 1319, 4621, 34361, 1, 24781, 10253, 58379, 7039, 35363, 59219, 178501, 25621, 1, 36209, 3083, 60917, 2063, 1, 1, 186187, 11003, 62639, 2221, 1, 1549, 1, 4091, 1, 1, 194917, 65267, 196687, 1129, 1, 199357, 4657, 67049, 2377, 1097, 1, 204751, 205657, 1, 1, 208387, 69767, 1, 42227, 1, 2393, 213901, 1, 43151, 1, 1, 218551, 12911, 2939, 44273, 1697, 10631, 224197, 1217, 15073, 2083, 1, 1, 6569, 1847, 77279, 232801, 13751, 15649, 1, 33811, 2141, 238627, 1, 1, 241567, 242551, 11597, 9781, 49103, 1, 247501, 5779, 16633, 1, 5351, 4951, 253507, 1, 3407, 6257, 36793, 1091, 1, 52121, 2357, 262651, 263677, 2521, 53147, 266767, 1, 1, 1, 18061, 38851, 273001, 1, 1487, 55229, 92399, 4153, 39901, 18691, 11257, 282487, 2011, 6619, 3361, 2731, 16931, 1151, 1, 58211, 58427, 2273, 42043, 295387, 1, 59513, 2957, 2437, 1, 8627, 1, 17891, 1, 6007, 61493, 12343, 14747, 310801, 1, 1, 1, 8521, 105467, 45361, 1, 21319, 1193, 322051, 6337, 12973, 1, 108869, 1, 328897, 22003, 66239, 1, 15881, 1, 1429, 4493, 338137, 1, 113489, 1, 1, 114659, 1, 8447, 23167, 69737, 1, 117017, 352237, 1, 1, 1, 4519, 1, 71879, 1759, 1453, 1, 6173, 1433, 1, 4133, 123017, 1, 74297, 24847, 5581, 53593, 1, 75521, 75767, 1, 381301, 1381, 1, 77003, 22721, 3491, 3061, 77999, 26083, 1, 9157, 1, 1933, 15901, 132929, 400051, 57331, 26839, 1, 23831, 135467, 23981, 1, 3907, 10037, 8783, 1663, 3323, 1, 1, 1, 2179, 28123, 84629, 1, 141917, 25121, 12239, 1, 430987, 432301, 144539, 2351, 87251, 1, 1, 440227, 29437, 1, 5623, 148517, 63841, 17929, 1, 2137, 1, 3217, 90989, 13037, 1931, 1, 6871, 30781, 18523, 464437, 1, 1, 1, 31327, 471277, 27803, 158009, 1, 1, 159389, 1423, 480937, 1, 1, 1873, 1, 487897, 97859, 32713, 3019, 2557, 23567, 5839, 1, 9787, 3821, 501967, 1, 14423, 12347, 169217, 11839, 102101, 34129, 513367, 1, 172079, 20707, 1, 1, 4789, 1, 4999, 105269, 6359, 176417, 1, 106433, 1, 76441, 1621, 1, 107897, 108191, 3847, 543901, 4583, 36457, 6451, 549817, 1, 552787, 22171, 1, 15061, 1907, 1, 112349, 1, 188249, 1973, 9623, 37951, 1, 13309, 11251, 575317, 16481, 38557, 579877, 1, 194309, 2851, 1, 1, 1, 1, 1579, 3209, 595207, 11701, 1, 7057, 40093, 602947, 2663, 1, 121523, 1, 203579, 612301, 1, 1, 123401, 618577, 29531, 2477, 7333, 41659, 1, 628051, 5119, 3607, 1889, 211469, 1, 637597, 1, 128159, 91771, 214667, 5923, 129443, 1, 38261, 17623, 1831, 1, 131381, 1, 1, 661777, 1, 19001, 666667, 2503, 669937, 26863, 1, 674857, 96643, 13297, 1721, 8017, 227699, 684751, 1, 6553, 1, 6343, 231017, 694717, 1, 1, 99961, 701401, 1789, 2389, 8311, 1, 1, 101641, 1, 142973, 17477, 239417, 719947, 1, 1, 1, 726751, 242819, 146033, 1, 244529, 1, 737017, 2897, 148091, 15791, 3701, 1999, 1, 49939, 750817, 7451, 251429, 30241, 151553, 1, 761251, 8573, 2999, 3739, 45191, 1, 110251, 2309, 10337, 21001, 3691, 260189, 156467, 1, 261959, 19211, 1, 52747, 158597, 46751, 1, 1, 1, 1, 1, 805501, 4561, 23117, 1, 270899, 814501, 10333, 54541, 32797, 1, 274517, 1, 165437, 3251, 830827, 7639, 1, 167261, 167627, 6829, 841801, 1, 11273, 1, 2017, 1, 852847, 3637, 3359, 858397, 7229, 1, 2579, 1, 7817, 869551, 10499, 8317, 1, 877027, 1, 1, 176531, 1, 126631, 52253, 296729, 2099, 178793, 4457, 897751, 128521, 60103, 1, 905347, 302417, 1, 1, 1, 914887, 1, 306239, 10831, 184511, 18127, 1, 21589, 62017, 186437, 934117, 1, 1, 1, 62791, 1, 3041, 315899, 189929, 2293, 2671, 955501, 3313, 1, 7691, 2729, 4073, 138181, 193847, 64747, 973177, 975151, 325709, 3319, 28031, 327689, 1, 58061, 13187, 1, 14821, 47381, 24317, 199799,

6. Sequence of the polynom (only primes)

5, 3, 7, 37, 17, 67, 127, 151, 59, 41, 47, 89, 43, 337, 83, 457, 167, 547, 751, 269, 173, 1051, 1117, 79, 251, 1327, 467, 211, 311, 109, 101, 1801, 719, 2251, 2347, 163, 509, 2647, 131, 2857, 593, 3187, 3301, 1259, 4027, 277, 857, 631, 193, 331, 5107, 257, 1109, 1949, 353, 421, 6637, 2267, 6967, 1427, 487, 7477, 1093, 2609, 1601, 1637, 2789, 503, 8737, 1823, 227, 3167, 9697, 1979, 673, 1471, 10501, 3779, 11551, 2441, 4217, 2621, 13567, 373, 4679, 571, 2903, 4919, 2143, 1033, 941, 5417, 971, 479, 293, 17551, 5939, 3617, 3671, 887, 461, 1297, 3947, 541, 499, 21751, 7349, 647, 7649, 23251, 23557, 967, 521, 1181, 25117, 5087, 1553, 773, 5477, 9239, 28051, 28387, 383, 5813, 4201, 30097, 6089, 2053, 31147, 379, 1303, 10979, 33301, 2269, 983, 419, 11717, 35527, 991, 12479, 7643, 757, 907, 39397, 8039, 40597, 881, 8363, 563, 6091, 43051, 14489, 8861, 877, 3067, 46867, 15767, 47737, 463, 49057, 839, 16649, 10079, 10169, 17099, 7393, 3511, 1307, 54517, 1571, 3697, 55927, 56401, 18959, 11471, 2777, 587, 797, 709, 60757, 1201, 8821, 63247, 1723, 21419, 12953, 21929, 66301, 613, 677, 69427, 823, 1511, 71551, 24029, 14633, 1811, 74797, 5023, 25667, 77557, 919, 1049, 79801, 16187, 16301, 82651, 83227, 2411, 84967, 28517, 86137, 3469, 5821, 5171, 1747, 17939, 18059, 739, 2473, 92107, 883, 3733, 93937, 31517, 1613, 19157, 6427, 97651, 1171, 100801, 1361, 20543, 103357, 34667, 21059, 1009, 107251, 35969, 21713, 36629, 929, 111217, 7459, 22511, 113227, 37967, 114577, 2861, 23873, 5717, 7103, 121447, 123547, 17851, 761, 42839, 25847, 1013, 1301, 3571, 26717, 7901, 135787, 9151, 138007, 1559, 46499, 4007, 1277, 142501, 143257, 9601, 5791, 1223, 48767, 29567, 9907, 149377, 150151, 7187, 30341, 30497, 1087, 1213, 2311, 157207, 52667, 9341, 23143, 54269, 6577, 55079, 3533, 166867, 1597, 169327, 1319, 4621, 34361, 24781, 10253, 58379, 7039, 35363, 59219, 178501, 25621, 36209, 3083, 60917, 2063, 186187, 11003, 62639, 2221, 1549, 4091, 194917, 65267, 196687, 1129, 199357, 4657, 67049, 2377, 1097, 204751, 205657, 208387, 69767, 42227, 2393, 213901, 43151, 218551, 12911, 2939, 44273, 1697, 10631, 224197, 1217, 15073, 2083, 6569, 1847, 77279, 232801, 13751, 15649, 33811, 2141, 238627, 241567, 242551, 11597, 9781, 49103, 247501, 5779, 16633, 5351, 4951, 253507, 3407, 6257, 36793, 1091, 52121, 2357, 262651, 263677, 2521, 53147, 266767, 18061, 38851, 273001, 1487, 55229, 92399, 4153, 39901, 18691, 11257, 282487, 2011, 6619, 3361, 2731, 16931, 1151, 58211, 58427, 2273, 42043, 295387, 59513, 2957, 2437, 8627, 17891, 6007, 61493, 12343, 14747, 310801, 8521, 105467, 45361, 21319, 1193, 322051, 6337, 12973, 108869, 328897, 22003, 66239, 15881, 1429, 4493, 338137, 113489, 114659, 8447, 23167, 69737, 117017, 352237, 4519, 71879, 1759, 1453, 6173, 1433, 4133, 123017, 74297, 24847, 5581, 53593, 75521, 75767, 381301, 1381, 77003, 22721, 3491, 3061, 77999, 26083, 9157, 1933, 15901, 132929, 400051, 57331, 26839, 23831, 135467, 23981, 3907, 10037, 8783, 1663, 3323, 2179, 28123, 84629, 141917, 25121, 12239, 430987, 432301, 144539, 2351, 87251, 440227, 29437, 5623, 148517, 63841, 17929, 2137, 3217, 90989, 13037, 1931, 6871, 30781, 18523, 464437, 31327, 471277, 27803, 158009, 159389, 1423, 480937, 1873, 487897, 97859, 32713, 3019, 2557, 23567, 5839, 9787, 3821, 501967, 14423, 12347, 169217, 11839, 102101, 34129, 513367, 172079, 20707, 4789, 4999, 105269, 6359, 176417, 106433, 76441, 1621, 107897, 108191, 3847, 543901, 4583, 36457, 6451, 549817, 552787, 22171, 15061, 1907, 112349, 188249, 1973, 9623, 37951, 13309, 11251, 575317, 16481, 38557, 579877, 194309, 2851, 1579, 3209, 595207, 11701, 7057, 40093, 602947, 2663, 121523, 203579, 612301, 123401, 618577, 29531, 2477, 7333, 41659, 628051, 5119, 3607, 1889, 211469, 637597, 128159, 91771, 214667, 5923, 129443, 38261, 17623, 1831, 131381, 661777, 19001, 666667, 2503, 669937, 26863, 674857, 96643, 13297, 1721, 8017, 227699, 684751, 6553, 6343, 231017, 694717, 99961, 701401, 1789, 2389, 8311, 101641, 142973, 17477, 239417, 719947, 726751, 242819, 146033, 244529, 737017, 2897, 148091, 15791, 3701, 1999, 49939, 750817, 7451, 251429, 30241, 151553, 761251, 8573, 2999, 3739, 45191, 110251, 2309, 10337, 21001, 3691, 260189, 156467, 261959, 19211, 52747, 158597, 46751, 805501, 4561, 23117, 270899, 814501, 10333, 54541, 32797, 274517, 165437, 3251, 830827, 7639, 167261, 167627, 6829, 841801, 11273, 2017, 852847, 3637, 3359, 858397, 7229, 2579, 7817, 869551, 10499, 8317, 877027, 176531, 126631, 52253, 296729, 2099, 178793, 4457, 897751, 128521, 60103, 905347, 302417, 914887, 306239, 10831, 184511, 18127, 21589, 62017, 186437, 934117, 62791, 3041, 315899, 189929, 2293, 2671, 955501, 3313, 7691, 2729, 4073, 138181, 193847, 64747, 973177, 975151, 325709, 3319, 28031, 327689, 58061, 13187, 14821, 47381, 24317, 199799,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+1x-5 and
the reducible primes which appear as divisor for the first time
p | x^2+1x-5 and p < x^2+1x-5

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)
1106510.6000000.5000000.6000000.0000000.0000000.000000
21007528470.7500000.2800000.75000012.5000005.60000047.000000
31.0007241605640.7240000.1600000.7240009.6533345.71428612.000000
410.0007.1801.1546.0260.7180000.1154000.7180009.9171277.21250010.684397
5100.00071.3338.91362.4200.7133300.0891300.7133309.9349587.72357010.358447
61.000.000709.04772.774636.2730.7090470.0727740.7090479.9399588.16492710.193416
710.000.0007.064.484616.6676.447.8170.7064480.0616670.7064489.9633518.47372710.133727
8100.000.00070.456.3025.343.81965.112.4830.7045630.0534380.7045639.9733128.66564810.098376
91.000.000.000703.097.45147.178.885655.918.5660.7030970.0471790.7030979.9791988.82868310.073623
1010.000.000.0007.019.663.877422.252.9746.597.410.9030.7019660.0422250.7019669.9839128.95004110.058277


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
386510.7500000.6250000.1250002.0000001.666667inf
41612750.7500000.4375000.3125002.0000001.4000005.000000
5322212100.6875000.3750000.3125001.8333331.7142862.000000
6644622240.7187500.3437500.3750002.0909091.8333332.400000
71289331620.7265620.2421880.4843752.0217391.4090912.583333
8256187531340.7304690.2070310.5234382.0107531.7096772.161290
95123721012710.7265620.1972660.5292971.9893051.9056602.022388
101.0247391625770.7216800.1582030.5634771.9865591.6039602.129151
112.0481.4812981.1830.7231450.1455080.5776372.0040601.8395062.050260
124.0962.9405542.3860.7177730.1352540.5825201.9851451.8590602.016906
138.1925.8919684.9230.7191160.1181640.6009522.0037421.7472922.063286
1416.38411.7271.7739.9540.7157590.1082150.6075441.9906641.8316122.021938
1532.76823.4183.27120.1470.7146610.0998230.6148381.9969301.8448962.024010
1665.53646.8126.05540.7570.7142940.0923920.6219021.9989751.8511162.022981
17131.07293.36211.41681.9460.7122960.0870970.6251981.9944031.8853842.010599
18262.144186.43621.352165.0840.7111970.0814510.6297451.9969151.8703572.014546
19524.288372.18340.344331.8390.7098830.0769500.6329331.9963041.8894722.010122
201.048.576743.48675.947667.5390.7090440.0724290.6366151.9976361.8824862.011635
212.097.1521.485.176144.0081.341.1680.7081870.0686680.6395191.9975841.8961642.009123
224.194.3042.966.777274.0912.692.6860.7073350.0653480.6419861.9975931.9033042.007717
238.388.6085.927.213523.3825.403.8310.7065790.0623920.6441871.9978631.9095192.006855
2416.777.21611.844.4781.000.16010.844.3180.7059860.0596140.6463721.9983221.9109562.006783
2533.554.43223.669.4751.913.79021.755.6850.7054050.0570350.6483701.9983551.9134842.006183
2667.108.86447.303.2053.670.51843.632.6870.7048730.0546950.6501781.9984901.9179312.005576
27134.217.72894.536.4667.053.65187.482.8150.7043520.0525540.6517981.9985211.9217052.004983
28268.435.456188.945.06613.575.926175.369.1400.7038750.0505740.6533011.9986481.9246672.004612
29536.870.912377.665.38026.157.141351.508.2390.7034570.0487210.6547351.9988101.9267302.004390
301.073.741.824754.901.11750.474.362704.426.7550.7030560.0470080.6560491.9988621.9296592.004012
312.147.483.6481.509.023.31697.526.7671.411.496.5490.7026940.0454140.6572791.9989681.9322042.003752
324.294.967.2963.016.576.011188.620.0312.827.955.9800.7023510.0439170.6584351.9990251.9340342.003516
338.589.934.5926.030.428.350365.239.6575.665.188.6930.7020340.0425190.6595151.9990971.9363782.003280
3417.179.869.18412.055.763.917707.940.07311.347.823.8440.7017380.0412080.6605301.9991561.9382892.003079


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
122010110
243110111
385310221
4167510223
532121012424
664222014846
7128312916979
82565351110151414
951210199120302526
101.024162160136454338
112.048298296173797472
124.0965545521145136142131
138.1929689661251238236243
1416.3841.7731.7711457434438444
1532.7683.2713.2691829807823812
1665.5366.0556.05311.5141.5191.5101.512
17131.07211.41611.41412.8322.8442.8592.881
18262.14421.35221.35015.2935.3515.3515.357
19524.28840.34440.342110.00010.19410.08010.070
201.048.57675.94775.945118.88219.15618.93718.972
212.097.152144.008144.006135.90736.01335.96236.126
224.194.304274.091274.089168.28468.58468.52168.702
238.388.608523.382523.3801130.762130.935130.751130.934
2416.777.2161.000.1601.000.1581250.046250.109249.739250.266
2533.554.4321.913.7901.913.7881478.585478.960477.614478.631
2667.108.8643.670.5183.670.5161916.895918.027916.609918.987
27134.217.7287.053.6517.053.64911.762.9971.764.1271.762.7181.763.809
28268.435.45613.575.92613.575.92413.394.1783.394.8533.392.7903.394.105
29536.870.91226.157.14126.157.13916.542.4866.540.3066.536.8826.537.467
301.073.741.82450.474.36250.474.360112.622.51812.618.89012.615.33212.617.622
312.147.483.64897.526.76797.526.765124.386.45624.383.74424.376.40724.380.160
324.294.967.296188.620.031188.620.029147.161.13447.153.70147.149.72647.155.470
338.589.934.592365.239.657365.239.655191.322.79691.303.48991.300.52391.312.849
3417.179.869.184707.940.073707.940.0711176.998.095176.978.835176.977.631176.985.512


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
381011000
4165053101
53210193322
664246184965
712862184415171515
8256134419332353235
95122718618563696871
101.024577224353139145152141
112.0481.183471712264312302305
124.0962.3869791.407573628588597
138.1924.9232.0382.8851.2001.2841.2021.237
1416.3849.9544.2545.7002.4382.5362.4842.496
1532.76820.1478.80911.3384.9905.0724.9985.087
1665.53640.75718.06622.69110.10810.22910.19810.222
17131.07281.94636.71745.22920.22020.56220.55320.611
18262.144165.08474.33490.75041.09341.40541.12441.462
19524.288331.839150.478181.36182.68682.97683.16883.009
201.048.576667.539304.641362.898166.608167.039166.840167.052
212.097.1521.341.168615.911725.257335.317335.313335.123335.415
224.194.3042.692.6861.243.7621.448.924673.515673.521673.213672.437
238.388.6085.403.8312.505.3042.898.5271.350.8991.351.2181.350.8121.350.902
2416.777.21610.844.3185.046.6375.797.6812.710.5052.711.5842.711.0372.711.192
2533.554.43221.755.68510.165.44411.590.2415.438.6675.439.1225.438.9655.438.931
2667.108.86443.632.68720.447.41023.185.27710.908.63510.907.16010.909.11510.907.777
27134.217.72887.482.81541.116.32946.366.48621.875.44921.866.87521.870.08821.870.403
28268.435.456175.369.14082.641.40092.727.74043.851.60143.836.94143.840.80543.839.793
29536.870.912351.508.239166.053.584185.454.65587.886.33987.872.51987.876.11187.873.270
301.073.741.824704.426.755333.511.277370.915.478176.111.651176.089.255176.109.966176.115.883
312.147.483.6481.411.496.549669.663.721741.832.828352.869.569352.861.668352.885.547352.879.765
324.294.967.2962.827.955.9801.344.200.3471.483.755.633706.985.872706.993.138706.999.116706.977.854
338.589.934.5925.665.188.6932.697.548.9422.967.639.7511.416.264.0281.416.315.3041.416.328.6671.416.280.694
3417.179.869.18411.347.823.8445.412.266.4975.935.557.3472.836.964.5222.836.970.5312.836.958.7042.836.930.087


8. Check for existing Integer Sequences by OEIS

Found in Database : 5, 3, 1, 7, 1, 1, 37, 17, 67, 1, 1, 127, 151, 59, 41, 47, 89, 43, 337, 1,
Found in Database : 5, 3, 7, 37, 17, 67, 127, 151, 59, 41, 47, 89, 43, 337, 83, 457, 167, 547, 751, 269, 173, 1051, 1117, 79, 251, 1327, 467, 211, 311,
Found in Database : 3, 5, 7, 17, 37, 41, 43, 47, 59, 67, 79, 83, 89, 101, 109, 127, 131,