Inhaltsverzeichnis

Development of
Algorithmic Constructions

06:39:53
Deutsch
29.Mar 2024

Polynom = x^2-212x+107

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) = 107 = 107
f(1) = 13 = 13
f(2) = 313 = 313
f(3) = 65 = 5*13
f(4) = 725 = 5*5*29
f(5) = 29 = 29
f(6) = 1129 = 1129
f(7) = 83 = 83
f(8) = 1525 = 5*5*61
f(9) = 215 = 5*43
f(10) = 1913 = 1913
f(11) = 263 = 263
f(12) = 2293 = 2293
f(13) = 155 = 5*31
f(14) = 2665 = 5*13*41
f(15) = 89 = 89
f(16) = 3029 = 13*233
f(17) = 401 = 401
f(18) = 3385 = 5*677
f(19) = 445 = 5*89
f(20) = 3733 = 3733
f(21) = 61 = 61
f(22) = 4073 = 4073
f(23) = 265 = 5*53
f(24) = 4405 = 5*881
f(25) = 571 = 571
f(26) = 4729 = 4729
f(27) = 611 = 13*47
f(28) = 5045 = 5*1009
f(29) = 325 = 5*5*13
f(30) = 5353 = 53*101
f(31) = 43 = 43
f(32) = 5653 = 5653
f(33) = 725 = 5*5*29
f(34) = 5945 = 5*29*41
f(35) = 761 = 761
f(36) = 6229 = 6229
f(37) = 199 = 199
f(38) = 6505 = 5*1301
f(39) = 415 = 5*83
f(40) = 6773 = 13*521
f(41) = 863 = 863
f(42) = 7033 = 13*541
f(43) = 895 = 5*179
f(44) = 7285 = 5*31*47
f(45) = 463 = 463
f(46) = 7529 = 7529
f(47) = 239 = 239
f(48) = 7765 = 5*1553
f(49) = 985 = 5*197
f(50) = 7993 = 7993
f(51) = 1013 = 1013
f(52) = 8213 = 43*191
f(53) = 65 = 5*13
f(54) = 8425 = 5*5*337
f(55) = 533 = 13*41
f(56) = 8629 = 8629
f(57) = 1091 = 1091
f(58) = 8825 = 5*5*353
f(59) = 1115 = 5*223
f(60) = 9013 = 9013
f(61) = 569 = 569
f(62) = 9193 = 29*317
f(63) = 145 = 5*29
f(64) = 9365 = 5*1873
f(65) = 1181 = 1181
f(66) = 9529 = 13*733
f(67) = 1201 = 1201
f(68) = 9685 = 5*13*149
f(69) = 305 = 5*61
f(70) = 9833 = 9833
f(71) = 619 = 619
f(72) = 9973 = 9973
f(73) = 1255 = 5*251
f(74) = 10105 = 5*43*47
f(75) = 1271 = 31*41
f(76) = 10229 = 53*193
f(77) = 643 = 643
f(78) = 10345 = 5*2069
f(79) = 325 = 5*5*13
f(80) = 10453 = 10453
f(81) = 1313 = 13*101
f(82) = 10553 = 61*173
f(83) = 1325 = 5*5*53
f(84) = 10645 = 5*2129
f(85) = 167 = 167
f(86) = 10729 = 10729
f(87) = 673 = 673
f(88) = 10805 = 5*2161
f(89) = 1355 = 5*271
f(90) = 10873 = 83*131
f(91) = 1363 = 29*47
f(92) = 10933 = 13*29*29
f(93) = 685 = 5*137
f(94) = 10985 = 5*13*13*13
f(95) = 43 = 43
f(96) = 11029 = 41*269
f(97) = 1381 = 1381
f(98) = 11065 = 5*2213
f(99) = 1385 = 5*277
f(100) = 11093 = 11093

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^2-212x+107

f(0)=107
f(1)=13
f(2)=313
f(3)=5
f(4)=29
f(5)=1
f(6)=1129
f(7)=83
f(8)=61
f(9)=43
f(10)=1913
f(11)=263
f(12)=2293
f(13)=31
f(14)=41
f(15)=89
f(16)=233
f(17)=401
f(18)=677
f(19)=1
f(20)=3733
f(21)=1
f(22)=4073
f(23)=53
f(24)=881
f(25)=571
f(26)=4729
f(27)=47
f(28)=1009
f(29)=1
f(30)=101
f(31)=1
f(32)=5653
f(33)=1
f(34)=1
f(35)=761
f(36)=6229
f(37)=199
f(38)=1301
f(39)=1
f(40)=521
f(41)=863
f(42)=541
f(43)=179
f(44)=1
f(45)=463
f(46)=7529
f(47)=239
f(48)=1553
f(49)=197
f(50)=7993
f(51)=1013
f(52)=191
f(53)=1
f(54)=337
f(55)=1
f(56)=8629
f(57)=1091
f(58)=353
f(59)=223
f(60)=9013
f(61)=569
f(62)=317
f(63)=1
f(64)=1873
f(65)=1181
f(66)=733
f(67)=1201
f(68)=149
f(69)=1
f(70)=9833
f(71)=619
f(72)=9973
f(73)=251
f(74)=1
f(75)=1
f(76)=193
f(77)=643
f(78)=2069
f(79)=1
f(80)=10453
f(81)=1
f(82)=173
f(83)=1
f(84)=2129
f(85)=167
f(86)=10729
f(87)=673
f(88)=2161
f(89)=271
f(90)=131
f(91)=1
f(92)=1
f(93)=137
f(94)=1
f(95)=1
f(96)=269
f(97)=1381
f(98)=2213
f(99)=277

b) Substitution of the polynom
The polynom f(x)=x^2-212x+107 could be written as f(y)= y^2-11129 with x=y+106

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-106
f'(x)>2x-213 with x > 105

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

107, 13, 313, 5, 29, 1, 1129, 83, 61, 43, 1913, 263, 2293, 31, 41, 89, 233, 401, 677, 1, 3733, 1, 4073, 53, 881, 571, 4729, 47, 1009, 1, 101, 1, 5653, 1, 1, 761, 6229, 199, 1301, 1, 521, 863, 541, 179, 1, 463, 7529, 239, 1553, 197, 7993, 1013, 191, 1, 337, 1, 8629, 1091, 353, 223, 9013, 569, 317, 1, 1873, 1181, 733, 1201, 149, 1, 9833, 619, 9973, 251, 1, 1, 193, 643, 2069, 1, 10453, 1, 173, 1, 2129, 167, 10729, 673, 2161, 271, 131, 1, 1, 137, 1, 1, 269, 1381, 2213, 277, 11093, 347, 11113, 139, 1, 1, 359, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 971, 1, 283, 1, 1867, 1, 1, 1, 1, 379, 3271, 439, 751, 1, 1, 281, 1, 1, 1051, 1, 1, 1, 1259, 1, 6827, 887, 1, 1, 1583, 1, 1, 547, 1, 1, 739, 1237, 1, 1, 431, 1, 1, 1459, 479, 307, 1, 1, 1, 1, 2767, 1, 499, 1, 3023, 1, 15767, 503, 16427, 419, 1, 2179, 1367, 1, 3691, 1, 467, 2437, 1, 1, 4111, 1307, 1, 1, 1, 1, 22727, 2887, 757, 1, 1, 1, 24971, 3169, 5147, 653, 2039, 1, 2099, 1, 1123, 3559, 28871, 3659, 1187, 1, 709, 1931, 31307, 1, 6427, 1, 32971, 2087, 6763, 1, 34667, 1, 35527, 1, 1, 1151, 1, 2357, 587, 1, 39047, 4937, 929, 1, 8171, 1291, 41771, 5279, 8539, 1, 43627, 1, 1087, 563, 9103, 5749, 46471, 5869, 1, 599, 48407, 1, 1, 1, 1, 6359, 1093, 1621, 1, 661, 1, 6737, 1327, 1373, 11087, 1, 1, 1, 11503, 1451, 58567, 1, 59627, 1, 1, 1, 1, 7789, 967, 1, 4919, 1, 2243, 1, 1, 1, 67271, 1, 13679, 1, 1, 1, 70667, 1, 1, 9049, 1697, 4597, 14827, 1, 75307, 1, 76487, 1, 1, 1223, 6067, 4967, 3203, 2017, 2803, 1, 82507, 1039, 16747, 659, 2741, 823, 1, 1, 1861, 2753, 1069, 1117, 1, 11329, 853, 11489, 1, 1, 7219, 2953, 1, 1, 1, 1, 97771, 769, 19819, 1, 3463, 12637, 683, 1, 1, 1, 104471, 1, 1, 2663, 2281, 13487, 108587, 1, 1, 6917, 1, 14009, 1, 2837, 1283, 1, 1, 727, 1, 1, 118471, 1, 827, 1, 121367, 1, 122827, 3089, 24859, 15629, 1, 7907, 821, 1, 128747, 16187, 1, 1, 2027, 1, 133271, 8377, 26959, 3389, 136327, 17137, 137867, 1733, 27883, 1, 2311, 1, 983, 3583, 2719, 1, 145687, 1831, 1, 1, 3631, 1, 1, 1, 11699, 2389, 1, 3863, 1, 1, 156971, 4931, 31723, 1993, 1153, 1549, 839, 1, 32719, 1, 1, 1, 33391, 1, 1231, 21187, 170347, 1, 2647, 1, 13367, 1, 35099, 4409, 5717, 11131, 937, 1, 1, 22699, 182471, 1, 36847, 1, 186007, 11681, 187787, 1, 7583, 1, 6599, 1, 7727, 1213, 1, 1, 15139, 4943, 39727, 1559, 1, 1, 941, 5081, 3347, 1, 206027, 1, 41579, 1, 209771, 26339, 42331, 1063, 1193, 6703, 215447, 1, 1499, 1, 1, 27529, 1, 2777, 5189, 1, 225067, 5651, 45403, 28499, 1, 3593, 46187, 1, 232907, 1, 7577, 5897, 1, 14867, 4507, 1, 1, 6047, 242887, 1, 18839, 1, 1, 15497, 248971, 31249, 1, 6301, 3049, 15881, 255127, 1601, 51439, 1, 259271, 2503, 1, 1, 1, 1, 265547, 1, 1, 33589, 2671, 16927, 1, 1, 1, 1, 1, 1, 55663, 8731, 2141, 17597, 56527, 1, 284807, 2749, 286987, 1, 1, 1, 291371, 36559, 11743, 1, 295787, 4639, 1279, 3739, 1, 1, 1, 1, 60943, 3823, 7487, 1, 10663, 7759, 62299, 39079, 7297, 1, 63211, 1, 318347, 39937, 320647, 1609, 64591, 1, 325271, 1, 65519, 8219, 1, 41387, 1, 1, 66923, 1, 1, 1, 67867, 8513, 11783, 1, 7321, 1, 13859, 3343, 1949, 43759, 14051, 2203, 353687, 1, 6719, 8933, 71707, 1, 27767, 22637, 5591, 1, 1, 45887, 368327, 9239, 74159, 5813, 12041, 1, 1, 1, 13043, 1, 8101, 1, 76651, 1, 385771, 1, 77659, 9739, 1, 12253, 30259, 4933, 6091, 1, 398471, 1, 80207, 1, 403607, 12653, 9907, 1, 1, 3943, 411371, 1, 1, 1, 1, 52237, 9749, 10513, 1, 1, 424471, 6653, 6571, 10711, 2543, 53887, 1, 2711, 1, 27277, 437771, 1, 1, 1, 443147, 2137, 445847, 1, 1, 56239, 5437, 1951, 1, 1423, 456727, 28631, 459467, 1, 1, 1, 1, 29147, 1, 1, 1789, 1, 473287, 11867, 1, 1, 478871, 2309, 19267, 1, 484487, 60737, 487307, 1, 1607, 15361, 1, 2131, 1871, 1, 1, 3907, 1, 6287, 1, 1, 1, 63589, 102031, 1, 513047, 8039, 1531, 1, 2531, 5003, 1667, 1, 1, 6577, 527627, 66137, 530567, 1, 106703, 1, 1, 16811, 1, 13523, 17497, 67987, 545387, 1709, 1, 34367, 5153, 69109, 1, 1, 1579, 2687, 9187, 1, 112687, 70619, 1, 70999, 113903, 1, 572567, 1, 44279, 1, 1, 1, 4441, 36457, 2207, 1, 587947, 2377, 2473, 2963, 118831, 1, 597271, 2879, 120079, 1, 12841, 1759, 2417, 7603, 121963, 1, 612971, 76819, 9479, 15443, 47639, 19403, 2297, 1, 1, 1, 1, 78809, 1, 1, 635287, 1, 1, 1, 128347, 1, 15731, 10103, 129643, 8123, 1, 81637, 654727, 1, 1, 41227, 50867, 5179, 132911, 3331, 2411, 1, 23143, 1, 3137, 42257, 1, 1, 1, 1, 684427, 1, 687767, 1, 1, 86599, 694471, 1, 1, 1, 53939, 1, 1, 17657, 28319, 2063, 711371, 1, 28591, 2239, 718187, 1, 1, 1, 145007, 1747, 728471, 1, 146383, 18341, 735367, 1, 738827, 9257, 4789, 23251, 57367, 1, 11527, 1, 752747, 11789, 756247, 1, 1, 95189, 763271, 95629, 153359, 1, 1, 1, 773867, 19391, 1, 1, 780971, 1, 1, 9829,

6. Sequence of the polynom (only primes)

107, 13, 313, 5, 29, 1129, 83, 61, 43, 1913, 263, 2293, 31, 41, 89, 233, 401, 677, 3733, 4073, 53, 881, 571, 4729, 47, 1009, 101, 5653, 761, 6229, 199, 1301, 521, 863, 541, 179, 463, 7529, 239, 1553, 197, 7993, 1013, 191, 337, 8629, 1091, 353, 223, 9013, 569, 317, 1873, 1181, 733, 1201, 149, 9833, 619, 9973, 251, 193, 643, 2069, 10453, 173, 2129, 167, 10729, 673, 2161, 271, 131, 137, 269, 1381, 2213, 277, 11093, 347, 11113, 139, 359, 971, 283, 1867, 379, 3271, 439, 751, 281, 1051, 1259, 6827, 887, 1583, 547, 739, 1237, 431, 1459, 479, 307, 2767, 499, 3023, 15767, 503, 16427, 419, 2179, 1367, 3691, 467, 2437, 4111, 1307, 22727, 2887, 757, 24971, 3169, 5147, 653, 2039, 2099, 1123, 3559, 28871, 3659, 1187, 709, 1931, 31307, 6427, 32971, 2087, 6763, 34667, 35527, 1151, 2357, 587, 39047, 4937, 929, 8171, 1291, 41771, 5279, 8539, 43627, 1087, 563, 9103, 5749, 46471, 5869, 599, 48407, 6359, 1093, 1621, 661, 6737, 1327, 1373, 11087, 11503, 1451, 58567, 59627, 7789, 967, 4919, 2243, 67271, 13679, 70667, 9049, 1697, 4597, 14827, 75307, 76487, 1223, 6067, 4967, 3203, 2017, 2803, 82507, 1039, 16747, 659, 2741, 823, 1861, 2753, 1069, 1117, 11329, 853, 11489, 7219, 2953, 97771, 769, 19819, 3463, 12637, 683, 104471, 2663, 2281, 13487, 108587, 6917, 14009, 2837, 1283, 727, 118471, 827, 121367, 122827, 3089, 24859, 15629, 7907, 821, 128747, 16187, 2027, 133271, 8377, 26959, 3389, 136327, 17137, 137867, 1733, 27883, 2311, 983, 3583, 2719, 145687, 1831, 3631, 11699, 2389, 3863, 156971, 4931, 31723, 1993, 1153, 1549, 839, 32719, 33391, 1231, 21187, 170347, 2647, 13367, 35099, 4409, 5717, 11131, 937, 22699, 182471, 36847, 186007, 11681, 187787, 7583, 6599, 7727, 1213, 15139, 4943, 39727, 1559, 941, 5081, 3347, 206027, 41579, 209771, 26339, 42331, 1063, 1193, 6703, 215447, 1499, 27529, 2777, 5189, 225067, 5651, 45403, 28499, 3593, 46187, 232907, 7577, 5897, 14867, 4507, 6047, 242887, 18839, 15497, 248971, 31249, 6301, 3049, 15881, 255127, 1601, 51439, 259271, 2503, 265547, 33589, 2671, 16927, 55663, 8731, 2141, 17597, 56527, 284807, 2749, 286987, 291371, 36559, 11743, 295787, 4639, 1279, 3739, 60943, 3823, 7487, 10663, 7759, 62299, 39079, 7297, 63211, 318347, 39937, 320647, 1609, 64591, 325271, 65519, 8219, 41387, 66923, 67867, 8513, 11783, 7321, 13859, 3343, 1949, 43759, 14051, 2203, 353687, 6719, 8933, 71707, 27767, 22637, 5591, 45887, 368327, 9239, 74159, 5813, 12041, 13043, 8101, 76651, 385771, 77659, 9739, 12253, 30259, 4933, 6091, 398471, 80207, 403607, 12653, 9907, 3943, 411371, 52237, 9749, 10513, 424471, 6653, 6571, 10711, 2543, 53887, 2711, 27277, 437771, 443147, 2137, 445847, 56239, 5437, 1951, 1423, 456727, 28631, 459467, 29147, 1789, 473287, 11867, 478871, 2309, 19267, 484487, 60737, 487307, 1607, 15361, 2131, 1871, 3907, 6287, 63589, 102031, 513047, 8039, 1531, 2531, 5003, 1667, 6577, 527627, 66137, 530567, 106703, 16811, 13523, 17497, 67987, 545387, 1709, 34367, 5153, 69109, 1579, 2687, 9187, 112687, 70619, 70999, 113903, 572567, 44279, 4441, 36457, 2207, 587947, 2377, 2473, 2963, 118831, 597271, 2879, 120079, 12841, 1759, 2417, 7603, 121963, 612971, 76819, 9479, 15443, 47639, 19403, 2297, 78809, 635287, 128347, 15731, 10103, 129643, 8123, 81637, 654727, 41227, 50867, 5179, 132911, 3331, 2411, 23143, 3137, 42257, 684427, 687767, 86599, 694471, 53939, 17657, 28319, 2063, 711371, 28591, 2239, 718187, 145007, 1747, 728471, 146383, 18341, 735367, 738827, 9257, 4789, 23251, 57367, 11527, 752747, 11789, 756247, 95189, 763271, 95629, 153359, 773867, 19391, 780971, 9829,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-212x+107 and
the reducible primes which appear as divisor for the first time
p | x^2-212x+107 and p < x^2-212x+107

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)
1109450.9000000.4000000.9000000.0000000.0000000.000000
21007419550.7400000.1900000.7400008.2222224.75000011.000000
31.0005101183920.5100000.1180000.5100006.8918926.2105267.127273
410.0006.2758685.4070.6275000.0868000.62750012.3039227.35593213.793367
5100.00065.1816.83558.3460.6518100.0683500.65181010.3874107.87442410.790827
61.000.000661.24455.651605.5930.6612440.0556510.66124410.1447358.14206310.379340
710.000.0006.659.697470.5626.189.1350.6659700.0470560.66597010.0714668.45558910.219958
8100.000.00066.949.4424.075.06262.874.3800.6694940.0407510.66949410.0529268.65998910.158832
91.000.000.000672.174.32535.965.596636.208.7290.6721740.0359660.67217410.0400308.82577910.118728
1010.000.000.0006.743.126.566321.825.5256.421.301.0410.6743130.0321830.67431310.0318128.94815010.093074


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
244221.0000000.5000000.5000001.3333331.0000002.000000
387340.8750000.3750000.5000001.7500001.5000002.000000
416155100.9375000.3125000.6250002.1428571.6666672.500000
532269170.8125000.2812500.5312501.7333331.8000001.700000
6645114370.7968750.2187500.5781251.9615381.5555562.176471
71287720570.6015620.1562500.4453121.5098041.4285711.540541
82568924650.3476560.0937500.2539061.1558441.2000001.140351
9512223561670.4355470.1093750.3261722.5056182.3333332.569231
101.0245281234050.5156250.1201170.3955082.3677132.1964292.425150
112.0481.1612179440.5668950.1059570.4609382.1988641.7642282.330864
124.0962.4673892.0780.6022950.0949710.5073242.1248921.7926272.201271
138.1925.1057324.3730.6231690.0893550.5338132.0693151.8817482.104427
1416.38410.3801.3439.0370.6335450.0819700.5515752.0333011.8347002.066545
1532.76821.0492.49218.5570.6423650.0760500.5663152.0278421.8555472.053447
1665.53642.5934.68937.9040.6499180.0715480.5783692.0235171.8816212.042572
17131.07285.6808.71976.9610.6536870.0665210.5871662.0115981.8594582.030419
18262.144172.20916.291155.9180.6569250.0621450.5947802.0099091.8684482.025935
19524.288345.68130.674315.0070.6593340.0585060.6008282.0073341.8828802.020338
201.048.576693.39558.142635.2530.6612730.0554490.6058242.0058811.8954812.016631
212.097.1521.390.350110.4221.279.9280.6629710.0526530.6103172.0051341.8991782.014832
224.194.3042.786.759209.6712.577.0880.6644150.0499890.6144262.0043581.8988162.013463
238.388.6085.584.213399.5755.184.6380.6656900.0476330.6180572.0038381.9057242.011820
2416.777.21611.188.278762.61910.425.6590.6668730.0454560.6214182.0035551.9085752.010875
2533.554.43222.412.6221.459.74120.952.8810.6679480.0435040.6244442.0032241.9141162.009742
2667.108.86444.893.2092.799.41542.093.7940.6689610.0417150.6272462.0030321.9177482.008974
27134.217.72889.909.5295.377.31584.532.2140.6698780.0400640.6298142.0027421.9208712.008187
28268.435.456180.045.63810.352.208169.693.4300.6707220.0385650.6321572.0025201.9251632.007441
29536.870.912360.515.43319.946.333340.569.1000.6715120.0371530.6343592.0023561.9267712.006967
301.073.741.824721.822.56938.478.576683.343.9930.6722500.0358360.6364142.0021961.9291052.006477
312.147.483.6481.445.122.90574.334.3581.370.788.5470.6729380.0346150.6383232.0020471.9318382.006001
324.294.967.2962.893.022.455143.779.3032.749.243.1520.6735840.0334760.6401082.0019211.9342242.005592
338.589.934.5925.791.212.165278.379.9175.512.832.2480.6741860.0324080.6417782.0017861.9361612.005218
3417.179.869.18411.592.162.701539.549.92111.052.612.7800.6747530.0314060.6433472.0016821.9381792.004888


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
122111100
242111100
383212100
4165323110
5329635130
664141047160
71282014610190
82562416810491
95125628281020917
101.02412351721053951
112.048217791381099999
124.096389136253101849186
138.192732250482103619352
1416.3841.343448895106669658
1532.7682.4928091.683101.23391.240
1665.5364.6891.5523.137102.29492.376
17131.0728.7192.9205.799104.31994.381
18262.14416.2915.41410.877108.11198.161
19524.28830.67410.14920.5251015.273915.382
201.048.57658.14219.29438.8481028.998929.125
212.097.152110.42236.80773.6151055.104955.299
224.194.304209.67170.059139.61210104.4949105.158
238.388.608399.575133.405266.17010199.3969200.160
2416.777.216762.619254.237508.38210380.7249381.876
2533.554.4321.459.741486.784972.95710729.1289730.594
2667.108.8642.799.415932.5731.866.842101.398.39291.401.004
27134.217.7285.377.3151.791.2723.586.043102.687.29492.690.002
28268.435.45610.352.2083.448.6156.903.593105.174.99595.177.194
29536.870.91219.946.3336.646.37613.299.957109.971.81699.974.498
301.073.741.82438.478.57612.824.85025.653.7261019.237.377919.241.180
312.147.483.64874.334.35824.775.16149.559.1971037.163.594937.170.745
324.294.967.296143.779.30347.925.51295.853.7911071.882.350971.896.934
338.589.934.592278.379.91792.798.082185.581.83510139.180.7369139.199.162
3417.179.869.184539.549.921179.847.427359.702.49410269.777.1279269.772.775


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
121100010
242110020
384220130
41610373331
532175126461
664371126136117
71285721361810209
825665263918142112
9512167828534454543
101.0244052042017911780129
112.048944478466201274187282
124.0962.0781.0661.012428618403629
138.1924.3732.2512.1229241.2858931.271
1416.3849.0374.6674.3701.9082.6051.8922.632
1532.76818.5579.5109.0473.9445.3373.9555.321
1665.53637.90419.37018.5348.16310.7268.23710.778
17131.07276.96139.46637.49516.76921.58816.94121.663
18262.144155.91879.78576.13334.50243.38034.54143.495
19524.288315.007161.131153.87670.45187.00970.62286.925
201.048.576635.253324.932310.321143.074174.568143.495174.116
212.097.1521.279.928652.958626.970290.811349.467290.984348.666
224.194.3042.577.0881.313.8481.263.240589.445700.087589.447698.109
238.388.6085.184.6382.639.1492.545.4891.191.9341.401.2041.192.6521.398.848
2416.777.21610.425.6595.300.7445.124.9152.408.0792.805.7542.408.5372.803.289
2533.554.43220.952.88110.645.43510.307.4464.856.9945.616.8174.862.4545.616.616
2667.108.86442.093.79421.369.35320.724.4419.797.10611.245.7099.806.24411.244.735
27134.217.72884.532.21442.885.39241.646.82219.750.96522.515.26319.753.52922.512.457
28268.435.456169.693.43086.032.37483.661.05639.771.49845.078.27739.771.87645.071.779
29536.870.912340.569.100172.575.975167.993.12580.035.36290.247.08980.046.69190.239.958
301.073.741.824683.343.993346.093.630337.250.363161.008.723180.654.683161.018.399180.662.188
312.147.483.6481.370.788.547693.952.230676.836.317323.758.984361.654.300323.751.221361.624.042
324.294.967.2962.749.243.1521.391.134.1941.358.108.958650.725.062723.904.156650.724.469723.889.465
338.589.934.5925.512.832.2482.788.273.8232.724.558.4251.307.495.1531.448.912.7061.307.535.9011.448.888.488
3417.179.869.18411.052.612.7805.587.986.9655.464.625.8152.626.283.4292.900.001.3812.626.354.9602.899.973.010


8. Check for existing Integer Sequences by OEIS

Found in Database : 107, 13, 313, 5, 29, 1, 1129, 83, 61, 43, 1913, 263, 2293, 31, 41, 89, 233, 401, 677, 1,
Found in Database : 107, 13, 313, 5, 29, 1129, 83, 61, 43, 1913, 263, 2293, 31, 41, 89, 233, 401, 677, 3733, 4073, 53, 881, 571, 4729, 47, 1009, 101, 5653, 761, 6229, 199, 1301,
Found in Database : 5, 13, 29, 31, 41, 43, 47, 53, 61, 83, 89, 101, 107, 131, 137, 139, 149,