Inhaltsverzeichnis

Development of
Algorithmic Constructions

08:50:58
Deutsch
29.Mar 2024

Polynom = x^2+1x-13

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) = 13 = 13
f(1) = 11 = 11
f(2) = 7 = 7
f(3) = 1 = 1
f(4) = 7 = 7
f(5) = 17 = 17
f(6) = 29 = 29
f(7) = 43 = 43
f(8) = 59 = 59
f(9) = 77 = 7*11
f(10) = 97 = 97
f(11) = 119 = 7*17
f(12) = 143 = 11*13
f(13) = 169 = 13*13
f(14) = 197 = 197
f(15) = 227 = 227
f(16) = 259 = 7*37
f(17) = 293 = 293
f(18) = 329 = 7*47
f(19) = 367 = 367
f(20) = 407 = 11*37
f(21) = 449 = 449
f(22) = 493 = 17*29
f(23) = 539 = 7*7*11
f(24) = 587 = 587
f(25) = 637 = 7*7*13
f(26) = 689 = 13*53
f(27) = 743 = 743
f(28) = 799 = 17*47
f(29) = 857 = 857
f(30) = 917 = 7*131
f(31) = 979 = 11*89
f(32) = 1043 = 7*149
f(33) = 1109 = 1109
f(34) = 1177 = 11*107
f(35) = 1247 = 29*43
f(36) = 1319 = 1319
f(37) = 1393 = 7*199
f(38) = 1469 = 13*113
f(39) = 1547 = 7*13*17
f(40) = 1627 = 1627
f(41) = 1709 = 1709
f(42) = 1793 = 11*163
f(43) = 1879 = 1879
f(44) = 1967 = 7*281
f(45) = 2057 = 11*11*17
f(46) = 2149 = 7*307
f(47) = 2243 = 2243
f(48) = 2339 = 2339
f(49) = 2437 = 2437
f(50) = 2537 = 43*59
f(51) = 2639 = 7*13*29
f(52) = 2743 = 13*211
f(53) = 2849 = 7*11*37
f(54) = 2957 = 2957
f(55) = 3067 = 3067
f(56) = 3179 = 11*17*17
f(57) = 3293 = 37*89
f(58) = 3409 = 7*487
f(59) = 3527 = 3527
f(60) = 3647 = 7*521
f(61) = 3769 = 3769
f(62) = 3893 = 17*229
f(63) = 4019 = 4019
f(64) = 4147 = 11*13*29
f(65) = 4277 = 7*13*47
f(66) = 4409 = 4409
f(67) = 4543 = 7*11*59
f(68) = 4679 = 4679
f(69) = 4817 = 4817
f(70) = 4957 = 4957
f(71) = 5099 = 5099
f(72) = 5243 = 7*7*107
f(73) = 5389 = 17*317
f(74) = 5537 = 7*7*113
f(75) = 5687 = 11*11*47
f(76) = 5839 = 5839
f(77) = 5993 = 13*461
f(78) = 6149 = 11*13*43
f(79) = 6307 = 7*17*53
f(80) = 6467 = 29*223
f(81) = 6629 = 7*947
f(82) = 6793 = 6793
f(83) = 6959 = 6959
f(84) = 7127 = 7127
f(85) = 7297 = 7297
f(86) = 7469 = 7*11*97
f(87) = 7643 = 7643
f(88) = 7819 = 7*1117
f(89) = 7997 = 11*727
f(90) = 8177 = 13*17*37
f(91) = 8359 = 13*643
f(92) = 8543 = 8543
f(93) = 8729 = 7*29*43
f(94) = 8917 = 37*241
f(95) = 9107 = 7*1301
f(96) = 9299 = 17*547
f(97) = 9493 = 11*863
f(98) = 9689 = 9689
f(99) = 9887 = 9887
f(100) = 10087 = 7*11*131

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-13

f(0)=13
f(1)=11
f(2)=7
f(3)=1
f(4)=1
f(5)=17
f(6)=29
f(7)=43
f(8)=59
f(9)=1
f(10)=97
f(11)=1
f(12)=1
f(13)=1
f(14)=197
f(15)=227
f(16)=37
f(17)=293
f(18)=47
f(19)=367
f(20)=1
f(21)=449
f(22)=1
f(23)=1
f(24)=587
f(25)=1
f(26)=53
f(27)=743
f(28)=1
f(29)=857
f(30)=131
f(31)=89
f(32)=149
f(33)=1109
f(34)=107
f(35)=1
f(36)=1319
f(37)=199
f(38)=113
f(39)=1
f(40)=1627
f(41)=1709
f(42)=163
f(43)=1879
f(44)=281
f(45)=1
f(46)=307
f(47)=2243
f(48)=2339
f(49)=2437
f(50)=1
f(51)=1
f(52)=211
f(53)=1
f(54)=2957
f(55)=3067
f(56)=1
f(57)=1
f(58)=487
f(59)=3527
f(60)=521
f(61)=3769
f(62)=229
f(63)=4019
f(64)=1
f(65)=1
f(66)=4409
f(67)=1
f(68)=4679
f(69)=4817
f(70)=4957
f(71)=5099
f(72)=1
f(73)=317
f(74)=1
f(75)=1
f(76)=5839
f(77)=461
f(78)=1
f(79)=1
f(80)=223
f(81)=947
f(82)=6793
f(83)=6959
f(84)=7127
f(85)=7297
f(86)=1
f(87)=7643
f(88)=1117
f(89)=727
f(90)=1
f(91)=643
f(92)=8543
f(93)=1
f(94)=241
f(95)=1301
f(96)=547
f(97)=863
f(98)=9689
f(99)=9887

b) Substitution of the polynom
The polynom f(x)=x^2+1x-13 could be written as f(y)= y^2-13.25 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

13, 11, 7, 1, 1, 17, 29, 43, 59, 1, 97, 1, 1, 1, 197, 227, 37, 293, 47, 367, 1, 449, 1, 1, 587, 1, 53, 743, 1, 857, 131, 89, 149, 1109, 107, 1, 1319, 199, 113, 1, 1627, 1709, 163, 1879, 281, 1, 307, 2243, 2339, 2437, 1, 1, 211, 1, 2957, 3067, 1, 1, 487, 3527, 521, 3769, 229, 4019, 1, 1, 4409, 1, 4679, 4817, 4957, 5099, 1, 317, 1, 1, 5839, 461, 1, 1, 223, 947, 6793, 6959, 7127, 7297, 1, 7643, 1117, 727, 1, 643, 8543, 1, 241, 1301, 547, 863, 9689, 9887, 1, 10289, 1499, 823, 839, 11117, 11329, 1, 1069, 1, 12197, 1129, 269, 757, 1871, 13327, 1, 1061, 14029, 1297, 1, 1, 1, 311, 911, 15737, 271, 439, 2357, 1289, 1, 467, 331, 1619, 18077, 2621, 433, 2699, 661, 19447, 19727, 1, 1, 1583, 1, 21157, 1, 1279, 22039, 3191, 22637, 1, 2113, 23549, 23857, 1, 1, 24793, 1, 541, 25747, 1, 26393, 347, 1, 3911, 1, 967, 1, 1, 593, 29399, 607, 30089, 2767, 1811, 31139, 409, 31849, 1, 1123, 1, 1, 1, 4861, 1, 4967, 1, 3229, 35897, 36277, 5237, 2179, 5347, 2909, 2939, 1, 38993, 1, 3617, 5741, 1097, 40993, 41399, 431, 1, 907, 1, 3343, 1, 4027, 1, 6449, 45569, 6571, 2731, 997, 47293, 4339, 983, 3739, 1, 1151, 49939, 50387, 1753, 1, 877, 7457, 4787, 53117, 1, 1, 599, 4229, 1, 1, 56393, 1, 57347, 751, 58309, 1, 1, 1013, 60257, 4673, 673, 577, 523, 1459, 5749, 63743, 1367, 1, 65267, 9397, 66293, 66809, 5179, 1, 9767, 6263, 1, 619, 1, 4177, 2467, 1471, 72617, 1493, 73699, 5711, 1, 75337, 1, 6949, 647, 77549, 2111, 811, 79229, 11399, 1, 1051, 6269, 1, 683, 83219, 11971, 84377, 1, 653, 2003, 5101, 7937, 1, 88493, 1, 6899, 1, 937, 91493, 1, 92707, 13331, 8539, 94543, 2213, 8707, 1, 1, 1, 98269, 98897, 99527, 2707, 1, 101429, 2083, 9337, 103349, 103993, 104639, 1, 1, 15227, 107243, 1, 108557, 733, 1427, 110543, 15887, 6581, 1, 2633, 8761, 1259, 10477, 16561, 4021, 10663, 117979, 2239, 1, 120049, 1, 121439, 3301, 859, 1, 17749, 1, 1, 1181, 127079, 2719, 128509, 18461, 1, 1697, 1, 10163, 929, 1, 2741, 135043, 1, 1, 4733, 137999, 12613, 19927, 140237, 1831, 10903, 1, 143249, 1, 20681, 1, 20899, 1, 147827, 8741, 1, 1, 150919, 1667, 1, 153259, 154043, 1447, 1, 1, 1321, 1, 1637, 5503, 160387, 23027, 1, 1789, 1, 14947, 165229, 166043, 1, 167677, 24071, 169319, 5867, 1, 1153, 1, 1213, 3557, 175129, 941, 176807, 3011, 1, 1, 25741, 1, 181889, 1, 1, 2027, 991, 26597, 187043, 187909, 1, 953, 1601, 4451, 1, 193147, 194027, 1, 15061, 28097, 197567, 28351, 1223, 11779, 201139, 18367, 1, 4337, 2659, 12097, 15889, 15959, 208379, 1031, 210209, 30161, 1, 212969, 1999, 1, 1, 5039, 4441, 16811, 16883, 971, 1, 2887, 223243, 32027, 1, 226087, 2551, 227993, 32707, 1543, 1, 1049, 21163, 233759, 234727, 3061, 8161, 1997, 5077, 239597, 2129, 241559, 34649, 1, 2687, 2531, 22409, 6689, 1, 1229, 2341, 1, 2837, 4783, 1361, 255517, 2819, 1801, 1, 259577, 1, 261619, 262643, 5381, 264697, 1, 266759, 1, 24439, 20759, 1, 1, 1, 6373, 275087, 276137, 1, 39749, 9631, 1, 1, 1733, 1283, 21893, 1103, 6101, 41117, 26263, 1, 4933, 26557, 41887, 294293, 42197, 10223, 1, 22973, 299743, 3907, 17761, 43291, 1, 305243, 306349, 2347, 2593, 5843, 1531, 23993, 1, 1, 315269, 1, 2131, 1, 3593, 1, 1, 1, 1, 29587, 1, 1483, 1, 330037, 8951, 1, 333493, 47807, 335807, 1597, 1, 7219, 2861, 2389, 3767, 1, 345143, 1553, 20441, 49811, 3931, 1, 352229, 8219, 32237, 355799, 3923, 1, 51341, 21211, 361789, 12517, 1, 52201, 1861, 1, 3449, 10007, 1, 28669, 1, 10139, 7681, 34327, 13063, 380059, 2039, 54647, 383767, 55001, 386249, 387493, 1759, 1, 5081, 392489, 56249, 1, 8431, 397517, 13751, 57149, 401309, 1, 1, 2833, 1, 407669, 1, 1, 58787, 412793, 1, 14323, 11261, 59707, 38113, 60077, 1, 1, 24967, 9901, 61007, 4813, 61381, 430979, 1, 39419, 434927, 1, 39779, 1, 33863, 441547, 1951, 444209, 63649, 1, 5821, 449557, 450899, 41113, 15641, 3823, 35099, 1, 458993, 460349, 461707, 1, 66347, 12589, 6067, 468527, 1, 1, 472643, 5209, 1, 68111, 2557, 479543, 480929, 1, 1607, 2137, 69499, 487889, 8293, 1, 37853, 1, 494899, 70901, 45247, 499129, 1579, 29527, 10273, 504797, 10331, 3407, 46279, 1, 1, 1, 514793, 2543, 1, 519107, 520549, 521993, 1, 47717, 4423, 527789, 3701, 40823, 1, 76231, 18451, 76649, 11447, 1, 49177, 31907, 77699, 1, 78121, 42179, 42293, 32429, 6211, 79181, 1787, 7237, 19267, 560239, 1, 1, 80677, 5011, 1, 43789, 4357, 15467, 52163, 1, 33931, 1, 579869, 581393, 1, 584447, 1, 1, 6473, 1, 6653, 1, 4919, 1, 3037, 85691, 10193, 1, 4057, 606049, 1, 1, 2029, 55663, 5737, 615427, 616997, 1, 36479, 88817, 21493, 56807, 2969, 48311, 1, 1907, 90401, 1, 17189, 2269, 1, 91541, 5309, 1, 645599, 3461, 1721, 50033, 1901, 653659, 1, 2273, 2953, 60013, 661769, 94771, 60457, 1, 1, 1777, 51659, 673207, 1, 676493, 8807, 15809, 1, 62099, 1, 98057, 6089, 2663, 4091, 1, 3491, 1, 99721, 699719, 9109, 41357, 704747, 706427, 708109, 101399, 711479, 1, 4999, 716549, 24767, 65449, 103091, 723337, 103577, 16901, 3181, 730157, 43051, 1, 1, 1, 67157, 2237, 1, 7669, 3673, 15901, 1, 750809, 1, 754279, 3137, 1, 1, 6397, 762989, 1, 766487, 1, 1, 4127, 110501, 775267, 70639, 778793, 1, 8597, 784097, 1, 2803, 789419, 4231, 7411, 113537, 1, 1933, 800117, 1, 1, 61961, 115327, 809087, 10531, 812689, 814493, 74209, 22111, 1, 48337, 1, 22307, 63629, 1, 1, 6997, 834469, 10861, 1, 839959, 841793, 843629, 1, 1, 1, 1, 1, 1, 77867, 1, 16231, 1, 50821, 865817, 867679, 869543, 11317, 30113, 1, 6133, 878893, 20483, 3257, 7433, 23957, 1, 1, 81097, 1, 52697, 1, 69203, 9907, 31153, 905339, 15377, 21143, 18593, 82997, 18671, 1, 1, 9491, 1, 10159, 71261, 1, 1, 932177, 84919, 3049, 2851, 85447, 1, 943799, 4241, 1, 4297, 10457, 953539, 12409, 1, 33083, 1, 1, 3727, 967259, 138461, 971197, 973169, 75011, 6833, 139871, 981077, 1, 33967, 987029, 16763, 2423, 3299, 1, 142427, 1,

6. Sequence of the polynom (only primes)

13, 11, 7, 17, 29, 43, 59, 97, 197, 227, 37, 293, 47, 367, 449, 587, 53, 743, 857, 131, 89, 149, 1109, 107, 1319, 199, 113, 1627, 1709, 163, 1879, 281, 307, 2243, 2339, 2437, 211, 2957, 3067, 487, 3527, 521, 3769, 229, 4019, 4409, 4679, 4817, 4957, 5099, 317, 5839, 461, 223, 947, 6793, 6959, 7127, 7297, 7643, 1117, 727, 643, 8543, 241, 1301, 547, 863, 9689, 9887, 10289, 1499, 823, 839, 11117, 11329, 1069, 12197, 1129, 269, 757, 1871, 13327, 1061, 14029, 1297, 311, 911, 15737, 271, 439, 2357, 1289, 467, 331, 1619, 18077, 2621, 433, 2699, 661, 19447, 19727, 1583, 21157, 1279, 22039, 3191, 22637, 2113, 23549, 23857, 24793, 541, 25747, 26393, 347, 3911, 967, 593, 29399, 607, 30089, 2767, 1811, 31139, 409, 31849, 1123, 4861, 4967, 3229, 35897, 36277, 5237, 2179, 5347, 2909, 2939, 38993, 3617, 5741, 1097, 40993, 41399, 431, 907, 3343, 4027, 6449, 45569, 6571, 2731, 997, 47293, 4339, 983, 3739, 1151, 49939, 50387, 1753, 877, 7457, 4787, 53117, 599, 4229, 56393, 57347, 751, 58309, 1013, 60257, 4673, 673, 577, 523, 1459, 5749, 63743, 1367, 65267, 9397, 66293, 66809, 5179, 9767, 6263, 619, 4177, 2467, 1471, 72617, 1493, 73699, 5711, 75337, 6949, 647, 77549, 2111, 811, 79229, 11399, 1051, 6269, 683, 83219, 11971, 84377, 653, 2003, 5101, 7937, 88493, 6899, 937, 91493, 92707, 13331, 8539, 94543, 2213, 8707, 98269, 98897, 99527, 2707, 101429, 2083, 9337, 103349, 103993, 104639, 15227, 107243, 108557, 733, 1427, 110543, 15887, 6581, 2633, 8761, 1259, 10477, 16561, 4021, 10663, 117979, 2239, 120049, 121439, 3301, 859, 17749, 1181, 127079, 2719, 128509, 18461, 1697, 10163, 929, 2741, 135043, 4733, 137999, 12613, 19927, 140237, 1831, 10903, 143249, 20681, 20899, 147827, 8741, 150919, 1667, 153259, 154043, 1447, 1321, 1637, 5503, 160387, 23027, 1789, 14947, 165229, 166043, 167677, 24071, 169319, 5867, 1153, 1213, 3557, 175129, 941, 176807, 3011, 25741, 181889, 2027, 991, 26597, 187043, 187909, 953, 1601, 4451, 193147, 194027, 15061, 28097, 197567, 28351, 1223, 11779, 201139, 18367, 4337, 2659, 12097, 15889, 15959, 208379, 1031, 210209, 30161, 212969, 1999, 5039, 4441, 16811, 16883, 971, 2887, 223243, 32027, 226087, 2551, 227993, 32707, 1543, 1049, 21163, 233759, 234727, 3061, 8161, 1997, 5077, 239597, 2129, 241559, 34649, 2687, 2531, 22409, 6689, 1229, 2341, 2837, 4783, 1361, 255517, 2819, 1801, 259577, 261619, 262643, 5381, 264697, 266759, 24439, 20759, 6373, 275087, 276137, 39749, 9631, 1733, 1283, 21893, 1103, 6101, 41117, 26263, 4933, 26557, 41887, 294293, 42197, 10223, 22973, 299743, 3907, 17761, 43291, 305243, 306349, 2347, 2593, 5843, 1531, 23993, 315269, 2131, 3593, 29587, 1483, 330037, 8951, 333493, 47807, 335807, 1597, 7219, 2861, 2389, 3767, 345143, 1553, 20441, 49811, 3931, 352229, 8219, 32237, 355799, 3923, 51341, 21211, 361789, 12517, 52201, 1861, 3449, 10007, 28669, 10139, 7681, 34327, 13063, 380059, 2039, 54647, 383767, 55001, 386249, 387493, 1759, 5081, 392489, 56249, 8431, 397517, 13751, 57149, 401309, 2833, 407669, 58787, 412793, 14323, 11261, 59707, 38113, 60077, 24967, 9901, 61007, 4813, 61381, 430979, 39419, 434927, 39779, 33863, 441547, 1951, 444209, 63649, 5821, 449557, 450899, 41113, 15641, 3823, 35099, 458993, 460349, 461707, 66347, 12589, 6067, 468527, 472643, 5209, 68111, 2557, 479543, 480929, 1607, 2137, 69499, 487889, 8293, 37853, 494899, 70901, 45247, 499129, 1579, 29527, 10273, 504797, 10331, 3407, 46279, 514793, 2543, 519107, 520549, 521993, 47717, 4423, 527789, 3701, 40823, 76231, 18451, 76649, 11447, 49177, 31907, 77699, 78121, 42179, 42293, 32429, 6211, 79181, 1787, 7237, 19267, 560239, 80677, 5011, 43789, 4357, 15467, 52163, 33931, 579869, 581393, 584447, 6473, 6653, 4919, 3037, 85691, 10193, 4057, 606049, 2029, 55663, 5737, 615427, 616997, 36479, 88817, 21493, 56807, 2969, 48311, 1907, 90401, 17189, 2269, 91541, 5309, 645599, 3461, 1721, 50033, 1901, 653659, 2273, 2953, 60013, 661769, 94771, 60457, 1777, 51659, 673207, 676493, 8807, 15809, 62099, 98057, 6089, 2663, 4091, 3491, 99721, 699719, 9109, 41357, 704747, 706427, 708109, 101399, 711479, 4999, 716549, 24767, 65449, 103091, 723337, 103577, 16901, 3181, 730157, 43051, 67157, 2237, 7669, 3673, 15901, 750809, 754279, 3137, 6397, 762989, 766487, 4127, 110501, 775267, 70639, 778793, 8597, 784097, 2803, 789419, 4231, 7411, 113537, 1933, 800117, 61961, 115327, 809087, 10531, 812689, 814493, 74209, 22111, 48337, 22307, 63629, 6997, 834469, 10861, 839959, 841793, 843629, 77867, 16231, 50821, 865817, 867679, 869543, 11317, 30113, 6133, 878893, 20483, 3257, 7433, 23957, 81097, 52697, 69203, 9907, 31153, 905339, 15377, 21143, 18593, 82997, 18671, 9491, 10159, 71261, 932177, 84919, 3049, 2851, 85447, 943799, 4241, 4297, 10457, 953539, 12409, 33083, 3727, 967259, 138461, 971197, 973169, 75011, 6833, 139871, 981077, 33967, 987029, 16763, 2423, 3299, 142427,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+1x-13 and
the reducible primes which appear as divisor for the first time
p | x^2+1x-13 and p < x^2+1x-13

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)
1108800.8000000.8000000.0000000.0000000.0000000.000000
21007044260.7000000.4400000.2600008.7500005.500000inf
31.0007282424860.7280000.2420000.48600010.4000005.50000018.692308
410.0007.2161.7075.5090.7216000.1707000.5509009.9120877.05371911.335391
5100.00071.51513.27758.2380.7151500.1327700.5823809.9106157.77797310.571428
61.000.000711.376108.783602.5930.7113760.1087830.6025939.9472278.19334210.347075
710.000.0007.084.291919.1146.165.1770.7084290.0919110.6165189.9585748.44905910.231080
8100.000.00070.621.4757.970.13562.651.3400.7062150.0797010.6265139.9687428.67154110.162131
91.000.000.000704.598.49870.335.984634.262.5140.7045990.0703360.6342639.9771158.82494310.123687
1010.000.000.0007.033.411.961629.416.8426.403.995.1190.7033410.0629420.6404009.9821568.94871710.096758


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
387700.8750000.8750000.0000002.3333332.333333-nan
416111010.6875000.6250000.0625001.5714291.428571inf
532221750.6875000.5312500.1562502.0000001.7000005.000000
6644530150.7031250.4687500.2343752.0454551.7647063.000000
71289251410.7187500.3984380.3203122.0444451.7000002.733333
8256184821020.7187500.3203120.3984382.0000001.6078432.487805
95123751462290.7324220.2851560.4472662.0380431.7804882.245098
101.0247442494950.7265620.2431640.4833981.9840001.7054802.161572
112.0481.4854491.0360.7250980.2192380.5058591.9959681.8032132.092929
124.0962.9757992.1760.7263180.1950680.5312502.0033671.7795102.100386
138.1925.9141.4314.4830.7219240.1746830.5472411.9878991.7909892.060202
1416.38411.7792.6679.1120.7189330.1627810.5561521.9917151.8637322.032568
1532.76823.5014.95718.5440.7171940.1512760.5659181.9951611.8586432.035119
1665.53646.9149.14737.7670.7158510.1395720.5762791.9962561.8452692.036616
17131.07293.72916.96976.7600.7150960.1294630.5856321.9978901.8551442.032462
18262.144187.13831.750155.3880.7138750.1211170.5927581.9965861.8710592.024336
19524.288373.50959.967313.5420.7124120.1143780.5980341.9959011.8887242.017801
201.048.576745.802113.603632.1990.7112520.1083400.6029121.9967441.8944252.016314
212.097.1521.490.157214.9731.275.1840.7105620.1025070.6080551.9980601.8923182.017061
224.194.3042.975.364409.2272.566.1370.7093820.0975670.6118151.9966781.9036202.012366
238.388.6085.944.225780.1905.164.0350.7086070.0930060.6156011.9978141.9064972.012377
2416.777.21611.875.6931.490.68910.385.0040.7078460.0888520.6189941.9978541.9106742.011025
2533.554.43223.727.9382.855.28520.872.6530.7071480.0850940.6220541.9980261.9154132.009884
2667.108.86447.415.1375.476.47741.938.6600.7065410.0816060.6249351.9982831.9180142.009264
27134.217.72894.755.44010.518.38484.237.0560.7059830.0783680.6276151.9984221.9206482.008578
28268.435.456189.375.32620.236.922169.138.4040.7054780.0753880.6300901.9985691.9239572.007886
29536.870.912378.492.98938.997.885339.495.1040.7049980.0726390.6323591.9986391.9270662.007203
301.073.741.824756.508.98075.248.031681.260.9490.7045540.0700800.6344741.9987401.9295412.006689
312.147.483.6481.512.153.837145.367.2651.366.786.5720.7041520.0676920.6364601.9988581.9318412.006260
324.294.967.2963.022.672.526281.175.3322.741.497.1940.7037710.0654660.6383051.9989191.9342412.005798
338.589.934.5926.042.299.640544.431.7035.497.867.9370.7034160.0633800.6400361.9989921.9362712.005425
3417.179.869.18412.078.876.6051.055.266.49211.023.610.1130.7030830.0614250.6416591.9990531.9382902.005070
3534.359.738.36824.147.071.2482.047.338.80722.099.732.4410.7027720.0595850.6431871.9991161.9401152.004764
3668.719.476.73648.274.055.5093.975.721.29644.298.334.2130.7024800.0578540.6446261.9991681.9418972.004474


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
387341321
41610462431
532175125543
66430102061086
712851173414121213
825682295325182019
9512146539339373634
101.0242498716264577058
112.048449153296121103108117
124.096799277522216198184201
138.1921.431480951375363335358
1416.3842.6679121.755694670621682
1532.7684.9571.6743.2831.2751.2381.1981.246
1665.5369.1473.1076.0402.3072.2832.2562.301
17131.07216.9695.70911.2604.2294.2224.2204.298
18262.14431.75010.61321.1377.9217.9607.8568.013
19524.28859.96719.99839.96915.02915.02714.86415.047
201.048.576113.60337.89875.70528.42328.39728.21828.565
212.097.152214.97371.671143.30253.70853.69753.59453.974
224.194.304409.227136.594272.633102.363102.230101.963102.671
238.388.608780.190260.481519.709194.900194.915195.366195.009
2416.777.2161.490.689497.553993.136373.082372.437372.753372.417
2533.554.4322.855.285952.1801.903.105714.726713.439713.768713.352
2667.108.8645.476.4771.825.5493.650.9281.369.7231.368.7321.369.8101.368.212
27134.217.72810.518.3843.506.3717.012.0132.630.6722.628.8192.629.7732.629.120
28268.435.45620.236.9226.746.35013.490.5725.062.0145.058.3335.058.0315.058.544
29536.870.91238.997.88513.000.13625.997.7499.753.8869.746.3069.746.9419.750.752
301.073.741.82475.248.03125.084.10250.163.92918.816.74518.810.78718.810.88818.809.611
312.147.483.648145.367.26548.457.37396.909.89236.345.30236.343.30736.341.89536.336.761
324.294.967.296281.175.33293.730.835187.444.49770.301.21670.292.04770.297.07170.284.998
338.589.934.592544.431.703181.487.158362.944.545136.123.368136.100.437136.109.193136.098.705
3417.179.869.1841.055.266.492351.755.413703.511.079263.833.063263.801.800263.822.221263.809.408
3534.359.738.3682.047.338.807682.451.9081.364.886.899511.855.883511.808.383511.838.209511.836.332
3668.719.476.7363.975.721.2961.325.244.2752.650.477.021993.956.936993.908.368993.934.461993.921.531


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
4161100010
5325141121
66415784533
7128412021791213
8256102525020282628
951222911311648655858
101.024495237258113128134120
112.0481.036510526265270258243
124.0962.1761.0671.109536562533545
138.1924.4832.1862.2971.1201.0941.1301.139
1416.3849.1124.4684.6442.2822.2332.3022.295
1532.76818.5449.1919.3534.6274.6104.6644.643
1665.53637.76718.71019.0579.3999.4319.4979.440
17131.07276.76038.28138.47919.12119.26419.21819.157
18262.144155.38877.62377.76538.82238.83238.76038.974
19524.288313.542156.260157.28278.56578.39278.22678.359
201.048.576632.199315.327316.872158.518157.651157.774158.256
212.097.1521.275.184635.908639.276318.742318.274318.908319.260
224.194.3042.566.1371.279.7501.286.387641.293641.708641.545641.591
238.388.6085.164.0352.574.9712.589.0641.290.3471.291.7671.291.7721.290.149
2416.777.21610.385.0045.179.3915.205.6132.596.2002.596.7212.596.0482.596.035
2533.554.43220.872.65310.412.36110.460.2925.217.5125.219.5525.219.1705.216.419
2667.108.86441.938.66020.927.77721.010.88310.485.29410.485.21410.486.25410.481.898
27134.217.72884.237.05642.031.34642.205.71021.060.04221.057.59721.063.40921.056.008
28268.435.456169.138.40484.396.51984.741.88542.281.07542.283.53642.289.96742.283.826
29536.870.912339.495.104169.422.759170.072.34584.880.24384.865.18884.881.23684.868.437
301.073.741.824681.260.949339.998.240341.262.709170.331.457170.298.098170.319.777170.311.617
312.147.483.6481.366.786.572682.162.528684.624.044341.700.518341.695.808341.702.299341.687.947
324.294.967.2962.741.497.1941.368.351.9681.373.145.226685.379.279685.356.860685.386.141685.374.914
338.589.934.5925.497.867.9372.744.255.9422.753.611.9951.374.441.8921.374.440.8771.374.494.2921.374.490.876
3417.179.869.18411.023.610.1135.502.698.9835.520.911.1302.755.888.9092.755.919.1052.755.870.1922.755.931.907
3534.359.738.36822.099.732.44111.032.101.72211.067.630.7195.524.936.4635.524.944.0505.524.837.7925.525.014.136
3668.719.476.73644.298.334.21322.114.466.03022.183.868.18311.074.560.69811.074.679.73111.074.450.23311.074.643.551


8. Check for existing Integer Sequences by OEIS

Found in Database : 13, 11, 7, 1, 1, 17, 29, 43, 59, 1, 97, 1, 1, 1, 197, 227, 37, 293, 47, 367,
Found in Database : 13, 11, 7, 17, 29, 43, 59, 97, 197, 227, 37, 293, 47, 367, 449, 587, 53, 743, 857, 131, 89, 149, 1109, 107, 1319, 199, 113,
Found in Database : 7, 11, 13, 17, 29, 37, 43, 47, 53, 59, 89, 97, 107, 113, 131, 149,