Inhaltsverzeichnis

Development of
Algorithmic Constructions

05:35:30
Deutsch
29.Mar 2024

Polynom = x^2-164x-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) = 337 = 337
f(3) = 31 = 31
f(4) = 653 = 653
f(5) = 101 = 101
f(6) = 961 = 31*31
f(7) = 139 = 139
f(8) = 1261 = 13*97
f(9) = 11 = 11
f(10) = 1553 = 1553
f(11) = 53 = 53
f(12) = 1837 = 11*167
f(13) = 247 = 13*19
f(14) = 2113 = 2113
f(15) = 281 = 281
f(16) = 2381 = 2381
f(17) = 157 = 157
f(18) = 2641 = 19*139
f(19) = 173 = 173
f(20) = 2893 = 11*263
f(21) = 377 = 13*29
f(22) = 3137 = 3137
f(23) = 407 = 11*37
f(24) = 3373 = 3373
f(25) = 109 = 109
f(26) = 3601 = 13*277
f(27) = 29 = 29
f(28) = 3821 = 3821
f(29) = 491 = 491
f(30) = 4033 = 37*109
f(31) = 517 = 11*47
f(32) = 4237 = 19*223
f(33) = 271 = 271
f(34) = 4433 = 11*13*31
f(35) = 283 = 283
f(36) = 4621 = 4621
f(37) = 589 = 19*31
f(38) = 4801 = 4801
f(39) = 611 = 13*47
f(40) = 4973 = 4973
f(41) = 79 = 79
f(42) = 5137 = 11*467
f(43) = 163 = 163
f(44) = 5293 = 67*79
f(45) = 671 = 11*61
f(46) = 5441 = 5441
f(47) = 689 = 13*53
f(48) = 5581 = 5581
f(49) = 353 = 353
f(50) = 5713 = 29*197
f(51) = 361 = 19*19
f(52) = 5837 = 13*449
f(53) = 737 = 11*67
f(54) = 5953 = 5953
f(55) = 751 = 751
f(56) = 6061 = 11*19*29
f(57) = 191 = 191
f(58) = 6161 = 61*101
f(59) = 97 = 97
f(60) = 6253 = 13*13*37
f(61) = 787 = 787
f(62) = 6337 = 6337
f(63) = 797 = 797
f(64) = 6413 = 11*11*53
f(65) = 403 = 13*31
f(66) = 6481 = 6481
f(67) = 407 = 11*37
f(68) = 6541 = 31*211
f(69) = 821 = 821
f(70) = 6593 = 19*347
f(71) = 827 = 827
f(72) = 6637 = 6637
f(73) = 13 = 13
f(74) = 6673 = 6673
f(75) = 209 = 11*19
f(76) = 6701 = 6701
f(77) = 839 = 839
f(78) = 6721 = 11*13*47
f(79) = 841 = 29*29
f(80) = 6733 = 6733
f(81) = 421 = 421
f(82) = 6737 = 6737
f(83) = 421 = 421
f(84) = 6733 = 6733
f(85) = 841 = 29*29
f(86) = 6721 = 11*13*47
f(87) = 839 = 839
f(88) = 6701 = 6701
f(89) = 209 = 11*19
f(90) = 6673 = 6673
f(91) = 13 = 13
f(92) = 6637 = 6637
f(93) = 827 = 827
f(94) = 6593 = 19*347
f(95) = 821 = 821
f(96) = 6541 = 31*211
f(97) = 407 = 11*37
f(98) = 6481 = 6481
f(99) = 403 = 13*31
f(100) = 6413 = 11*11*53

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^2-164x-13

f(0)=13
f(1)=11
f(2)=337
f(3)=31
f(4)=653
f(5)=101
f(6)=1
f(7)=139
f(8)=97
f(9)=1
f(10)=1553
f(11)=53
f(12)=167
f(13)=19
f(14)=2113
f(15)=281
f(16)=2381
f(17)=157
f(18)=1
f(19)=173
f(20)=263
f(21)=29
f(22)=3137
f(23)=37
f(24)=3373
f(25)=109
f(26)=277
f(27)=1
f(28)=3821
f(29)=491
f(30)=1
f(31)=47
f(32)=223
f(33)=271
f(34)=1
f(35)=283
f(36)=4621
f(37)=1
f(38)=4801
f(39)=1
f(40)=4973
f(41)=79
f(42)=467
f(43)=163
f(44)=67
f(45)=61
f(46)=5441
f(47)=1
f(48)=5581
f(49)=353
f(50)=197
f(51)=1
f(52)=449
f(53)=1
f(54)=5953
f(55)=751
f(56)=1
f(57)=191
f(58)=1
f(59)=1
f(60)=1
f(61)=787
f(62)=6337
f(63)=797
f(64)=1
f(65)=1
f(66)=6481
f(67)=1
f(68)=211
f(69)=821
f(70)=347
f(71)=827
f(72)=6637
f(73)=1
f(74)=6673
f(75)=1
f(76)=6701
f(77)=839
f(78)=1
f(79)=1
f(80)=6733
f(81)=421
f(82)=6737
f(83)=1
f(84)=1
f(85)=1
f(86)=1
f(87)=1
f(88)=1
f(89)=1
f(90)=1
f(91)=1
f(92)=1
f(93)=1
f(94)=1
f(95)=1
f(96)=1
f(97)=1
f(98)=1
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-164x-13 could be written as f(y)= y^2-6737 with x=y+82

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-82
f'(x)>2x-165 with x > 82

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, 337, 31, 653, 101, 1, 139, 97, 1, 1553, 53, 167, 19, 2113, 281, 2381, 157, 1, 173, 263, 29, 3137, 37, 3373, 109, 277, 1, 3821, 491, 1, 47, 223, 271, 1, 283, 4621, 1, 4801, 1, 4973, 79, 467, 163, 67, 61, 5441, 1, 5581, 353, 197, 1, 449, 1, 5953, 751, 1, 191, 1, 1, 1, 787, 6337, 797, 1, 1, 6481, 1, 211, 821, 347, 827, 6637, 1, 6673, 1, 6701, 839, 1, 1, 6733, 421, 6737, 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, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 659, 1, 1, 1, 1, 193, 1, 239, 2099, 1, 1, 1, 1, 383, 251, 433, 1, 1, 4079, 1, 409, 1, 379, 643, 1, 349, 5807, 1, 569, 811, 6719, 1, 7187, 1, 1, 1, 8147, 1049, 1, 1, 1, 587, 877, 619, 10163, 1303, 10687, 1, 863, 359, 1069, 1, 397, 1, 677, 1, 463, 857, 13999, 1, 1, 1, 523, 1933, 1433, 1, 1259, 521, 16979, 2161, 17599, 2239, 1657, 1, 1451, 1, 19507, 1, 1061, 1, 1, 661, 21487, 1, 599, 1, 1, 1, 23539, 1493, 24239, 1, 1, 3163, 2333, 3253, 26387, 1, 577, 859, 2143, 3529, 28607, 3623, 29363, 1, 641, 1907, 1, 3911, 401, 1, 32467, 1, 1, 1, 1, 1, 2683, 1, 1231, 1, 36527, 2309, 37363, 4723, 2939, 439, 1, 617, 1, 1, 40787, 5153, 683, 1, 1, 2687, 1, 1, 1, 509, 45247, 1, 1, 1, 47087, 743, 1549, 1, 1, 1, 1, 1, 2677, 3209, 51827, 503, 1427, 6661, 4889, 1, 54767, 1, 55763, 541, 56767, 7159, 3041, 3643, 4523, 1, 2063, 1, 1, 7673, 1997, 1951, 1, 1, 64019, 8069, 1, 631, 607, 1, 67247, 1, 1847, 1, 69439, 673, 1, 1, 71663, 1, 1, 1, 1, 9311, 1597, 1, 76207, 4799, 1, 9743, 4133, 1, 79699, 1, 2609, 1, 82067, 10333, 1571, 953, 84467, 1, 7789, 5393, 1297, 10939, 6779, 11093, 89363, 1, 8237, 2851, 947, 1051, 1, 11719, 2551, 5939, 1, 1, 3343, 1109, 3169, 1, 9049, 1, 100847, 1, 1, 12853, 1, 1, 733, 1, 2003, 1, 107507, 13523, 108863, 13693, 1, 1733, 111599, 1, 112979, 1093, 1, 757, 115763, 1, 117167, 1, 1, 1, 10909, 1, 3917, 1, 727, 1931, 1, 15629, 1877, 1, 127219, 1, 1, 8089, 11833, 16363, 2801, 1, 4591, 1, 134639, 4231, 12377, 1, 1, 1, 139187, 8747, 1291, 1, 1, 941, 143807, 1, 2383, 4567, 1, 1, 11423, 18661, 1, 18859, 151667, 1, 13933, 9629, 1, 1, 937, 19661, 1, 1, 1, 1, 161363, 1, 12539, 20479, 14969, 10343, 166319, 1, 167987, 1, 1, 21313, 1, 5381, 173039, 1, 887, 1, 1747, 1, 9377, 1, 6203, 1, 181619, 22811, 1, 23029, 1, 1453, 2789, 5867, 3559, 23689, 911, 23911, 14783, 1097, 6257, 1, 1201, 1, 197567, 24809, 199379, 1, 1163, 1579, 18457, 1, 1, 1, 2131, 1, 1, 13093, 1, 1, 11173, 2423, 214163, 3361, 16619, 6781, 4637, 27361, 219839, 1, 221747, 1, 20333, 1, 1, 28319, 227519, 1, 229459, 1, 1, 3631, 1, 2663, 1693, 1, 1, 1, 1, 15017, 1, 2753, 243263, 1607, 2027, 1, 247279, 7759, 13121, 1, 2591, 31543, 1, 1223, 1, 1, 257459, 1, 19963, 32569, 261587, 1, 13877, 1, 265747, 33349, 1873, 1, 269939, 16937, 1669, 1, 274163, 34403, 25117, 1, 1, 1, 1787, 1, 282707, 1867, 7699, 1153, 22079, 1637, 1, 18143, 26489, 36559, 1, 36833, 22751, 9277, 297967, 4673, 1, 2897, 302399, 3449, 16033, 1, 6529, 19249, 309107, 1, 311359, 1, 313619, 2459, 1, 9907, 8599, 1, 10337, 1, 4817, 1, 2273, 1, 327347, 3733, 329663, 3181, 1, 1, 334319, 2621, 5519, 1, 1289, 3271, 31033, 1, 343727, 21557, 1, 43411, 18341, 1, 1, 5501, 353263, 1, 1, 1439, 358079, 1, 1, 1, 362927, 2069, 1319, 45823, 1, 1, 10007, 1, 372719, 5843, 7079, 1, 1, 47363, 380147, 1, 1601, 23993, 385139, 48299, 29819, 1, 3863, 1, 5861, 1, 1, 1709, 397759, 1609, 400307, 1321, 402863, 1, 36857, 50839, 11027, 4651, 31583, 1, 413167, 1619, 1, 1409, 1, 1, 1, 26393, 1, 26557, 22433, 4111, 428863, 53773, 9181, 6763, 1361, 1, 436819, 1, 14177, 55103, 1, 1, 1801, 1, 1, 5101, 450239, 2971, 41177, 14197, 35051, 1, 1, 1, 461119, 4447, 42169, 29077, 12611, 2659, 469363, 1, 472127, 1, 474899, 1, 1, 1, 2843, 1, 43933, 1, 5011, 30467, 488879, 30643, 1, 1, 1, 1, 26177, 1, 1, 3919, 1, 3319, 505919, 63419, 13751, 1, 3259, 32069, 1613, 64499, 1, 1753, 520339, 1, 18043, 1, 1543, 1, 40699, 6029, 28001, 33343, 1, 2579, 3221, 1, 540863, 6163, 543827, 17041, 4519, 1, 1, 68909, 1, 1, 42751, 1, 1, 1, 1667, 1, 1, 2441, 1, 1, 10771, 1, 15511, 1, 2129, 1, 1, 1913, 583087, 1, 4217, 5651, 31013, 73849, 53849, 1, 1, 1, 1, 75013, 20747, 75403, 604787, 37897, 1, 3463, 1, 76579, 55837, 1, 617363, 1, 1, 19441, 623699, 1663, 56989, 6043, 630067, 1, 633263, 39679, 1, 1697, 639679, 4219, 642899, 1831, 1, 1, 1, 81373, 652607, 1, 655859, 1, 1, 1, 60217, 83003, 1, 7583, 668947, 1, 35381, 21059, 675539, 84649, 1, 1, 1, 1, 1, 1, 1, 86311, 1, 86729, 1, 21787, 63533, 1, 1, 1, 1, 88411, 708979, 44417, 37493, 3433, 1, 1, 719167, 1, 1, 11317, 725999, 22741, 3457, 91393, 732863, 91823, 1, 1, 739759, 1, 1, 1, 1, 93553, 750163, 23497, 753647, 1, 3923, 7297, 69149, 1, 12527, 47869, 59051, 2531, 24877, 1823, 70429, 1, 40961, 1, 60139, 1, 785363, 1, 788927, 7603, 1, 4513, 1, 1, 1, 100183, 1, 7741, 27823, 1, 3229, 1, 5693, 1, 817727, 1, 1, 51449, 825007, 1, 63743, 103811, 26849, 9479,

6. Sequence of the polynom (only primes)

13, 11, 337, 31, 653, 101, 139, 97, 1553, 53, 167, 19, 2113, 281, 2381, 157, 173, 263, 29, 3137, 37, 3373, 109, 277, 3821, 491, 47, 223, 271, 283, 4621, 4801, 4973, 79, 467, 163, 67, 61, 5441, 5581, 353, 197, 449, 5953, 751, 191, 787, 6337, 797, 6481, 211, 821, 347, 827, 6637, 6673, 6701, 839, 6733, 421, 6737, 659, 193, 239, 2099, 383, 251, 433, 4079, 409, 379, 643, 349, 5807, 569, 811, 6719, 7187, 8147, 1049, 587, 877, 619, 10163, 1303, 10687, 863, 359, 1069, 397, 677, 463, 857, 13999, 523, 1933, 1433, 1259, 521, 16979, 2161, 17599, 2239, 1657, 1451, 19507, 1061, 661, 21487, 599, 23539, 1493, 24239, 3163, 2333, 3253, 26387, 577, 859, 2143, 3529, 28607, 3623, 29363, 641, 1907, 3911, 401, 32467, 2683, 1231, 36527, 2309, 37363, 4723, 2939, 439, 617, 40787, 5153, 683, 2687, 509, 45247, 47087, 743, 1549, 2677, 3209, 51827, 503, 1427, 6661, 4889, 54767, 55763, 541, 56767, 7159, 3041, 3643, 4523, 2063, 7673, 1997, 1951, 64019, 8069, 631, 607, 67247, 1847, 69439, 673, 71663, 9311, 1597, 76207, 4799, 9743, 4133, 79699, 2609, 82067, 10333, 1571, 953, 84467, 7789, 5393, 1297, 10939, 6779, 11093, 89363, 8237, 2851, 947, 1051, 11719, 2551, 5939, 3343, 1109, 3169, 9049, 100847, 12853, 733, 2003, 107507, 13523, 108863, 13693, 1733, 111599, 112979, 1093, 757, 115763, 117167, 10909, 3917, 727, 1931, 15629, 1877, 127219, 8089, 11833, 16363, 2801, 4591, 134639, 4231, 12377, 139187, 8747, 1291, 941, 143807, 2383, 4567, 11423, 18661, 18859, 151667, 13933, 9629, 937, 19661, 161363, 12539, 20479, 14969, 10343, 166319, 167987, 21313, 5381, 173039, 887, 1747, 9377, 6203, 181619, 22811, 23029, 1453, 2789, 5867, 3559, 23689, 911, 23911, 14783, 1097, 6257, 1201, 197567, 24809, 199379, 1163, 1579, 18457, 2131, 13093, 11173, 2423, 214163, 3361, 16619, 6781, 4637, 27361, 219839, 221747, 20333, 28319, 227519, 229459, 3631, 2663, 1693, 15017, 2753, 243263, 1607, 2027, 247279, 7759, 13121, 2591, 31543, 1223, 257459, 19963, 32569, 261587, 13877, 265747, 33349, 1873, 269939, 16937, 1669, 274163, 34403, 25117, 1787, 282707, 1867, 7699, 1153, 22079, 1637, 18143, 26489, 36559, 36833, 22751, 9277, 297967, 4673, 2897, 302399, 3449, 16033, 6529, 19249, 309107, 311359, 313619, 2459, 9907, 8599, 10337, 4817, 2273, 327347, 3733, 329663, 3181, 334319, 2621, 5519, 1289, 3271, 31033, 343727, 21557, 43411, 18341, 5501, 353263, 1439, 358079, 362927, 2069, 1319, 45823, 10007, 372719, 5843, 7079, 47363, 380147, 1601, 23993, 385139, 48299, 29819, 3863, 5861, 1709, 397759, 1609, 400307, 1321, 402863, 36857, 50839, 11027, 4651, 31583, 413167, 1619, 1409, 26393, 26557, 22433, 4111, 428863, 53773, 9181, 6763, 1361, 436819, 14177, 55103, 1801, 5101, 450239, 2971, 41177, 14197, 35051, 461119, 4447, 42169, 29077, 12611, 2659, 469363, 472127, 474899, 2843, 43933, 5011, 30467, 488879, 30643, 26177, 3919, 3319, 505919, 63419, 13751, 3259, 32069, 1613, 64499, 1753, 520339, 18043, 1543, 40699, 6029, 28001, 33343, 2579, 3221, 540863, 6163, 543827, 17041, 4519, 68909, 42751, 1667, 2441, 10771, 15511, 2129, 1913, 583087, 4217, 5651, 31013, 73849, 53849, 75013, 20747, 75403, 604787, 37897, 3463, 76579, 55837, 617363, 19441, 623699, 1663, 56989, 6043, 630067, 633263, 39679, 1697, 639679, 4219, 642899, 1831, 81373, 652607, 655859, 60217, 83003, 7583, 668947, 35381, 21059, 675539, 84649, 86311, 86729, 21787, 63533, 88411, 708979, 44417, 37493, 3433, 719167, 11317, 725999, 22741, 3457, 91393, 732863, 91823, 739759, 93553, 750163, 23497, 753647, 3923, 7297, 69149, 12527, 47869, 59051, 2531, 24877, 1823, 70429, 40961, 60139, 785363, 788927, 7603, 4513, 100183, 7741, 27823, 3229, 5693, 817727, 51449, 825007, 63743, 103811, 26849, 9479,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-164x-13 and
the reducible primes which appear as divisor for the first time
p | x^2-164x-13 and p < x^2-164x-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)
1109810.9000000.8000000.1000000.0000000.0000000.000000
21005444100.5400000.4400000.1000006.0000005.50000010.000000
31.0005263152110.5260000.3150000.2110009.7407417.15909121.100000
410.0006.3642.3753.9890.6364000.2375000.39890012.0988607.53968218.905212
5100.00065.91118.24047.6710.6591100.1824000.47671010.3568517.68000011.950614
61.000.000666.138147.184518.9540.6661380.1471840.51895410.1066288.06929810.886157
710.000.0006.700.0371.232.2525.467.7850.6700040.1232250.54677910.0580318.37218810.536165
8100.000.00067.303.22310.593.95456.709.2690.6730320.1059400.56709310.0452028.59723010.371525
91.000.000.000675.317.02592.960.017582.357.0080.6753170.0929600.58235710.0339488.77481710.269168
1010.000.000.0006.771.293.607828.410.6635.942.882.9440.6771290.0828410.59428810.0268368.91147310.204880


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
245501.2500001.2500000.0000001.6666671.666667-nan
388711.0000000.8750000.1250001.6000001.400000inf
416141220.8750000.7500000.1250001.7500001.7142862.000000
532241950.7500000.5937500.1562501.7142861.5833332.500000
664423480.6562500.5312500.1250001.7500001.7894741.600000
71285444100.4218750.3437500.0781251.2857141.2941181.250000
82569773240.3789060.2851560.0937501.7962961.6590912.400000
9512237162750.4628910.3164060.1464842.4432992.2191783.125000
101.0245383222160.5253910.3144530.2109382.2700421.9876542.880000
112.0481.2015926090.5864260.2890620.2973632.2323421.8385092.819444
124.0962.5181.1011.4170.6147460.2687990.3459472.0965861.8597972.326765
138.1925.1742.0133.1610.6315920.2457280.3858642.0548061.8283382.230769
1416.38410.5493.6476.9020.6438600.2225950.4212652.0388481.8117242.183486
1532.76821.3736.74314.6300.6522520.2057800.4464722.0260691.8489172.119675
1665.53643.07212.49330.5790.6572270.1906280.4665992.0152531.8527362.090157
17131.07286.51923.32263.1970.6600880.1779330.4821552.0087061.8668052.066680
18262.144173.68343.458130.2250.6625480.1657790.4967692.0074551.8633912.060620
19524.288348.49181.596266.8950.6646940.1556320.5090622.0064771.8775832.049491
201.048.576698.620153.785544.8350.6662560.1466610.5195952.0047001.8847122.041383
212.097.1521.399.805290.4961.109.3090.6674790.1385190.5289602.0036711.8889752.036046
224.194.3042.804.888550.6632.254.2250.6687370.1312880.5374492.0037711.8955962.032098
238.388.6085.618.2791.046.7324.571.5470.6697510.1247800.5449712.0030311.9008582.027991
2416.777.21611.253.8901.994.2369.259.6540.6707840.1188660.5519182.0030851.9052022.025497
2533.554.43222.539.4643.806.99418.732.4700.6717280.1134570.5582712.0028151.9089992.023021
2667.108.86445.135.2277.285.34037.849.8870.6725670.1085600.5640072.0024981.9136732.020550
27134.217.72890.378.33713.968.25476.410.0830.6733710.1040720.5692992.0023901.9173102.018766
28268.435.456180.949.87026.828.304154.121.5660.6740910.0999430.5741482.0021381.9206632.017032
29536.870.912362.257.55251.614.811310.642.7410.6747570.0961400.5786172.0019771.9238942.015570
301.073.741.824725.182.91399.438.706625.744.2070.6753790.0926100.5827702.0018431.9265542.014353
312.147.483.6481.451.622.820191.849.1301.259.773.6900.6759650.0893370.5866282.0017331.9293212.013241
324.294.967.2962.905.590.845370.604.6212.534.986.2240.6765110.0862880.5902222.0016161.9317502.012255
338.589.934.5925.815.583.087716.739.5615.098.843.5260.6770230.0834390.5935842.0015151.9339742.011389
3417.179.869.18411.639.421.3491.387.698.86810.251.722.4810.6775030.0807750.5967292.0014191.9361272.010598
3534.359.738.36823.294.388.7742.689.542.87920.604.845.8950.6779560.0782760.5996802.0013361.9381322.009891
3668.719.476.73646.618.061.7435.217.738.22441.400.323.5190.6783820.0759280.6024542.0012571.9400092.009252


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
123211110
245321121
387431231
41612574251
5321981153101
664341816106144
7128442321137195
825673383518192115
9512162699326523648
101.0243221441785111258101
112.0485922553378921495194
124.0961.101482619156411152382
138.1922.0139051.108278752275708
1416.3843.6471.6342.0135281.3434991.277
1532.7686.7433.0603.6839512.4419252.426
1665.53612.4935.6716.8221.7384.5181.6814.556
17131.07223.32210.51712.8053.1498.4603.1598.554
18262.14443.45819.58223.8765.83215.9185.86415.844
19524.28881.59636.77544.82110.87130.04710.89329.785
201.048.576153.78569.30284.48320.36456.63320.43656.352
212.097.152290.496130.507159.98938.520107.01238.335106.629
224.194.304550.663247.083303.58072.442203.26072.442202.519
238.388.6081.046.732469.573577.159137.245385.902137.810385.775
2416.777.2161.994.236894.4271.099.809260.711736.048261.312736.165
2533.554.4323.806.9941.708.2422.098.752497.1301.405.844497.4691.406.551
2667.108.8647.285.3403.266.6984.018.642949.9572.692.294950.0502.693.039
27134.217.72813.968.2546.262.8377.705.4171.818.6075.165.8461.819.7575.164.044
28268.435.45626.828.30412.023.28814.805.0163.489.3229.925.0323.487.7309.926.220
29536.870.91251.614.81123.121.60128.493.2106.701.80419.108.9616.698.85019.105.196
301.073.741.82499.438.70644.531.83954.906.86712.891.53336.830.74312.887.72336.828.707
312.147.483.648191.849.13085.890.531105.958.59924.839.58471.085.17224.834.55971.089.815
324.294.967.296370.604.621165.873.960204.730.66147.918.620137.380.53247.911.981137.393.488
338.589.934.592716.739.561320.727.002396.012.55992.571.921265.783.52992.568.414265.815.697
3417.179.869.1841.387.698.868620.834.881766.863.987179.029.888514.798.860179.032.151514.837.969
3534.359.738.3682.689.542.8791.203.022.4901.486.520.389346.632.600998.102.810346.662.967998.144.502
3668.719.476.7365.217.738.2242.333.371.1612.884.367.063671.870.6371.936.970.480671.923.3671.936.973.740


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
381101000
4162111001
5325321013
6648352123
712810462323
82562410146666
951275324320192115
101.0242169312360545646
112.048609298311162144155148
124.0961.417672745358337371351
138.1923.1611.5171.644818752787804
1416.3846.9023.3853.5171.7691.6931.7191.721
1532.76814.6307.2057.4253.6963.5743.7043.656
1665.53630.57915.19615.3837.6867.5697.8157.509
17131.07263.19731.45831.73916.03015.56816.12415.475
18262.144130.22564.71865.50732.92232.09733.18032.026
19524.288266.895132.525134.37067.58366.03467.45465.824
201.048.576544.835271.062273.773137.698134.824137.845134.468
212.097.1521.109.309551.960557.349280.080274.447281.137273.645
224.194.3042.254.2251.121.6691.132.556568.953557.760570.641556.871
238.388.6084.571.5472.275.7402.295.8071.154.8661.130.2811.155.4631.130.937
2416.777.2169.259.6544.612.0934.647.5612.337.6512.292.3402.337.1802.292.483
2533.554.43218.732.4709.333.1289.399.3424.724.8514.639.1394.727.9234.640.557
2667.108.86437.849.88718.858.78518.991.1029.540.9959.379.9319.547.6069.381.355
27134.217.72876.410.08338.085.43938.324.64419.254.17418.946.99519.262.93718.945.977
28268.435.456154.121.56676.833.56177.288.00538.828.25738.231.10538.833.56238.228.642
29536.870.912310.642.741154.882.538155.760.20378.236.93577.088.82578.238.53377.078.448
301.073.741.824625.744.207312.021.298313.722.909157.545.473155.337.632157.535.932155.325.170
312.147.483.6481.259.773.690628.276.188631.497.502317.078.314312.829.437317.074.285312.791.654
324.294.967.2962.534.986.2241.264.386.1001.270.600.124637.879.601629.626.838637.886.784629.593.001
338.589.934.5925.098.843.5262.543.387.6502.555.455.8761.282.690.7831.266.733.4411.282.720.9371.266.698.365
3417.179.869.18410.251.722.4815.114.170.6375.137.551.8442.578.383.1962.547.443.7472.578.456.1502.547.439.388
3534.359.738.36820.604.845.89510.279.688.13210.325.157.7635.181.136.4295.121.278.8735.181.244.5165.121.186.077
3668.719.476.73641.400.323.51920.656.205.25020.744.118.26910.408.164.47910.292.053.25110.408.291.20710.291.814.582


8. Check for existing Integer Sequences by OEIS

Found in Database : 13, 11, 337, 31, 653, 101, 1, 139, 97, 1, 1553, 53, 167, 19, 2113, 281, 2381, 157, 1, 173,
Found in Database : 13, 11, 337, 31, 653, 101, 139, 97, 1553, 53, 167, 19, 2113, 281, 2381, 157, 173, 263, 29, 3137, 37, 3373, 109, 277, 3821, 491, 47, 223, 271, 283, 4621, 4801,
Found in Database : 11, 13, 19, 29, 31, 37, 47, 53, 61, 67, 79, 97, 101, 109, 139,