Inhaltsverzeichnis

Development of
Algorithmic Constructions

19:10:20
Deutsch
28.Mar 2024

Polynom = x^2+114x-11

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) = 11 = 11
f(1) = 13 = 13
f(2) = 221 = 13*17
f(3) = 85 = 5*17
f(4) = 461 = 461
f(5) = 73 = 73
f(6) = 709 = 709
f(7) = 209 = 11*19
f(8) = 965 = 5*193
f(9) = 137 = 137
f(10) = 1229 = 1229
f(11) = 341 = 11*31
f(12) = 1501 = 19*79
f(13) = 205 = 5*41
f(14) = 1781 = 13*137
f(15) = 481 = 13*37
f(16) = 2069 = 2069
f(17) = 277 = 277
f(18) = 2365 = 5*11*43
f(19) = 629 = 17*37
f(20) = 2669 = 17*157
f(21) = 353 = 353
f(22) = 2981 = 11*271
f(23) = 785 = 5*157
f(24) = 3301 = 3301
f(25) = 433 = 433
f(26) = 3629 = 19*191
f(27) = 949 = 13*73
f(28) = 3965 = 5*13*61
f(29) = 517 = 11*47
f(30) = 4309 = 31*139
f(31) = 1121 = 19*59
f(32) = 4661 = 59*79
f(33) = 605 = 5*11*11
f(34) = 5021 = 5021
f(35) = 1301 = 1301
f(36) = 5389 = 17*317
f(37) = 697 = 17*41
f(38) = 5765 = 5*1153
f(39) = 1489 = 1489
f(40) = 6149 = 11*13*43
f(41) = 793 = 13*61
f(42) = 6541 = 31*211
f(43) = 1685 = 5*337
f(44) = 6941 = 11*631
f(45) = 893 = 19*47
f(46) = 7349 = 7349
f(47) = 1889 = 1889
f(48) = 7765 = 5*1553
f(49) = 997 = 997
f(50) = 8189 = 19*431
f(51) = 2101 = 11*191
f(52) = 8621 = 37*233
f(53) = 1105 = 5*13*17
f(54) = 9061 = 13*17*41
f(55) = 2321 = 11*211
f(56) = 9509 = 37*257
f(57) = 1217 = 1217
f(58) = 9965 = 5*1993
f(59) = 2549 = 2549
f(60) = 10429 = 10429
f(61) = 1333 = 31*43
f(62) = 10901 = 11*991
f(63) = 2785 = 5*557
f(64) = 11381 = 19*599
f(65) = 1453 = 1453
f(66) = 11869 = 11*13*83
f(67) = 3029 = 13*233
f(68) = 12365 = 5*2473
f(69) = 1577 = 19*83
f(70) = 12869 = 17*757
f(71) = 3281 = 17*193
f(72) = 13381 = 13381
f(73) = 1705 = 5*11*31
f(74) = 13901 = 13901
f(75) = 3541 = 3541
f(76) = 14429 = 47*307
f(77) = 1837 = 11*167
f(78) = 14965 = 5*41*73
f(79) = 3809 = 13*293
f(80) = 15509 = 13*1193
f(81) = 1973 = 1973
f(82) = 16061 = 16061
f(83) = 4085 = 5*19*43
f(84) = 16621 = 11*1511
f(85) = 2113 = 2113
f(86) = 17189 = 17189
f(87) = 4369 = 17*257
f(88) = 17765 = 5*11*17*19
f(89) = 2257 = 37*61
f(90) = 18349 = 59*311
f(91) = 4661 = 59*79
f(92) = 18941 = 13*31*47
f(93) = 2405 = 5*13*37
f(94) = 19541 = 19541
f(95) = 4961 = 11*11*41
f(96) = 20149 = 20149
f(97) = 2557 = 2557
f(98) = 20765 = 5*4153
f(99) = 5269 = 11*479
f(100) = 21389 = 73*293

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+114x-11

f(0)=11
f(1)=13
f(2)=17
f(3)=5
f(4)=461
f(5)=73
f(6)=709
f(7)=19
f(8)=193
f(9)=137
f(10)=1229
f(11)=31
f(12)=79
f(13)=41
f(14)=1
f(15)=37
f(16)=2069
f(17)=277
f(18)=43
f(19)=1
f(20)=157
f(21)=353
f(22)=271
f(23)=1
f(24)=3301
f(25)=433
f(26)=191
f(27)=1
f(28)=61
f(29)=47
f(30)=139
f(31)=59
f(32)=1
f(33)=1
f(34)=5021
f(35)=1301
f(36)=317
f(37)=1
f(38)=1153
f(39)=1489
f(40)=1
f(41)=1
f(42)=211
f(43)=337
f(44)=631
f(45)=1
f(46)=7349
f(47)=1889
f(48)=1553
f(49)=997
f(50)=431
f(51)=1
f(52)=233
f(53)=1
f(54)=1
f(55)=1
f(56)=257
f(57)=1217
f(58)=1993
f(59)=2549
f(60)=10429
f(61)=1
f(62)=991
f(63)=557
f(64)=599
f(65)=1453
f(66)=83
f(67)=1
f(68)=2473
f(69)=1
f(70)=757
f(71)=1
f(72)=13381
f(73)=1
f(74)=13901
f(75)=3541
f(76)=307
f(77)=167
f(78)=1
f(79)=293
f(80)=1193
f(81)=1973
f(82)=16061
f(83)=1
f(84)=1511
f(85)=2113
f(86)=17189
f(87)=1
f(88)=1
f(89)=1
f(90)=311
f(91)=1
f(92)=1
f(93)=1
f(94)=19541
f(95)=1
f(96)=20149
f(97)=2557
f(98)=4153
f(99)=479

b) Substitution of the polynom
The polynom f(x)=x^2+114x-11 could be written as f(y)= y^2-3260 with x=y-57

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+57
f'(x)>2x+113

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

11, 13, 17, 5, 461, 73, 709, 19, 193, 137, 1229, 31, 79, 41, 1, 37, 2069, 277, 43, 1, 157, 353, 271, 1, 3301, 433, 191, 1, 61, 47, 139, 59, 1, 1, 5021, 1301, 317, 1, 1153, 1489, 1, 1, 211, 337, 631, 1, 7349, 1889, 1553, 997, 431, 1, 233, 1, 1, 1, 257, 1217, 1993, 2549, 10429, 1, 991, 557, 599, 1453, 83, 1, 2473, 1, 757, 1, 13381, 1, 13901, 3541, 307, 167, 1, 293, 1193, 1973, 16061, 1, 1511, 2113, 17189, 1, 1, 1, 311, 1, 1, 1, 19541, 1, 20149, 2557, 4153, 479, 1, 2713, 1, 1117, 1, 1, 163, 1, 4793, 3037, 2239, 1, 25301, 641, 25981, 6581, 26669, 1, 421, 1, 28069, 1, 1693, 1, 29501, 3733, 1, 7649, 563, 3917, 857, 617, 227, 821, 239, 1, 829, 4297, 409, 1, 1871, 4493, 36341, 1, 2857, 1, 1, 223, 7753, 1, 1, 1, 487, 1021, 1, 613, 2477, 1, 661, 571, 1069, 503, 44701, 1, 2399, 523, 1499, 1, 9473, 1, 1, 937, 263, 1, 50101, 12641, 4639, 1, 547, 13109, 1, 6673, 53861, 1, 4217, 1, 1187, 1279, 11353, 1, 1, 14561, 58741, 1481, 5431, 15061, 4673, 1, 1123, 15569, 1697, 1, 3359, 3217, 1753, 743, 3877, 977, 1, 1, 5233, 281, 1607, 1741, 70181, 17681, 1, 1, 1, 18229, 6679, 1, 5737, 1, 1, 9533, 76829, 1759, 1, 9817, 331, 1811, 967, 1, 81421, 1, 6353, 1, 1523, 21089, 1, 1, 1, 4337, 1, 10993, 1213, 719, 1381, 1, 90989, 22901, 92221, 1, 4919, 1, 94709, 701, 1129, 1, 8839, 941, 7577, 4957, 1, 12553, 1, 1, 347, 1, 103669, 2371, 1721, 1, 1, 1, 2503, 13537, 1, 27409, 2347, 13873, 10151, 1, 113021, 1, 10399, 2213, 1, 14557, 3779, 1733, 367, 1, 1, 30161, 121349, 1, 1, 30869, 1, 1201, 125621, 6317, 11551, 15973, 128509, 32309, 1, 1, 787, 1, 383, 1, 10337, 1, 7151, 17077, 1, 1, 138869, 1, 140381, 7057, 3461, 1049, 1, 1, 1, 18217, 1, 36821, 148061, 1, 953, 1979, 151189, 1, 30553, 2953, 1, 1, 9173, 1, 1, 19793, 2609, 39989, 1, 1063, 162389, 40801, 1, 1, 8719, 41621, 167309, 21017, 1, 1, 10037, 21433, 4007, 1, 173981, 1, 13513, 44129, 1867, 22277, 1, 1451, 1, 1, 1, 45841, 10837, 1361, 2861, 3593, 1, 2143, 4621, 1, 4447, 1, 3271, 2551, 38953, 1, 196549, 3797, 1, 1, 1, 50261, 1669, 25357, 491, 51169, 205589, 1, 6691, 947, 16097, 1, 1, 1, 1, 26737, 12637, 1, 1559, 5441, 1, 54881, 220469, 2129, 1, 55829, 1637, 1, 6113, 1, 1, 1, 6217, 1, 2729, 2647, 1, 4517, 1447, 1, 237821, 1, 21799, 30097, 48353, 60689, 22159, 30593, 245741, 1, 1, 1, 249749, 1, 1171, 1663, 253789, 5791, 5443, 6421, 257861, 1061, 19993, 1, 1, 1777, 264029, 1949, 1423, 1, 1, 1, 270269, 67829, 1, 1, 1, 68881, 276581, 1, 278701, 69941, 9059, 1, 3329, 4177, 25919, 1, 1163, 1109, 1, 36313, 3691, 3851, 1433, 36857, 295949, 1, 1, 7481, 1913, 1, 1, 37957, 60953, 1627, 8297, 2027, 28111, 1, 5279, 1, 1, 2539, 4861, 3049, 318229, 79841, 1, 1, 322781, 1723, 1, 1, 1, 82129, 17351, 3181, 25537, 16657, 30391, 2207, 10859, 84449, 6163, 1, 1181, 85621, 343661, 1, 619, 607, 348389, 1, 1, 1, 353149, 44293, 355541, 17837, 18839, 44893, 1, 1, 5581, 45497, 33199, 659, 367621, 9221, 370061, 92821, 8663, 1, 3947, 94049, 29033, 1, 1, 1, 1, 1, 1, 1, 7043, 1567, 1, 1, 35671, 1, 1, 99041, 397429, 1, 1, 1, 23677, 2969, 21319, 1847, 407621, 647, 1, 1, 2663, 1, 1, 104161, 417941, 1, 38231, 1, 22271, 3121, 5009, 1, 1, 1, 431021, 21617, 433661, 4943, 436309, 2957, 87793, 1, 1, 1, 1, 1, 26293, 1, 40879, 56377, 90473, 3659, 1, 1, 7759, 2087, 2411, 4441, 2741, 10559, 2273, 58417, 468709, 1, 27733, 11821, 2269, 118901, 476989, 59797, 1, 1, 10267, 60493, 11287, 24337, 1, 5563, 490949, 1483, 1, 1, 1, 1, 1, 1, 502261, 125921, 1, 63317, 5347, 1, 46439, 64033, 1, 1, 1, 1, 30557, 11839, 104473, 1597, 1657, 11971, 27799, 13241, 531101, 1, 534029, 1, 751, 134609, 9151, 1, 2903, 1601, 7477, 1, 1873, 2927, 1, 6287, 1, 1, 557741, 1, 4093, 1, 29671, 70657, 1, 1, 1, 71413, 572821, 1, 4027, 72173, 13463, 145109, 1, 72937, 1, 13331, 1, 14741, 591181, 1, 2689, 1, 3229, 149729, 600469, 75253, 1, 1, 19571, 1, 55439, 1, 2851, 1, 1, 1, 619261, 1, 1, 2137, 625589, 7127, 125753, 2671, 10711, 79193, 48857, 1, 58031, 1, 641549, 160789, 1, 1879, 811, 1, 651221, 859, 1, 1, 50593, 1, 1, 1, 10889, 83233, 667501, 33457, 670781, 84053, 1, 1, 1, 6529, 61879, 3967, 35999, 1, 22171, 4201, 690629, 7867, 138793, 173909, 3613, 1, 53897, 35117, 41413, 5189, 1, 177269, 12923, 89057, 3739, 9419, 1, 17981, 55457, 1, 19577, 1931, 2467, 1, 731189, 1, 1, 3347, 1, 1, 741509, 14293, 1, 93337, 3581, 1, 751901, 1, 1, 1, 1, 3067, 8969, 1, 58913, 1, 9739, 38557, 2141, 8803, 776429, 1, 3319, 1, 10733, 1, 71551, 1, 1, 1, 1, 99497, 159553, 199889, 801349, 100393, 804941, 1, 17203, 101293, 62473, 1, 1, 102197, 819389, 1, 48413, 1, 1, 207121, 830309, 1, 1, 16073, 64433, 104933, 44279, 42157, 6079, 9623, 22937, 1, 170473, 1, 1, 214481, 66137, 1, 21061, 4603, 78839, 108637, 4051, 7039, 7229, 1, 14401, 44017, 1, 8501, 1, 1, 177953, 1, 893549, 1, 2711, 22481, 901141, 3701, 10903, 113357, 1, 1, 1, 1, 83311, 1, 54133, 6067, 924109, 6257, 185593, 10567, 30059, 17957, 1, 2131, 939581, 5741, 15991, 2003, 189473, 12491, 5087, 1, 20323, 47857, 1, 9241, 2089, 241249, 193393, 3907, 970909, 22111, 974861, 24421, 978821, 22291, 982789, 1, 1, 6029, 26777, 4003, 8221, 1, 26993, 125093, 91159, 1051, 1, 126097, 5981, 19477, 1014821, 2311, 1, 15013, 1, 1, 1, 257249, 1, 1, 1, 3989, 1, 1, 1, 261329, 1, 2791, 1051469, 15493, 1, 1, 1059701, 1, 1, 1, 1, 1, 1, 1, 12967, 53917, 4637, 2879, 2293, 1, 12809, 1, 7643, 8831, 1097141, 27481, 1101341, 14519, 1105549, 1, 221953, 1,

6. Sequence of the polynom (only primes)

11, 13, 17, 5, 461, 73, 709, 19, 193, 137, 1229, 31, 79, 41, 37, 2069, 277, 43, 157, 353, 271, 3301, 433, 191, 61, 47, 139, 59, 5021, 1301, 317, 1153, 1489, 211, 337, 631, 7349, 1889, 1553, 997, 431, 233, 257, 1217, 1993, 2549, 10429, 991, 557, 599, 1453, 83, 2473, 757, 13381, 13901, 3541, 307, 167, 293, 1193, 1973, 16061, 1511, 2113, 17189, 311, 19541, 20149, 2557, 4153, 479, 2713, 1117, 163, 4793, 3037, 2239, 25301, 641, 25981, 6581, 26669, 421, 28069, 1693, 29501, 3733, 7649, 563, 3917, 857, 617, 227, 821, 239, 829, 4297, 409, 1871, 4493, 36341, 2857, 223, 7753, 487, 1021, 613, 2477, 661, 571, 1069, 503, 44701, 2399, 523, 1499, 9473, 937, 263, 50101, 12641, 4639, 547, 13109, 6673, 53861, 4217, 1187, 1279, 11353, 14561, 58741, 1481, 5431, 15061, 4673, 1123, 15569, 1697, 3359, 3217, 1753, 743, 3877, 977, 5233, 281, 1607, 1741, 70181, 17681, 18229, 6679, 5737, 9533, 76829, 1759, 9817, 331, 1811, 967, 81421, 6353, 1523, 21089, 4337, 10993, 1213, 719, 1381, 90989, 22901, 92221, 4919, 94709, 701, 1129, 8839, 941, 7577, 4957, 12553, 347, 103669, 2371, 1721, 2503, 13537, 27409, 2347, 13873, 10151, 113021, 10399, 2213, 14557, 3779, 1733, 367, 30161, 121349, 30869, 1201, 125621, 6317, 11551, 15973, 128509, 32309, 787, 383, 10337, 7151, 17077, 138869, 140381, 7057, 3461, 1049, 18217, 36821, 148061, 953, 1979, 151189, 30553, 2953, 9173, 19793, 2609, 39989, 1063, 162389, 40801, 8719, 41621, 167309, 21017, 10037, 21433, 4007, 173981, 13513, 44129, 1867, 22277, 1451, 45841, 10837, 1361, 2861, 3593, 2143, 4621, 4447, 3271, 2551, 38953, 196549, 3797, 50261, 1669, 25357, 491, 51169, 205589, 6691, 947, 16097, 26737, 12637, 1559, 5441, 54881, 220469, 2129, 55829, 1637, 6113, 6217, 2729, 2647, 4517, 1447, 237821, 21799, 30097, 48353, 60689, 22159, 30593, 245741, 249749, 1171, 1663, 253789, 5791, 5443, 6421, 257861, 1061, 19993, 1777, 264029, 1949, 1423, 270269, 67829, 68881, 276581, 278701, 69941, 9059, 3329, 4177, 25919, 1163, 1109, 36313, 3691, 3851, 1433, 36857, 295949, 7481, 1913, 37957, 60953, 1627, 8297, 2027, 28111, 5279, 2539, 4861, 3049, 318229, 79841, 322781, 1723, 82129, 17351, 3181, 25537, 16657, 30391, 2207, 10859, 84449, 6163, 1181, 85621, 343661, 619, 607, 348389, 353149, 44293, 355541, 17837, 18839, 44893, 5581, 45497, 33199, 659, 367621, 9221, 370061, 92821, 8663, 3947, 94049, 29033, 7043, 1567, 35671, 99041, 397429, 23677, 2969, 21319, 1847, 407621, 647, 2663, 104161, 417941, 38231, 22271, 3121, 5009, 431021, 21617, 433661, 4943, 436309, 2957, 87793, 26293, 40879, 56377, 90473, 3659, 7759, 2087, 2411, 4441, 2741, 10559, 2273, 58417, 468709, 27733, 11821, 2269, 118901, 476989, 59797, 10267, 60493, 11287, 24337, 5563, 490949, 1483, 502261, 125921, 63317, 5347, 46439, 64033, 30557, 11839, 104473, 1597, 1657, 11971, 27799, 13241, 531101, 534029, 751, 134609, 9151, 2903, 1601, 7477, 1873, 2927, 6287, 557741, 4093, 29671, 70657, 71413, 572821, 4027, 72173, 13463, 145109, 72937, 13331, 14741, 591181, 2689, 3229, 149729, 600469, 75253, 19571, 55439, 2851, 619261, 2137, 625589, 7127, 125753, 2671, 10711, 79193, 48857, 58031, 641549, 160789, 1879, 811, 651221, 859, 50593, 10889, 83233, 667501, 33457, 670781, 84053, 6529, 61879, 3967, 35999, 22171, 4201, 690629, 7867, 138793, 173909, 3613, 53897, 35117, 41413, 5189, 177269, 12923, 89057, 3739, 9419, 17981, 55457, 19577, 1931, 2467, 731189, 3347, 741509, 14293, 93337, 3581, 751901, 3067, 8969, 58913, 9739, 38557, 2141, 8803, 776429, 3319, 10733, 71551, 99497, 159553, 199889, 801349, 100393, 804941, 17203, 101293, 62473, 102197, 819389, 48413, 207121, 830309, 16073, 64433, 104933, 44279, 42157, 6079, 9623, 22937, 170473, 214481, 66137, 21061, 4603, 78839, 108637, 4051, 7039, 7229, 14401, 44017, 8501, 177953, 893549, 2711, 22481, 901141, 3701, 10903, 113357, 83311, 54133, 6067, 924109, 6257, 185593, 10567, 30059, 17957, 2131, 939581, 5741, 15991, 2003, 189473, 12491, 5087, 20323, 47857, 9241, 2089, 241249, 193393, 3907, 970909, 22111, 974861, 24421, 978821, 22291, 982789, 6029, 26777, 4003, 8221, 26993, 125093, 91159, 1051, 126097, 5981, 19477, 1014821, 2311, 15013, 257249, 3989, 261329, 2791, 1051469, 15493, 1059701, 12967, 53917, 4637, 2879, 2293, 12809, 7643, 8831, 1097141, 27481, 1101341, 14519, 1105549, 221953,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+114x-11 and
the reducible primes which appear as divisor for the first time
p | x^2+114x-11 and p < x^2+114x-11

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)
11011471.1000000.4000001.1000000.0000000.0000000.000000
21006915540.6900000.1500000.6900006.2727273.7500007.714286
31.0006291105190.6290000.1100000.6290009.1159427.3333339.611111
410.0006.4707865.6840.6470000.0786000.64700010.2861687.14545410.951831
5100.00065.7836.08359.7000.6578300.0608300.65783010.1673887.73918610.503167
61.000.000663.53649.905613.6310.6635360.0499050.66353610.0867408.20401110.278576
710.000.0006.678.430420.4826.257.9480.6678430.0420480.66784310.0649108.42564910.198227
8100.000.00067.111.4783.644.08063.467.3980.6711150.0364410.67111510.0489908.66643510.141887
91.000.000.000673.578.02032.157.157641.420.8630.6735780.0321570.67357810.0367048.82449210.106304
1010.000.000.0006.755.475.008287.698.9026.467.776.1060.6755470.0287700.67554710.0292398.94665210.083513


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)
123121.5000000.5000001.0000000.0000000.0000000.000000
245231.2500000.5000000.7500001.6666672.0000001.500000
389361.1250000.3750000.7500001.8000001.5000002.000000
416165111.0000000.3125000.6875001.7777781.6666671.833333
532266200.8125000.1875000.6250001.6250001.2000001.818182
664489390.7500000.1406250.6093751.8461541.5000001.950000
71288721660.6796880.1640620.5156251.8125002.3333331.692308
8256167321350.6523440.1250000.5273441.9195401.5238102.045455
9512326582680.6367190.1132810.5234381.9520961.8125001.985185
101.0246451125330.6298830.1093750.5205081.9785281.9310341.988806
112.0481.3011931.1080.6352540.0942380.5410162.0170541.7232142.078799
124.0962.6163562.2600.6386720.0869140.5517582.0107611.8445602.039711
138.1925.2826494.6330.6447750.0792240.5655522.0191131.8230342.050000
1416.38410.6441.2119.4330.6496580.0739140.5757452.0151461.8659482.036046
1532.76821.4222.24119.1810.6537480.0683900.5853582.0125891.8505372.033393
1665.53642.9744.17938.7950.6557310.0637660.5919652.0060681.8647922.022574
17131.07286.3117.83878.4730.6585010.0597990.5987012.0084471.8755682.022761
18262.144173.07814.623158.4550.6602400.0557820.6044582.0052831.8656542.019230
19524.288347.09827.595319.5030.6620370.0526330.6094042.0054431.8870962.016364
201.048.576695.88252.183643.6990.6636450.0497660.6138792.0048571.8910312.014688
212.097.1521.395.01398.5101.296.5030.6651940.0469730.6182212.0046691.8877802.014145
224.194.3042.795.330187.4992.607.8310.6664590.0447030.6217552.0038021.9033502.011435
238.388.6085.600.280357.0195.243.2610.6676050.0425600.6250452.0034421.9041122.010583
2416.777.21611.218.788682.14210.536.6460.6686920.0406590.6280332.0032551.9106602.009560
2533.554.43222.470.6141.304.83521.165.7790.6696760.0388870.6307892.0029451.9128502.008778
2667.108.86445.003.9712.503.97642.499.9950.6706110.0373120.6332992.0027921.9189982.007958
27134.217.72890.123.2804.807.78185.315.4990.6714710.0358210.6356502.0025631.9200592.007424
28268.435.456180.454.7329.252.891171.201.8410.6722460.0344700.6377772.0023101.9245662.006691
29536.870.912361.296.20217.828.568343.467.6340.6729670.0332080.6397582.0021431.9268112.006215
301.073.741.824723.321.33434.404.300688.917.0340.6736450.0320420.6416042.0020181.9297292.005770
312.147.483.6481.448.004.02966.456.6521.381.547.3770.6742790.0309460.6433332.0018821.9316382.005390
324.294.967.2962.898.560.563128.526.6852.770.033.8780.6748740.0299250.6449492.0017631.9339932.005023
338.589.934.5925.801.907.643248.864.9955.553.042.6480.6754310.0289720.6464592.0016511.9362902.004684
3417.179.869.18411.612.819.687482.376.32411.130.443.3630.6759550.0280780.6478772.0015521.9383052.004387


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
121010100
242020110
383120120
4165140140
5326240150
6649360180
71282171401200
825632122001310
951258193901570
101.0241123775011110
112.04819362131011920
124.096356114242013550
138.192649207442016480
1416.3841.211397814011.2100
1532.7682.2417461.495012.2400
1665.5364.1791.4022.777014.1780
17131.0727.8382.6085.230017.8370
18262.14414.6234.8759.7480114.6220
19524.28827.5959.22018.3750127.5940
201.048.57652.18317.35834.8250152.1820
212.097.15298.51032.79365.7170198.5090
224.194.304187.49962.551124.94801187.4980
238.388.608357.019119.037237.98201357.0180
2416.777.216682.142227.351454.79101682.1410
2533.554.4321.304.835434.786870.049011.304.8340
2667.108.8642.503.976834.4611.669.515012.503.9750
27134.217.7284.807.7811.601.8773.205.904014.807.7800
28268.435.4569.252.8913.082.9116.169.980019.252.8900
29536.870.91217.828.5685.942.75311.885.8150117.828.5670
301.073.741.82434.404.30011.466.15722.938.1430134.404.2990
312.147.483.64866.456.65222.150.60244.306.0500166.456.6510
324.294.967.296128.526.68542.845.43085.681.25501128.526.6840
338.589.934.592248.864.99582.950.435165.914.56001248.864.9940
3417.179.869.184482.376.324160.785.495321.590.82901482.376.3230


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
122111010
243211020
386513120
41611835132
532201557364
664392316164118
71286638282462313
8256135736252144128
9512268138130106327852
101.02453327825519075156112
112.0481.108574534366182332228
124.0962.2601.1571.103729416643472
138.1924.6332.3182.3151.4488721.2891.024
1416.3849.4334.7214.7122.9051.8412.5602.127
1532.76819.1819.6399.5425.7573.9005.2254.299
1665.53638.79519.45119.34411.5358.04010.3968.824
17131.07278.47339.28839.18523.07816.46720.92618.002
18262.144158.45579.51378.94246.20733.83141.83536.582
19524.288319.503160.681158.82292.33368.65784.07474.439
201.048.576643.699323.481320.218183.779139.971169.332150.617
212.097.1521.296.503651.724644.779367.373284.453340.088304.589
224.194.3042.607.8311.311.1431.296.688733.850576.497682.574614.910
238.388.6085.243.2612.634.8632.608.3981.466.6151.166.7561.369.5121.240.378
2416.777.21610.536.6465.295.3765.241.2702.933.6832.358.5262.744.6112.499.826
2533.554.43221.165.77910.635.16510.530.6145.860.8454.764.9975.506.0735.033.864
2667.108.86442.499.99521.347.96421.152.03111.716.6929.617.87111.035.74510.129.687
27134.217.72885.315.49942.847.32642.468.17323.422.57819.398.41922.115.91320.378.589
28268.435.456171.201.84185.959.25385.242.58846.820.43839.097.56344.312.76940.971.071
29536.870.912343.467.634172.420.382171.047.25293.607.93978.738.85688.781.41182.339.428
301.073.741.824688.917.034345.797.444343.119.590187.151.924158.477.615177.866.547165.420.948
312.147.483.6481.381.547.377693.354.794688.192.583374.191.787318.824.491356.266.783332.264.316
324.294.967.2962.770.033.8781.389.993.5321.380.040.346748.190.112641.206.425713.541.387667.095.954
338.589.934.5925.553.042.6482.786.074.1412.766.968.5071.496.016.6221.289.003.0241.428.920.8461.339.102.156
3417.179.869.18411.130.443.3635.583.771.2505.546.672.1132.991.309.8112.590.308.3312.861.475.3952.687.349.826


8. Check for existing Integer Sequences by OEIS

Found in Database : 11, 13, 17, 5, 461, 73, 709, 19, 193, 137, 1229, 31, 79, 41, 1, 37, 2069, 277, 43, 1,
Found in Database : 11, 13, 17, 5, 461, 73, 709, 19, 193, 137, 1229, 31, 79, 41, 37, 2069, 277, 43, 157, 353, 271, 3301, 433, 191, 61, 47, 139, 59, 5021, 1301, 317, 1153, 1489,
Found in Database : 5, 11, 13, 17, 19, 31, 37, 41, 43, 47, 59, 61, 73, 79, 83, 137, 139,