Inhaltsverzeichnis

Development of
Algorithmic Constructions

16:31:55
Deutsch
29.Mar 2024

Polynom = x^2+44x-61

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) = 61 = 61
f(1) = 1 = 1
f(2) = 31 = 31
f(3) = 5 = 5
f(4) = 131 = 131
f(5) = 23 = 23
f(6) = 239 = 239
f(7) = 37 = 37
f(8) = 355 = 5*71
f(9) = 13 = 13
f(10) = 479 = 479
f(11) = 17 = 17
f(12) = 611 = 13*47
f(13) = 85 = 5*17
f(14) = 751 = 751
f(15) = 103 = 103
f(16) = 899 = 29*31
f(17) = 61 = 61
f(18) = 1055 = 5*211
f(19) = 71 = 71
f(20) = 1219 = 23*53
f(21) = 163 = 163
f(22) = 1391 = 13*107
f(23) = 185 = 5*37
f(24) = 1571 = 1571
f(25) = 13 = 13
f(26) = 1759 = 1759
f(27) = 29 = 29
f(28) = 1955 = 5*17*23
f(29) = 257 = 257
f(30) = 2159 = 17*127
f(31) = 283 = 283
f(32) = 2371 = 2371
f(33) = 155 = 5*31
f(34) = 2591 = 2591
f(35) = 169 = 13*13
f(36) = 2819 = 2819
f(37) = 367 = 367
f(38) = 3055 = 5*13*47
f(39) = 397 = 397
f(40) = 3299 = 3299
f(41) = 107 = 107
f(42) = 3551 = 53*67
f(43) = 115 = 5*23
f(44) = 3811 = 37*103
f(45) = 493 = 17*29
f(46) = 4079 = 4079
f(47) = 527 = 17*31
f(48) = 4355 = 5*13*67
f(49) = 281 = 281
f(50) = 4639 = 4639
f(51) = 299 = 13*23
f(52) = 4931 = 4931
f(53) = 635 = 5*127
f(54) = 5231 = 5231
f(55) = 673 = 673
f(56) = 5539 = 29*191
f(57) = 89 = 89
f(58) = 5855 = 5*1171
f(59) = 47 = 47
f(60) = 6179 = 37*167
f(61) = 793 = 13*61
f(62) = 6511 = 17*383
f(63) = 835 = 5*167
f(64) = 6851 = 13*17*31
f(65) = 439 = 439
f(66) = 7199 = 23*313
f(67) = 461 = 461
f(68) = 7555 = 5*1511
f(69) = 967 = 967
f(70) = 7919 = 7919
f(71) = 1013 = 1013
f(72) = 8291 = 8291
f(73) = 265 = 5*53
f(74) = 8671 = 13*23*29
f(75) = 277 = 277
f(76) = 9059 = 9059
f(77) = 1157 = 13*89
f(78) = 9455 = 5*31*61
f(79) = 1207 = 17*71
f(80) = 9859 = 9859
f(81) = 629 = 17*37
f(82) = 10271 = 10271
f(83) = 655 = 5*131
f(84) = 10691 = 10691
f(85) = 1363 = 29*47
f(86) = 11119 = 11119
f(87) = 1417 = 13*109
f(88) = 11555 = 5*2311
f(89) = 23 = 23
f(90) = 11999 = 13*13*71
f(91) = 191 = 191
f(92) = 12451 = 12451
f(93) = 1585 = 5*317
f(94) = 12911 = 12911
f(95) = 1643 = 31*53
f(96) = 13379 = 17*787
f(97) = 851 = 23*37
f(98) = 13855 = 5*17*163
f(99) = 881 = 881
f(100) = 14339 = 13*1103

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+44x-61

f(0)=61
f(1)=1
f(2)=31
f(3)=5
f(4)=131
f(5)=23
f(6)=239
f(7)=37
f(8)=71
f(9)=13
f(10)=479
f(11)=17
f(12)=47
f(13)=1
f(14)=751
f(15)=103
f(16)=29
f(17)=1
f(18)=211
f(19)=1
f(20)=53
f(21)=163
f(22)=107
f(23)=1
f(24)=1571
f(25)=1
f(26)=1759
f(27)=1
f(28)=1
f(29)=257
f(30)=127
f(31)=283
f(32)=2371
f(33)=1
f(34)=2591
f(35)=1
f(36)=2819
f(37)=367
f(38)=1
f(39)=397
f(40)=3299
f(41)=1
f(42)=67
f(43)=1
f(44)=1
f(45)=1
f(46)=4079
f(47)=1
f(48)=1
f(49)=281
f(50)=4639
f(51)=1
f(52)=4931
f(53)=1
f(54)=5231
f(55)=673
f(56)=191
f(57)=89
f(58)=1171
f(59)=1
f(60)=167
f(61)=1
f(62)=383
f(63)=1
f(64)=1
f(65)=439
f(66)=313
f(67)=461
f(68)=1511
f(69)=967
f(70)=7919
f(71)=1013
f(72)=8291
f(73)=1
f(74)=1
f(75)=277
f(76)=9059
f(77)=1
f(78)=1
f(79)=1
f(80)=9859
f(81)=1
f(82)=10271
f(83)=1
f(84)=10691
f(85)=1
f(86)=11119
f(87)=109
f(88)=2311
f(89)=1
f(90)=1
f(91)=1
f(92)=12451
f(93)=317
f(94)=12911
f(95)=1
f(96)=787
f(97)=1
f(98)=1
f(99)=881

b) Substitution of the polynom
The polynom f(x)=x^2+44x-61 could be written as f(y)= y^2-545 with x=y-22

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+22
f'(x)>2x+43

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

61, 1, 31, 5, 131, 23, 239, 37, 71, 13, 479, 17, 47, 1, 751, 103, 29, 1, 211, 1, 53, 163, 107, 1, 1571, 1, 1759, 1, 1, 257, 127, 283, 2371, 1, 2591, 1, 2819, 367, 1, 397, 3299, 1, 67, 1, 1, 1, 4079, 1, 1, 281, 4639, 1, 4931, 1, 5231, 673, 191, 89, 1171, 1, 167, 1, 383, 1, 1, 439, 313, 461, 1511, 967, 7919, 1013, 8291, 1, 1, 277, 9059, 1, 1, 1, 9859, 1, 10271, 1, 10691, 1, 11119, 109, 2311, 1, 1, 1, 12451, 317, 12911, 1, 787, 1, 1, 881, 1103, 1823, 14831, 1, 15331, 487, 337, 503, 3271, 1, 16879, 2143, 757, 1, 619, 1, 1423, 2347, 1, 2417, 853, 311, 331, 1, 20771, 2633, 1, 2707, 4391, 1, 1327, 1429, 1, 587, 643, 1, 24419, 773, 5011, 1, 829, 3253, 2027, 1, 27011, 1709, 1, 1, 1, 1, 409, 3673, 2287, 1, 30431, 1, 31139, 1, 1, 4027, 32579, 1, 33311, 421, 2003, 1, 1, 4397, 547, 1123, 36319, 1, 1279, 937, 37871, 4783, 577, 2441, 607, 1, 1, 1, 613, 1, 1, 661, 42719, 1, 1, 1, 1531, 431, 1223, 571, 3547, 2909, 1, 5927, 563, 6037, 1, 1, 1601, 1, 1, 6373, 51439, 499, 1, 3301, 53279, 3359, 2357, 1367, 1, 1, 56099, 1, 11411, 1, 4463, 1, 58991, 1487, 59971, 3779, 1297, 1, 12391, 1, 1, 7933, 1, 1, 3823, 1, 617, 8317, 13411, 8447, 68099, 4289, 69151, 1, 1151, 1, 5483, 1, 1, 1, 1, 1, 74531, 1877, 1427, 1, 5903, 4831, 677, 1, 78979, 1, 80111, 2017, 2621, 2557, 1, 2593, 983, 809, 84719, 10663, 6607, 1, 87071, 5479, 88259, 1, 17891, 11257, 90659, 1, 1, 1, 733, 1, 1, 11867, 659, 6011, 96799, 6089, 1381, 2467, 2113, 1, 4373, 3163, 1567, 3203, 6067, 12973, 6143, 1, 4597, 1, 3691, 1, 1667, 13627, 1637, 1061, 3581, 349, 1, 883, 1697, 1, 23011, 1, 2477, 1, 1931, 1481, 1, 14983, 3889, 1, 24391, 3833, 1, 3877, 124771, 3137, 1, 1, 7507, 1, 1, 8111, 3527, 1, 5737, 1, 1499, 1, 4651, 1, 27271, 17137, 1, 1019, 1, 1, 140831, 8849, 3847, 1, 28771, 18077, 1, 4567, 1373, 1, 5119, 811, 2459, 401, 1783, 9521, 9007, 9619, 154691, 1, 1, 1, 12143, 1, 31891, 1, 161059, 20233, 162671, 1, 164291, 1, 12763, 1, 1, 1619, 169199, 1, 170851, 1, 1033, 5417, 7573, 1, 35171, 1699, 177539, 11149, 1, 2251, 1, 1, 182639, 22937, 36871, 1447, 1, 1, 14447, 1, 1, 1831, 191299, 12011, 38611, 1, 194819, 1439, 6779, 4937, 1217, 1, 200159, 1, 1, 25357, 5507, 25583, 1, 1, 1, 1, 1, 26267, 1, 26497, 1009, 1, 9337, 1, 1, 1, 218479, 27427, 44071, 13831, 1, 1, 224131, 1, 17387, 1669, 2213, 1, 45971, 7213, 231779, 1, 8059, 5867, 18127, 1, 4483, 1, 47911, 1, 14207, 30313, 14323, 1, 245471, 3851, 1889, 2389, 49891, 31307, 1, 509, 5393, 3181, 1069, 32063, 257519, 1901, 1, 1, 20123, 1, 263651, 1, 265711, 33343, 1, 1, 1741, 16931, 271939, 34123, 3079, 1, 1, 1, 1259, 1091, 1193, 1213, 1733, 1, 9181, 3571, 286751, 17989, 1, 541, 58211, 1, 10111, 1, 3319, 1, 1, 1, 13033, 37607, 1, 1, 2953, 19079, 23567, 7687, 13417, 38713, 310819, 2437, 1, 4909, 1, 1, 1879, 1, 319811, 1543, 322079, 20201, 64871, 1, 6163, 40973, 328931, 2063, 331231, 1, 333539, 1, 5167, 1, 10909, 1, 9203, 4271, 2053, 43003, 5659, 1493, 5347, 5449, 349919, 1, 1, 8837, 1, 44483, 1, 22391, 1, 22541, 15733, 3491, 364271, 9137, 1, 11497, 369119, 1, 74311, 2741, 373999, 1, 1, 4721, 29147, 1, 13151, 1, 76771, 1301, 386339, 1, 8273, 1, 391331, 1583, 23167, 1, 4663, 24851, 1, 1, 401411, 10067, 403951, 1, 406499, 12743, 3557, 12823, 31663, 51613, 1, 1, 1, 1, 18233, 1, 84391, 1, 424559, 53233, 3919, 1, 429791, 1, 1, 54217, 87011, 54547, 25747, 1, 25903, 5521, 443011, 1, 34283, 1, 89671, 1, 451039, 1, 12263, 1, 15739, 57223, 8663, 1693, 1, 1, 1, 58243, 1, 11717, 1, 1, 472799, 1, 95111, 59617, 10177, 1, 1609, 1, 28463, 2333, 28627, 61007, 1847, 61357, 5531, 15427, 495071, 1, 16061, 4801, 500719, 2729, 1, 1, 1451, 1867, 1627, 1, 17659, 2791, 10957, 8069, 1, 4057, 5851, 5021, 523631, 13127, 526531, 32999, 1, 33181, 6263, 1, 1, 1, 1, 3373, 41627, 1, 544099, 1, 1, 68567, 1, 34469, 552991, 1, 42767, 4099, 15107, 1, 1, 1, 564959, 1, 9311, 1, 570991, 71563, 5573, 2767, 1, 36161, 44623, 1, 34303, 1, 34483, 18367, 19009, 1, 118471, 1, 1, 74623, 20639, 1, 601631, 37699, 26293, 75787, 121571, 4481, 19709, 1, 11587, 1, 1, 1, 3671, 77747, 1, 1, 21611, 1, 629891, 15787, 2237, 79333, 2879, 1, 7523, 1, 1907, 2777, 3061, 16187, 1, 1, 652319, 1777, 131111, 1, 1, 2663, 1, 1, 665311, 1, 668579, 83777, 134371, 2903, 1, 42299, 2269, 8501, 681731, 6571, 685039, 85837, 4441, 21563, 1, 1, 40883, 17417, 2203, 1, 701699, 43961, 10847, 44171, 3709, 2399, 22961, 17837, 715171, 1, 718559, 1, 1, 1, 725359, 6991, 728771, 1, 732191, 45869, 4513, 1, 147811, 1, 742499, 1789, 745951, 4673, 3391, 93893, 1, 94327, 6577, 47381, 6971, 47599, 11393, 1, 1, 1, 33493, 1, 1, 12119, 10949, 1, 3701, 1, 784451, 49139, 787999, 3797, 1, 1, 1973, 1, 798691, 5003, 21683, 25127, 805859, 100957, 1, 4409, 1, 50929, 1, 1, 1, 102763, 28411, 103217, 165511, 12959, 15683, 1, 2971, 1, 36457, 1, 64783, 1, 1, 52981, 849539, 3433, 37097, 21377, 1, 1, 1, 26953, 172871, 8329, 1, 2939, 51283, 1, 51503, 1, 879299, 4789, 176611, 1, 886819, 1, 68507, 2789, 894371, 4871, 30971, 112507, 180391, 3323, 905759, 1, 1, 22787, 1, 1, 917219, 28723, 184211, 28843, 40213, 115853, 1, 1, 32159, 4493, 1, 1, 1, 117787, 944239, 118273, 20173, 2969, 1, 1, 955939, 119737, 14767, 1, 963779, 4643, 4049, 1, 971651, 7159, 975599, 1, 1, 1, 983519, 1, 987491, 1, 1, 124183, 16319, 2011, 4253, 62591, 1, 125683, 59263, 25237, 2683, 1, 1, 1, 203911, 1, 1, 128203, 1, 1, 1, 64609, 19543, 1, 207971, 1,

6. Sequence of the polynom (only primes)

61, 31, 5, 131, 23, 239, 37, 71, 13, 479, 17, 47, 751, 103, 29, 211, 53, 163, 107, 1571, 1759, 257, 127, 283, 2371, 2591, 2819, 367, 397, 3299, 67, 4079, 281, 4639, 4931, 5231, 673, 191, 89, 1171, 167, 383, 439, 313, 461, 1511, 967, 7919, 1013, 8291, 277, 9059, 9859, 10271, 10691, 11119, 109, 2311, 12451, 317, 12911, 787, 881, 1103, 1823, 14831, 15331, 487, 337, 503, 3271, 16879, 2143, 757, 619, 1423, 2347, 2417, 853, 311, 331, 20771, 2633, 2707, 4391, 1327, 1429, 587, 643, 24419, 773, 5011, 829, 3253, 2027, 27011, 1709, 409, 3673, 2287, 30431, 31139, 4027, 32579, 33311, 421, 2003, 4397, 547, 1123, 36319, 1279, 937, 37871, 4783, 577, 2441, 607, 613, 661, 42719, 1531, 431, 1223, 571, 3547, 2909, 5927, 563, 6037, 1601, 6373, 51439, 499, 3301, 53279, 3359, 2357, 1367, 56099, 11411, 4463, 58991, 1487, 59971, 3779, 1297, 12391, 7933, 3823, 617, 8317, 13411, 8447, 68099, 4289, 69151, 1151, 5483, 74531, 1877, 1427, 5903, 4831, 677, 78979, 80111, 2017, 2621, 2557, 2593, 983, 809, 84719, 10663, 6607, 87071, 5479, 88259, 17891, 11257, 90659, 733, 11867, 659, 6011, 96799, 6089, 1381, 2467, 2113, 4373, 3163, 1567, 3203, 6067, 12973, 6143, 4597, 3691, 1667, 13627, 1637, 1061, 3581, 349, 883, 1697, 23011, 2477, 1931, 1481, 14983, 3889, 24391, 3833, 3877, 124771, 3137, 7507, 8111, 3527, 5737, 1499, 4651, 27271, 17137, 1019, 140831, 8849, 3847, 28771, 18077, 4567, 1373, 5119, 811, 2459, 401, 1783, 9521, 9007, 9619, 154691, 12143, 31891, 161059, 20233, 162671, 164291, 12763, 1619, 169199, 170851, 1033, 5417, 7573, 35171, 1699, 177539, 11149, 2251, 182639, 22937, 36871, 1447, 14447, 1831, 191299, 12011, 38611, 194819, 1439, 6779, 4937, 1217, 200159, 25357, 5507, 25583, 26267, 26497, 1009, 9337, 218479, 27427, 44071, 13831, 224131, 17387, 1669, 2213, 45971, 7213, 231779, 8059, 5867, 18127, 4483, 47911, 14207, 30313, 14323, 245471, 3851, 1889, 2389, 49891, 31307, 509, 5393, 3181, 1069, 32063, 257519, 1901, 20123, 263651, 265711, 33343, 1741, 16931, 271939, 34123, 3079, 1259, 1091, 1193, 1213, 1733, 9181, 3571, 286751, 17989, 541, 58211, 10111, 3319, 13033, 37607, 2953, 19079, 23567, 7687, 13417, 38713, 310819, 2437, 4909, 1879, 319811, 1543, 322079, 20201, 64871, 6163, 40973, 328931, 2063, 331231, 333539, 5167, 10909, 9203, 4271, 2053, 43003, 5659, 1493, 5347, 5449, 349919, 8837, 44483, 22391, 22541, 15733, 3491, 364271, 9137, 11497, 369119, 74311, 2741, 373999, 4721, 29147, 13151, 76771, 1301, 386339, 8273, 391331, 1583, 23167, 4663, 24851, 401411, 10067, 403951, 406499, 12743, 3557, 12823, 31663, 51613, 18233, 84391, 424559, 53233, 3919, 429791, 54217, 87011, 54547, 25747, 25903, 5521, 443011, 34283, 89671, 451039, 12263, 15739, 57223, 8663, 1693, 58243, 11717, 472799, 95111, 59617, 10177, 1609, 28463, 2333, 28627, 61007, 1847, 61357, 5531, 15427, 495071, 16061, 4801, 500719, 2729, 1451, 1867, 1627, 17659, 2791, 10957, 8069, 4057, 5851, 5021, 523631, 13127, 526531, 32999, 33181, 6263, 3373, 41627, 544099, 68567, 34469, 552991, 42767, 4099, 15107, 564959, 9311, 570991, 71563, 5573, 2767, 36161, 44623, 34303, 34483, 18367, 19009, 118471, 74623, 20639, 601631, 37699, 26293, 75787, 121571, 4481, 19709, 11587, 3671, 77747, 21611, 629891, 15787, 2237, 79333, 2879, 7523, 1907, 2777, 3061, 16187, 652319, 1777, 131111, 2663, 665311, 668579, 83777, 134371, 2903, 42299, 2269, 8501, 681731, 6571, 685039, 85837, 4441, 21563, 40883, 17417, 2203, 701699, 43961, 10847, 44171, 3709, 2399, 22961, 17837, 715171, 718559, 725359, 6991, 728771, 732191, 45869, 4513, 147811, 742499, 1789, 745951, 4673, 3391, 93893, 94327, 6577, 47381, 6971, 47599, 11393, 33493, 12119, 10949, 3701, 784451, 49139, 787999, 3797, 1973, 798691, 5003, 21683, 25127, 805859, 100957, 4409, 50929, 102763, 28411, 103217, 165511, 12959, 15683, 2971, 36457, 64783, 52981, 849539, 3433, 37097, 21377, 26953, 172871, 8329, 2939, 51283, 51503, 879299, 4789, 176611, 886819, 68507, 2789, 894371, 4871, 30971, 112507, 180391, 3323, 905759, 22787, 917219, 28723, 184211, 28843, 40213, 115853, 32159, 4493, 117787, 944239, 118273, 20173, 2969, 955939, 119737, 14767, 963779, 4643, 4049, 971651, 7159, 975599, 983519, 987491, 124183, 16319, 2011, 4253, 62591, 125683, 59263, 25237, 2683, 203911, 128203, 64609, 19543, 207971,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+44x-61 and
the reducible primes which appear as divisor for the first time
p | x^2+44x-61 and p < x^2+44x-61

To avoid confusion with the number of primes:
I did not count the primes <= A
but I counted the primes appending the x and therefore the x <= A

ABCDEFGHIJK
exponent =log10 (x)<=xnumber of all primesnumber of primes p = f(x) number of primes p | f(x) C/xD/xE/xC(n) / C(n-1)D(n) / D(n-1)E(n) / E(n-1)
1109540.9000000.5000000.9000000.0000000.0000000.000000
21005925340.5900000.2500000.5900006.5555555.0000008.500000
31.0006381355030.6380000.1350000.63800010.8135605.40000014.794118
410.0006.5749805.5940.6574000.0980000.65740010.3040757.25925911.121272
5100.00066.4137.67758.7360.6641300.0767700.66413010.1023737.83367310.499822
61.000.000669.37063.393605.9770.6693700.0633930.66937010.0789008.25752310.316960
710.000.0006.727.380536.9916.190.3890.6727380.0536990.67273810.0503168.47082510.215551
8100.000.00067.525.0754.648.03562.877.0400.6752510.0464800.67525110.0373518.65570410.157204
91.000.000.000677.238.63441.004.271636.234.3630.6772390.0410040.67723910.0294408.82185110.118708
1010.000.000.0006.788.160.156366.945.6086.421.214.5480.6788160.0366950.67881610.0232918.94896010.092530


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
244311.0000000.7500000.2500002.0000001.500000inf
388441.0000000.5000000.5000002.0000001.3333334.000000
41612660.7500000.3750000.3750001.5000001.5000001.500000
532229130.6875000.2812500.4062501.8333331.5000002.166667
6643716210.5781250.2500000.3281251.6818181.7777781.615385
71288029510.6250000.2265620.3984382.1621621.8125002.428571
8256158461120.6171880.1796880.4375001.9750001.5862072.196079
9512323742490.6308590.1445310.4863282.0443041.6086962.223214
101.0246571375200.6416020.1337890.5078122.0340561.8513512.088353
112.0481.3282491.0790.6484380.1215820.5268552.0213091.8175182.075000
124.0962.6684512.2170.6513670.1101070.5412602.0090361.8112452.054680
138.1925.3628334.5290.6545410.1016850.5528562.0097451.8470072.042851
1416.38410.7891.5159.2740.6585080.0924680.5660402.0121221.8187272.047693
1532.76821.6842.81918.8650.6617430.0860290.5757142.0098251.8607262.034182
1665.53643.4625.22938.2330.6631770.0797880.5833892.0043351.8549132.026663
17131.07287.1299.84377.2860.6647420.0750960.5896452.0047171.8823872.021447
18262.144174.72918.641156.0880.6665380.0711100.5954282.0054061.8938332.019615
19524.288350.29034.977315.3130.6681250.0667130.6014122.0047621.8763482.020098
201.048.576701.91066.177635.7330.6693940.0631110.6062822.0037971.8920152.016196
212.097.1521.406.351125.5771.280.7740.6706000.0598800.6107212.0036061.8975932.014641
224.194.3042.816.651239.2262.577.4250.6715420.0570360.6145062.0028081.9050142.012396
238.388.6085.641.360456.0325.185.3280.6725030.0543630.6181392.0028611.9062812.011825
2416.777.21611.296.886869.96810.426.9180.6733470.0518540.6214932.0025111.9076912.010850
2533.554.43222.620.8781.665.29620.955.5820.6741550.0496300.6245252.0023991.9142042.009758
2667.108.86445.289.3633.193.43842.095.9250.6748640.0475860.6272782.0021051.9176402.008817
27134.217.72890.669.9996.134.13884.535.8610.6755440.0457030.6298412.0020151.9208572.008172
28268.435.456181.507.49911.798.935169.708.5640.6761680.0439540.6322142.0018481.9234872.007533
29536.870.912363.330.82022.734.835340.595.9850.6767560.0423470.6344092.0017401.9268552.006947
301.073.741.824727.238.69643.867.886683.370.8100.6772940.0408550.6364392.0015881.9295452.006397
312.147.483.6481.455.564.26884.749.2611.370.815.0070.6778000.0394640.6383352.0014941.9319202.005961
324.294.967.2962.913.159.124163.925.3782.749.233.7460.6782730.0381670.6401062.0013951.9342402.005547
338.589.934.5925.830.168.234317.407.0355.512.761.1990.6787210.0369510.6417702.0013221.9362902.005199
3417.179.869.18411.667.555.435615.248.94411.052.306.4910.6791410.0358120.6433292.0012381.9383592.004858


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
122200011
243210111
384220112
4166330114
5329540315
664166100619
7128291118013115
8256461531021124
9512742549036137
101.0241374691068168
112.0482499015901181130
124.09645115729402181232
138.19283329553804091423
1416.3841.51551999607331781
1532.7682.8199691.85001.39811.420
1665.5365.2291.7543.47502.59812.630
17131.0729.8433.2616.58204.94614.896
18262.14418.6416.20312.43809.29319.347
19524.28834.97711.63223.345017.423117.553
201.048.57666.17721.99744.180033.057133.119
212.097.152125.57741.91683.661062.815162.761
224.194.304239.22679.836159.3900119.5871119.638
238.388.608456.032152.136303.8960227.8241228.207
2416.777.216869.968290.029579.9390435.0431434.924
2533.554.4321.665.296555.2281.110.0680832.5551832.740
2667.108.8643.193.4381.064.9462.128.49201.596.48311.596.954
27134.217.7286.134.1382.045.9234.088.21503.066.79513.067.342
28268.435.45611.798.9353.933.4597.865.47605.898.72815.900.206
29536.870.91222.734.8357.577.90115.156.934011.367.676111.367.158
301.073.741.82443.867.88614.622.46129.245.425021.937.602121.930.283
312.147.483.64884.749.26128.249.51656.499.745042.375.856142.373.404
324.294.967.296163.925.37854.634.197109.291.181081.960.844181.964.533
338.589.934.592317.407.035105.796.314211.610.7210158.703.4501158.703.584
3417.179.869.184615.248.944205.078.822410.170.1220307.621.3381307.627.605


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
241010010
384130022
4166240024
53213671435
6642110113549
71285128239101022
8256112615118282739
951224913611346765671
101.02452027424698159112151
112.0481.079566513208322242307
124.0962.2171.1471.070470638499610
138.1924.5292.3492.1801.0091.2431.0231.254
1416.3849.2744.7784.4962.1512.5232.1182.482
1532.76818.8659.6789.1874.3825.0584.3935.032
1665.53638.23319.45918.7748.92810.1958.96610.144
17131.07277.28639.57137.71518.13820.42918.19220.527
18262.144156.08879.48676.60236.70541.34036.84441.199
19524.288315.313160.963154.35074.52083.05874.69683.039
201.048.576635.733324.208311.525150.865167.123151.015166.730
212.097.1521.280.774652.227628.547305.209335.676305.024334.865
224.194.3042.577.4251.310.6921.266.733615.922673.492615.854672.157
238.388.6085.185.3282.633.4252.551.9031.242.6121.351.1251.241.5681.350.023
2416.777.21610.426.9185.293.2235.133.6952.504.2012.711.8652.502.5242.708.328
2533.554.43220.955.58210.625.12310.330.4595.043.0105.437.0655.040.3605.435.147
2667.108.86442.095.92521.332.27120.763.65410.147.42610.903.01910.146.95210.898.528
27134.217.72884.535.86142.815.85841.720.00320.412.21821.858.85620.413.37221.851.415
28268.435.456169.708.56485.911.97183.796.59341.036.65343.823.32741.041.58943.806.995
29536.870.912340.595.985172.345.559168.250.42682.472.92087.830.06282.476.64887.816.355
301.073.741.824683.370.810345.613.351337.757.459165.689.697175.994.596165.700.763175.985.754
312.147.483.6481.370.815.007692.990.753677.824.254332.777.425352.608.216332.808.603352.620.763
324.294.967.2962.749.233.7461.389.269.9501.359.963.796668.153.079706.456.858668.159.156706.464.653
338.589.934.5925.512.761.1992.784.737.6192.728.023.5801.341.133.3841.415.234.4131.341.142.2641.415.251.138
3417.179.869.18411.052.306.4915.581.040.0175.471.266.4742.691.386.6282.834.770.9172.691.319.4402.834.829.506


8. Check for existing Integer Sequences by OEIS

Found in Database : 61, 1, 31, 5, 131, 23, 239, 37, 71, 13, 479, 17, 47, 1, 751, 103, 29, 1, 211, 1,
Found in Database : 61, 31, 5, 131, 23, 239, 37, 71, 13, 479, 17, 47, 751, 103, 29, 211, 53, 163, 107, 1571, 1759, 257, 127, 283, 2371, 2591, 2819, 367, 397,
Found in Database : 5, 13, 17, 23, 29, 31, 37, 47, 53, 61, 67, 71, 89, 103, 107, 109, 127, 131,