Inhaltsverzeichnis

Development of
Algorithmic Constructions

20:52:20
Deutsch
28.Mar 2024

Polynom = x^2-52x+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) = 5 = 5
f(2) = 89 = 89
f(3) = 17 = 17
f(4) = 181 = 181
f(5) = 7 = 7
f(6) = 265 = 5*53
f(7) = 19 = 19
f(8) = 341 = 11*31
f(9) = 47 = 47
f(10) = 409 = 409
f(11) = 55 = 5*11
f(12) = 469 = 7*67
f(13) = 31 = 31
f(14) = 521 = 521
f(15) = 17 = 17
f(16) = 565 = 5*113
f(17) = 73 = 73
f(18) = 601 = 601
f(19) = 77 = 7*11
f(20) = 629 = 17*37
f(21) = 5 = 5
f(22) = 649 = 11*59
f(23) = 41 = 41
f(24) = 661 = 661
f(25) = 83 = 83
f(26) = 665 = 5*7*19
f(27) = 83 = 83
f(28) = 661 = 661
f(29) = 41 = 41
f(30) = 649 = 11*59
f(31) = 5 = 5
f(32) = 629 = 17*37
f(33) = 77 = 7*11
f(34) = 601 = 601
f(35) = 73 = 73
f(36) = 565 = 5*113
f(37) = 17 = 17
f(38) = 521 = 521
f(39) = 31 = 31
f(40) = 469 = 7*67
f(41) = 55 = 5*11
f(42) = 409 = 409
f(43) = 47 = 47
f(44) = 341 = 11*31
f(45) = 19 = 19
f(46) = 265 = 5*53
f(47) = 7 = 7
f(48) = 181 = 181
f(49) = 17 = 17
f(50) = 89 = 89
f(51) = 5 = 5
f(52) = 11 = 11
f(53) = 1 = 1
f(54) = 119 = 7*17
f(55) = 11 = 11
f(56) = 235 = 5*47
f(57) = 37 = 37
f(58) = 359 = 359
f(59) = 53 = 53
f(60) = 491 = 491
f(61) = 35 = 5*7
f(62) = 631 = 631
f(63) = 11 = 11
f(64) = 779 = 19*41
f(65) = 107 = 107
f(66) = 935 = 5*11*17
f(67) = 127 = 127
f(68) = 1099 = 7*157
f(69) = 37 = 37
f(70) = 1271 = 31*41
f(71) = 85 = 5*17
f(72) = 1451 = 1451
f(73) = 193 = 193
f(74) = 1639 = 11*149
f(75) = 217 = 7*31
f(76) = 1835 = 5*367
f(77) = 121 = 11*11
f(78) = 2039 = 2039
f(79) = 67 = 67
f(80) = 2251 = 2251
f(81) = 295 = 5*59
f(82) = 2471 = 7*353
f(83) = 323 = 17*19
f(84) = 2699 = 2699
f(85) = 11 = 11
f(86) = 2935 = 5*587
f(87) = 191 = 191
f(88) = 3179 = 11*17*17
f(89) = 413 = 7*59
f(90) = 3431 = 47*73
f(91) = 445 = 5*89
f(92) = 3691 = 3691
f(93) = 239 = 239
f(94) = 3959 = 37*107
f(95) = 1 = 1
f(96) = 4235 = 5*7*11*11
f(97) = 547 = 547
f(98) = 4519 = 4519
f(99) = 583 = 11*53
f(100) = 4811 = 17*283

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^2-52x+11

f(0)=11
f(1)=5
f(2)=89
f(3)=17
f(4)=181
f(5)=7
f(6)=53
f(7)=19
f(8)=31
f(9)=47
f(10)=409
f(11)=1
f(12)=67
f(13)=1
f(14)=521
f(15)=1
f(16)=113
f(17)=73
f(18)=601
f(19)=1
f(20)=37
f(21)=1
f(22)=59
f(23)=41
f(24)=661
f(25)=83
f(26)=1
f(27)=1
f(28)=1
f(29)=1
f(30)=1
f(31)=1
f(32)=1
f(33)=1
f(34)=1
f(35)=1
f(36)=1
f(37)=1
f(38)=1
f(39)=1
f(40)=1
f(41)=1
f(42)=1
f(43)=1
f(44)=1
f(45)=1
f(46)=1
f(47)=1
f(48)=1
f(49)=1
f(50)=1
f(51)=1
f(52)=1
f(53)=1
f(54)=1
f(55)=1
f(56)=1
f(57)=1
f(58)=359
f(59)=1
f(60)=491
f(61)=1
f(62)=631
f(63)=1
f(64)=1
f(65)=107
f(66)=1
f(67)=127
f(68)=157
f(69)=1
f(70)=1
f(71)=1
f(72)=1451
f(73)=193
f(74)=149
f(75)=1
f(76)=367
f(77)=1
f(78)=2039
f(79)=1
f(80)=2251
f(81)=1
f(82)=353
f(83)=1
f(84)=2699
f(85)=1
f(86)=587
f(87)=191
f(88)=1
f(89)=1
f(90)=1
f(91)=1
f(92)=3691
f(93)=239
f(94)=1
f(95)=1
f(96)=1
f(97)=547
f(98)=4519
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-52x+11 could be written as f(y)= y^2-665 with x=y+26

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-26
f'(x)>2x-53 with x > 26

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, 5, 89, 17, 181, 7, 53, 19, 31, 47, 409, 1, 67, 1, 521, 1, 113, 73, 601, 1, 37, 1, 59, 41, 661, 83, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 359, 1, 491, 1, 631, 1, 1, 107, 1, 127, 157, 1, 1, 1, 1451, 193, 149, 1, 367, 1, 2039, 1, 2251, 1, 353, 1, 2699, 1, 587, 191, 1, 1, 1, 1, 3691, 239, 1, 1, 1, 547, 4519, 1, 283, 1, 269, 1, 5419, 1, 1, 1, 1, 389, 1, 1, 1, 863, 7079, 907, 1487, 1, 709, 499, 8171, 1, 503, 1093, 1277, 571, 1867, 1, 9739, 1, 10151, 1, 1, 337, 647, 701, 2287, 1, 1697, 1, 1, 1, 12791, 1, 13259, 241, 1, 1747, 1, 1, 313, 1, 1, 1933, 1429, 1997, 1, 1031, 16759, 1, 17291, 439, 1621, 1, 18379, 1, 541, 1201, 1, 2473, 20071, 509, 1, 1, 317, 673, 397, 2767, 1181, 2843, 1, 1, 23671, 1499, 1, 1, 4987, 1, 25579, 1619, 1543, 1, 26891, 1, 1, 1, 5647, 1, 1, 1, 29611, 1, 1783, 3833, 31019, 1, 577, 1, 4637, 373, 33191, 839, 33931, 1, 34679, 1, 1, 1, 683, 1, 3361, 467, 5393, 1, 2267, 1, 7867, 4967, 1, 1, 1321, 1, 41771, 5273, 1039, 1, 1, 2741, 44279, 1, 45131, 1, 1, 829, 997, 739, 9547, 3011, 48619, 6133, 643, 1249, 50411, 1, 1, 809, 1, 941, 1, 6703, 54091, 1, 487, 3469, 727, 7057, 1, 7177, 57899, 1, 3463, 1, 5441, 1, 733, 1, 1, 1, 1, 1, 953, 1609, 64871, 743, 65899, 593, 1217, 1, 67979, 8563, 69031, 1, 1, 2207, 6469, 4481, 14447, 827, 1, 1319, 74411, 937, 75511, 2377, 4507, 877, 2221, 9787, 1, 1, 1951, 1, 81131, 1459, 82279, 10357, 1, 1, 2729, 1, 12253, 1, 86951, 1, 1663, 1, 1051, 1, 2447, 11393, 1, 2309, 1, 5849, 13457, 2963, 19087, 12007, 1, 12163, 97931, 1, 99191, 1, 100459, 12637, 20347, 1, 14717, 1, 104311, 1, 9601, 1, 3449, 1, 21647, 1, 1231, 1, 1, 2789, 16033, 1283, 2143, 1, 1, 3613, 6121, 2089, 117671, 1, 1, 1871, 10949, 1, 1, 1, 123239, 15493, 4021, 1567, 1, 1, 2161, 1, 1, 853, 130379, 1, 1, 1657, 3251, 1523, 7927, 16937, 2477, 1223, 3359, 4327, 1907, 3499, 3803, 17683, 1847, 1117, 1, 821, 145259, 18253, 811, 1, 148331, 9319, 2237, 1, 977, 1, 1987, 1, 154571, 971, 156151, 1, 157739, 1, 2897, 1, 9467, 919, 3067, 1021, 1, 1, 165799, 1, 33487, 1, 169079, 1, 1, 4289, 1, 1, 9161, 1, 5021, 1, 1, 22283, 179111, 1, 180811, 1, 182519, 1, 36847, 1361, 185959, 1, 26813, 2357, 1013, 1, 5167, 24007, 1, 3461, 1723, 1, 1, 2467, 198251, 1, 1, 25117, 1091, 12671, 6569, 1, 5011, 1, 10909, 1, 19009, 6563, 42187, 13241, 1, 26713, 214631, 1, 19681, 1, 3701, 1, 2591, 27647, 1, 27883, 224011, 1, 1, 1289, 7349, 28597, 4177, 28837, 13627, 1, 233591, 1, 1, 1, 21589, 1, 6841, 1, 1, 15149, 12809, 1, 2293, 1, 247339, 1, 1061, 7823, 1, 31543, 2129, 6359, 255371, 2003, 257399, 1, 1, 4651, 1, 1, 8501, 3307, 1, 2083, 38237, 33587, 1, 1, 4057, 8527, 1, 1, 276011, 1, 278119, 34897, 1367, 17581, 1, 1, 6053, 1, 6991, 35963, 16987, 1, 1877, 18251, 293099, 3343, 295271, 1, 3863, 1, 2011, 1, 1, 1993, 304039, 5449, 2531, 1, 1, 1759, 3491, 38977, 8941, 1, 315179, 1, 317431, 1, 319691, 1, 29269, 40387, 3413, 1, 19207, 20479, 1, 1, 1, 1, 333419, 1901, 1, 1, 4073, 42403, 20023, 8539, 1, 1, 49297, 1, 6317, 43577, 1, 1, 1, 1, 354551, 11117, 1, 1, 71867, 1, 1667, 1, 1511, 4567, 21563, 1483, 19421, 1, 74287, 23291, 1, 5861, 1399, 9439, 1, 1, 1, 11953, 6977, 24061, 386219, 1, 388711, 9749, 2027, 1291, 8377, 12343, 11321, 4517, 398759, 1613, 1, 1, 403831, 3617, 13109, 50957, 1, 1, 37409, 25799, 1, 1, 21929, 1, 5743, 52567, 1, 1889, 24967, 1, 1, 10709, 1, 3169, 61757, 1, 2351, 13633, 1, 1, 1, 1, 6067, 1, 3943, 1, 1, 56197, 1741, 56533, 26683, 1, 3593, 7151, 41729, 8221, 92347, 1, 464459, 14557, 1, 5857, 1, 58913, 1, 5387, 95087, 1753, 478199, 2141, 11731, 1, 483751, 1, 486539, 1, 1, 30671, 2719, 1, 8389, 12409, 497771, 4457, 2677, 1, 1, 5737, 8581, 63463, 72733, 3191, 511991, 32089, 514859, 5867, 6091, 1, 1, 1, 523511, 1, 16981, 65983, 75617, 66347, 9677, 1, 535159, 3049, 14543, 1, 6079, 3989, 1, 1, 109387, 1, 4621, 1, 50261, 13859, 1, 17417, 6733, 5003, 1, 1, 51349, 70793, 33403, 1, 81553, 1, 573899, 4231, 6073, 72307, 1, 1, 583031, 7307, 53281, 73453, 589159, 1571, 16921, 1, 595319, 4663, 54401, 1, 1, 1, 604619, 1, 121547, 1, 1, 2069, 1, 1, 617131, 38669, 1, 19433, 1861, 11161, 626599, 1, 8627, 1973, 5231, 39659, 4783, 7247, 127867, 1, 1, 1, 645751, 1, 648971, 7393, 17627, 81727, 1, 20533, 1, 41269, 661931, 1, 14153, 1, 1, 1, 3631, 1913, 1, 2063, 21881, 1, 97373, 1, 12923, 1, 2333, 1, 62869, 12379, 1699, 8707, 36749, 1, 701579, 5171, 1831, 88327, 2939, 2017, 41863, 1, 4799, 12799, 5657, 90017, 4657, 4111, 1, 22717, 9463, 1, 43063, 1, 17939, 11519, 147787, 1, 67489, 1979, 18191, 1, 8419, 1, 2029, 11789, 151247, 94747, 759719, 1, 40169, 1, 1, 48029, 45307, 96497, 154747, 1, 3001, 1, 70981, 1, 25301, 8933, 787879, 1, 9311, 6197, 794999, 2621, 1, 1, 114593, 100493, 6659, 1, 1, 6337, 812939, 14549, 2423, 1, 74561, 1, 823799, 4691, 1, 103657, 1, 104113, 1, 10457, 838391, 1, 4363, 2851, 15377, 1999, 849419, 1, 1, 10687, 856811, 1, 78229, 1, 1, 1, 867959, 13591, 16447, 21839, 2347, 109663, 125597, 2503, 1, 3253, 1, 3583, 2243, 3187, 1, 56009, 1, 28123, 1, 112967, 1931, 10313, 909451, 1, 24683, 1, 1, 16411, 184187, 10487, 924779, 3407, 84421, 1, 133213, 116803, 25307, 6173, 4001, 29443, 85829, 8447,

6. Sequence of the polynom (only primes)

11, 5, 89, 17, 181, 7, 53, 19, 31, 47, 409, 67, 521, 113, 73, 601, 37, 59, 41, 661, 83, 359, 491, 631, 107, 127, 157, 1451, 193, 149, 367, 2039, 2251, 353, 2699, 587, 191, 3691, 239, 547, 4519, 283, 269, 5419, 389, 863, 7079, 907, 1487, 709, 499, 8171, 503, 1093, 1277, 571, 1867, 9739, 10151, 337, 647, 701, 2287, 1697, 12791, 13259, 241, 1747, 313, 1933, 1429, 1997, 1031, 16759, 17291, 439, 1621, 18379, 541, 1201, 2473, 20071, 509, 317, 673, 397, 2767, 1181, 2843, 23671, 1499, 4987, 25579, 1619, 1543, 26891, 5647, 29611, 1783, 3833, 31019, 577, 4637, 373, 33191, 839, 33931, 34679, 683, 3361, 467, 5393, 2267, 7867, 4967, 1321, 41771, 5273, 1039, 2741, 44279, 45131, 829, 997, 739, 9547, 3011, 48619, 6133, 643, 1249, 50411, 809, 941, 6703, 54091, 487, 3469, 727, 7057, 7177, 57899, 3463, 5441, 733, 953, 1609, 64871, 743, 65899, 593, 1217, 67979, 8563, 69031, 2207, 6469, 4481, 14447, 827, 1319, 74411, 937, 75511, 2377, 4507, 877, 2221, 9787, 1951, 81131, 1459, 82279, 10357, 2729, 12253, 86951, 1663, 1051, 2447, 11393, 2309, 5849, 13457, 2963, 19087, 12007, 12163, 97931, 99191, 100459, 12637, 20347, 14717, 104311, 9601, 3449, 21647, 1231, 2789, 16033, 1283, 2143, 3613, 6121, 2089, 117671, 1871, 10949, 123239, 15493, 4021, 1567, 2161, 853, 130379, 1657, 3251, 1523, 7927, 16937, 2477, 1223, 3359, 4327, 1907, 3499, 3803, 17683, 1847, 1117, 821, 145259, 18253, 811, 148331, 9319, 2237, 977, 1987, 154571, 971, 156151, 157739, 2897, 9467, 919, 3067, 1021, 165799, 33487, 169079, 4289, 9161, 5021, 22283, 179111, 180811, 182519, 36847, 1361, 185959, 26813, 2357, 1013, 5167, 24007, 3461, 1723, 2467, 198251, 25117, 1091, 12671, 6569, 5011, 10909, 19009, 6563, 42187, 13241, 26713, 214631, 19681, 3701, 2591, 27647, 27883, 224011, 1289, 7349, 28597, 4177, 28837, 13627, 233591, 21589, 6841, 15149, 12809, 2293, 247339, 1061, 7823, 31543, 2129, 6359, 255371, 2003, 257399, 4651, 8501, 3307, 2083, 38237, 33587, 4057, 8527, 276011, 278119, 34897, 1367, 17581, 6053, 6991, 35963, 16987, 1877, 18251, 293099, 3343, 295271, 3863, 2011, 1993, 304039, 5449, 2531, 1759, 3491, 38977, 8941, 315179, 317431, 319691, 29269, 40387, 3413, 19207, 20479, 333419, 1901, 4073, 42403, 20023, 8539, 49297, 6317, 43577, 354551, 11117, 71867, 1667, 1511, 4567, 21563, 1483, 19421, 74287, 23291, 5861, 1399, 9439, 11953, 6977, 24061, 386219, 388711, 9749, 2027, 1291, 8377, 12343, 11321, 4517, 398759, 1613, 403831, 3617, 13109, 50957, 37409, 25799, 21929, 5743, 52567, 1889, 24967, 10709, 3169, 61757, 2351, 13633, 6067, 3943, 56197, 1741, 56533, 26683, 3593, 7151, 41729, 8221, 92347, 464459, 14557, 5857, 58913, 5387, 95087, 1753, 478199, 2141, 11731, 483751, 486539, 30671, 2719, 8389, 12409, 497771, 4457, 2677, 5737, 8581, 63463, 72733, 3191, 511991, 32089, 514859, 5867, 6091, 523511, 16981, 65983, 75617, 66347, 9677, 535159, 3049, 14543, 6079, 3989, 109387, 4621, 50261, 13859, 17417, 6733, 5003, 51349, 70793, 33403, 81553, 573899, 4231, 6073, 72307, 583031, 7307, 53281, 73453, 589159, 1571, 16921, 595319, 4663, 54401, 604619, 121547, 2069, 617131, 38669, 19433, 1861, 11161, 626599, 8627, 1973, 5231, 39659, 4783, 7247, 127867, 645751, 648971, 7393, 17627, 81727, 20533, 41269, 661931, 14153, 3631, 1913, 2063, 21881, 97373, 12923, 2333, 62869, 12379, 1699, 8707, 36749, 701579, 5171, 1831, 88327, 2939, 2017, 41863, 4799, 12799, 5657, 90017, 4657, 4111, 22717, 9463, 43063, 17939, 11519, 147787, 67489, 1979, 18191, 8419, 2029, 11789, 151247, 94747, 759719, 40169, 48029, 45307, 96497, 154747, 3001, 70981, 25301, 8933, 787879, 9311, 6197, 794999, 2621, 114593, 100493, 6659, 6337, 812939, 14549, 2423, 74561, 823799, 4691, 103657, 104113, 10457, 838391, 4363, 2851, 15377, 1999, 849419, 10687, 856811, 78229, 867959, 13591, 16447, 21839, 2347, 109663, 125597, 2503, 3253, 3583, 2243, 3187, 56009, 28123, 112967, 1931, 10313, 909451, 24683, 16411, 184187, 10487, 924779, 3407, 84421, 133213, 116803, 25307, 6173, 4001, 29443, 85829, 8447,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-52x+11 and
the reducible primes which appear as divisor for the first time
p | x^2-52x+11 and p < x^2-52x+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)
11010461.0000000.4000001.0000000.0000000.0000000.000000
21006335280.6300000.3500000.6300006.3000008.7500004.666667
31.0009283925360.9280000.3920000.92800014.73015911.20000019.142857
410.0009.8883.9655.9230.9888000.3965000.98880010.65517210.11479611.050373
5100.00099.84139.69560.1460.9984100.3969500.99841010.09718910.01135010.154652
61.000.000999.791396.987602.8040.9997910.3969870.99979110.01383210.00093210.022346
710.000.0009.999.7423.969.9206.029.8220.9999740.3969920.99997410.00183210.00012610.002956
8100.000.00099.999.69639.699.24360.300.4530.9999970.3969920.99999710.00022810.00001110.000370
91.000.000.000999.999.647396.992.476603.007.1711.0000000.3969921.00000010.00002710.00000110.000044
1010.000.000.0009.999.999.5983.969.924.8066.030.074.7921.0000000.3969921.00000010.00000410.00000010.000006


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)
123211.5000001.0000000.5000000.0000000.0000000.000000
245321.2500000.7500000.5000001.6666671.5000002.000000
388351.0000000.3750000.6250001.6000001.0000002.500000
41613580.8125000.3125000.5000001.6250001.6666671.600000
5322310130.7187500.3125000.4062501.7692312.0000001.625000
6643721160.5781250.3281250.2500001.6086962.1000001.230769
71288646400.6718750.3593750.3125002.3243242.1904762.500000
8256205971080.8007810.3789060.4218752.3837212.1086962.700000
95124491992500.8769530.3886720.4882812.1902442.0515462.314815
101.0249524025500.9296880.3925780.5371092.1202672.0201012.200000
112.0481.9648081.1560.9589840.3945310.5644532.0630252.0099502.101818
124.0964.0011.6212.3800.9768070.3957520.5810552.0371692.0061882.058824
138.1928.0843.2484.8360.9868160.3964840.5903322.0204952.0037012.031933
1416.38416.2626.5009.7620.9925540.3967290.5958252.0116282.0012312.018610
1532.76832.63213.00319.6290.9958500.3968200.5990302.0066412.0004622.010756
1665.53665.38526.01239.3730.9976960.3969120.6007842.0037082.0004612.005859
17131.072130.90752.03078.8770.9987410.3969570.6017842.0020952.0002312.003327
18262.144261.964104.065157.8990.9993130.3969760.6023372.0011462.0000962.001838
19524.288524.092208.134315.9580.9996260.3969840.6026422.0006262.0000382.001013
201.048.5761.048.365416.272632.0930.9997990.3969880.6028112.0003452.0000192.000560
212.097.1522.096.928832.5491.264.3790.9998930.3969900.6029032.0001892.0000122.000305
224.194.3044.194.0661.665.1032.528.9630.9999430.3969910.6029522.0001002.0000062.000162
238.388.6088.388.3543.330.2095.058.1450.9999700.3969920.6029782.0000532.0000022.000087
2416.777.21616.776.9516.660.42310.116.5280.9999840.3969920.6029922.0000292.0000012.000047
2533.554.43233.554.15213.320.85220.233.3000.9999920.3969920.6029992.0000152.0000012.000024
2667.108.86467.108.57026.641.71040.466.8600.9999960.3969920.6030032.0000082.0000002.000013
27134.217.728134.217.41953.283.42480.933.9950.9999980.3969920.6030052.0000042.0000002.000007
28268.435.456268.435.131106.566.853161.868.2780.9999990.3969920.6030062.0000022.0000002.000004
29536.870.912536.870.571213.133.711323.736.8600.9999990.3969920.6030072.0000012.0000002.000002
301.073.741.8241.073.741.468426.267.426647.474.0421.0000000.3969920.6030072.0000012.0000002.000001
312.147.483.6482.147.483.277852.534.8571.294.948.4201.0000000.3969920.6030072.0000002.0000002.000000
324.294.967.2964.294.966.9111.705.069.7182.589.897.1931.0000000.3969920.6030072.0000002.0000002.000000
338.589.934.5928.589.934.1913.410.139.4415.179.794.7501.0000000.3969920.6030072.0000002.0000002.000000
3417.179.869.18417.179.868.7616.820.278.88410.359.589.8771.0000000.3969920.6030072.0000002.0000002.000000


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
122021100
243121110
383121110
4165233110
53210645140
6642111108373
7128462026816715
8256973760841741
951219971128892792
101.02440213826481937194
112.04880827453483977396
124.0961.6215451.07688037803
138.1923.2481.0872.16181.61771.616
1416.3846.5002.1714.32983.24273.243
1532.76813.0034.3398.66486.49476.494
1665.53626.0128.67517.337812.999712.998
17131.07252.03017.34734.683826.008726.007
18262.144104.06534.69269.373852.025752.025
19524.288208.13469.383138.7518104.0607104.059
201.048.576416.272138.762277.5108208.1287208.129
212.097.152832.549277.521555.0288416.2677416.267
224.194.3041.665.103555.0391.110.0648832.5447832.544
238.388.6083.330.2091.110.0742.220.13581.665.09871.665.096
2416.777.2166.660.4232.220.1454.440.27883.330.20473.330.204
2533.554.43213.320.8524.440.2888.880.56486.660.41976.660.418
2667.108.86426.641.7108.880.57417.761.136813.320.848713.320.847
27134.217.72853.283.42417.761.14635.522.278826.641.705726.641.704
28268.435.456106.566.85335.522.28971.044.564853.283.419753.283.419
29536.870.912213.133.71171.044.575142.089.1368106.566.8487106.566.848
301.073.741.824426.267.426142.089.146284.178.2808213.133.7057213.133.706
312.147.483.648852.534.857284.178.290568.356.5678426.267.4227426.267.420
324.294.967.2961.705.069.718568.356.5771.136.713.1418852.534.8517852.534.852
338.589.934.5923.410.139.4411.136.713.1512.273.426.29081.705.069.71371.705.069.713
3417.179.869.1846.820.278.8842.273.426.2994.546.852.58583.410.139.43573.410.139.434


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
121010010
242021010
385231121
4168352222
53213583532
664166105542
7128402020101479
8256108585022352328
951225014011050775469
101.024550306244111160117162
112.0481.156641515243342237334
124.0962.3801.3171.063495704491690
138.1924.8362.6842.1529971.4211.0041.414
1416.3849.7625.4114.3512.0172.8542.0272.864
1532.76819.62910.8858.7444.0715.7534.0615.744
1665.53639.37321.82917.5448.15911.5378.15811.519
17131.07278.87743.74235.13516.34923.09616.34823.084
18262.144157.89987.55470.34532.72346.22132.73346.222
19524.288315.958175.191140.76765.49092.48565.49492.489
201.048.576632.093350.467281.626131.023185.030131.024185.016
212.097.1521.264.379701.044563.335262.092370.096262.094370.097
224.194.3042.528.9631.402.1851.126.778524.232740.253524.237740.241
238.388.6085.058.1452.804.4822.253.6631.048.5241.480.5481.048.5121.480.561
2416.777.21610.116.5285.609.0834.507.4452.097.0872.961.1892.097.0942.961.158
2533.554.43220.233.30011.218.3099.014.9914.194.2245.922.4134.194.2565.922.407
2667.108.86440.466.86022.436.74818.030.1128.388.53811.844.9028.388.54311.844.877
27134.217.72880.933.99544.873.64236.060.35316.777.15423.689.85516.777.13723.689.849
28268.435.456161.868.27889.747.42772.120.85133.554.35847.379.78233.554.35847.379.780
29536.870.912323.736.860179.495.017144.241.84367.108.78394.759.64567.108.78994.759.643
301.073.741.824647.474.042358.990.194288.483.848134.217.645189.519.376134.217.647189.519.374
312.147.483.6481.294.948.420717.980.566576.967.854268.435.371379.038.833268.435.369379.038.847
324.294.967.2962.589.897.1931.435.961.3021.153.935.891536.870.827758.077.784536.870.820758.077.762
338.589.934.5925.179.794.7502.871.922.7942.307.871.9561.073.741.7361.516.155.6651.073.741.7291.516.155.620
3417.179.869.18410.359.589.8775.743.845.7784.615.744.0992.147.483.5503.032.311.3992.147.483.5553.032.311.373


8. Check for existing Integer Sequences by OEIS

Found in Database : 11, 5, 89, 17, 181, 7, 53, 19, 31, 47, 409, 1, 67, 1, 521, 1, 113, 73, 601, 1,
Found in Database : 11, 5, 89, 17, 181, 7, 53, 19, 31, 47, 409, 67, 521, 113, 73, 601, 37, 59, 41, 661, 83,
Found in Database : 5, 7, 11, 17, 19, 31, 37, 41, 47, 53, 59, 67, 73, 83, 89, 107, 113, 127, 149,