Inhaltsverzeichnis

Development of
Algorithmic Constructions

10:47:34
Deutsch
29.Mar 2024

Polynom = x^2+20x-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) = 33 = 3*11
f(3) = 29 = 29
f(4) = 85 = 5*17
f(5) = 57 = 3*19
f(6) = 145 = 5*29
f(7) = 89 = 89
f(8) = 213 = 3*71
f(9) = 125 = 5*5*5
f(10) = 289 = 17*17
f(11) = 165 = 3*5*11
f(12) = 373 = 373
f(13) = 209 = 11*19
f(14) = 465 = 3*5*31
f(15) = 257 = 257
f(16) = 565 = 5*113
f(17) = 309 = 3*103
f(18) = 673 = 673
f(19) = 365 = 5*73
f(20) = 789 = 3*263
f(21) = 425 = 5*5*17
f(22) = 913 = 11*83
f(23) = 489 = 3*163
f(24) = 1045 = 5*11*19
f(25) = 557 = 557
f(26) = 1185 = 3*5*79
f(27) = 629 = 17*37
f(28) = 1333 = 31*43
f(29) = 705 = 3*5*47
f(30) = 1489 = 1489
f(31) = 785 = 5*157
f(32) = 1653 = 3*19*29
f(33) = 869 = 11*79
f(34) = 1825 = 5*5*73
f(35) = 957 = 3*11*29
f(36) = 2005 = 5*401
f(37) = 1049 = 1049
f(38) = 2193 = 3*17*43
f(39) = 1145 = 5*229
f(40) = 2389 = 2389
f(41) = 1245 = 3*5*83
f(42) = 2593 = 2593
f(43) = 1349 = 19*71
f(44) = 2805 = 3*5*11*17
f(45) = 1457 = 31*47
f(46) = 3025 = 5*5*11*11
f(47) = 1569 = 3*523
f(48) = 3253 = 3253
f(49) = 1685 = 5*337
f(50) = 3489 = 3*1163
f(51) = 1805 = 5*19*19
f(52) = 3733 = 3733
f(53) = 1929 = 3*643
f(54) = 3985 = 5*797
f(55) = 2057 = 11*11*17
f(56) = 4245 = 3*5*283
f(57) = 2189 = 11*199
f(58) = 4513 = 4513
f(59) = 2325 = 3*5*5*31
f(60) = 4789 = 4789
f(61) = 2465 = 5*17*29
f(62) = 5073 = 3*19*89
f(63) = 2609 = 2609
f(64) = 5365 = 5*29*37
f(65) = 2757 = 3*919
f(66) = 5665 = 5*11*103
f(67) = 2909 = 2909
f(68) = 5973 = 3*11*181
f(69) = 3065 = 5*613
f(70) = 6289 = 19*331
f(71) = 3225 = 3*5*5*43
f(72) = 6613 = 17*389
f(73) = 3389 = 3389
f(74) = 6945 = 3*5*463
f(75) = 3557 = 3557
f(76) = 7285 = 5*31*47
f(77) = 3729 = 3*11*113
f(78) = 7633 = 17*449
f(79) = 3905 = 5*11*71
f(80) = 7989 = 3*2663
f(81) = 4085 = 5*19*43
f(82) = 8353 = 8353
f(83) = 4269 = 3*1423
f(84) = 8725 = 5*5*349
f(85) = 4457 = 4457
f(86) = 9105 = 3*5*607
f(87) = 4649 = 4649
f(88) = 9493 = 11*863
f(89) = 4845 = 3*5*17*19
f(90) = 9889 = 11*29*31
f(91) = 5045 = 5*1009
f(92) = 10293 = 3*47*73
f(93) = 5249 = 29*181
f(94) = 10705 = 5*2141
f(95) = 5457 = 3*17*107
f(96) = 11125 = 5*5*5*89
f(97) = 5669 = 5669
f(98) = 11553 = 3*3851
f(99) = 5885 = 5*11*107
f(100) = 11989 = 19*631

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+20x-11

f(0)=11
f(1)=5
f(2)=3
f(3)=29
f(4)=17
f(5)=19
f(6)=1
f(7)=89
f(8)=71
f(9)=1
f(10)=1
f(11)=1
f(12)=373
f(13)=1
f(14)=31
f(15)=257
f(16)=113
f(17)=103
f(18)=673
f(19)=73
f(20)=263
f(21)=1
f(22)=83
f(23)=163
f(24)=1
f(25)=557
f(26)=79
f(27)=37
f(28)=43
f(29)=47
f(30)=1489
f(31)=157
f(32)=1
f(33)=1
f(34)=1
f(35)=1
f(36)=401
f(37)=1049
f(38)=1
f(39)=229
f(40)=2389
f(41)=1
f(42)=2593
f(43)=1
f(44)=1
f(45)=1
f(46)=1
f(47)=523
f(48)=3253
f(49)=337
f(50)=1163
f(51)=1
f(52)=3733
f(53)=643
f(54)=797
f(55)=1
f(56)=283
f(57)=199
f(58)=4513
f(59)=1
f(60)=4789
f(61)=1
f(62)=1
f(63)=2609
f(64)=1
f(65)=919
f(66)=1
f(67)=2909
f(68)=181
f(69)=613
f(70)=331
f(71)=1
f(72)=389
f(73)=3389
f(74)=463
f(75)=3557
f(76)=1
f(77)=1
f(78)=449
f(79)=1
f(80)=2663
f(81)=1
f(82)=8353
f(83)=1423
f(84)=349
f(85)=4457
f(86)=607
f(87)=4649
f(88)=863
f(89)=1
f(90)=1
f(91)=1009
f(92)=1
f(93)=1
f(94)=2141
f(95)=107
f(96)=1
f(97)=5669
f(98)=3851
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+20x-11 could be written as f(y)= y^2-111 with x=y-10

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+10
f'(x)>2x+19

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, 3, 29, 17, 19, 1, 89, 71, 1, 1, 1, 373, 1, 31, 257, 113, 103, 673, 73, 263, 1, 83, 163, 1, 557, 79, 37, 43, 47, 1489, 157, 1, 1, 1, 1, 401, 1049, 1, 229, 2389, 1, 2593, 1, 1, 1, 1, 523, 3253, 337, 1163, 1, 3733, 643, 797, 1, 283, 199, 4513, 1, 4789, 1, 1, 2609, 1, 919, 1, 2909, 181, 613, 331, 1, 389, 3389, 463, 3557, 1, 1, 449, 1, 2663, 1, 8353, 1423, 349, 4457, 607, 4649, 863, 1, 1, 1009, 1, 1, 2141, 107, 1, 5669, 3851, 1, 631, 1, 12433, 6329, 859, 1, 1, 1, 727, 281, 433, 1453, 1, 2503, 1, 7757, 1051, 8009, 16273, 1, 1, 1, 1, 1, 1, 3019, 3677, 491, 6311, 1, 19489, 659, 1823, 10169, 1, 10457, 4241, 3583, 1, 1, 439, 2269, 22993, 353, 4721, 1087, 1, 12269, 857, 839, 359, 1, 1, 13229, 487, 4519, 499, 1, 9371, 569, 28789, 971, 29473, 877, 2011, 1, 6173, 1, 31573, 1, 1, 653, 33013, 5563, 397, 461, 1, 601, 3203, 1187, 1, 3637, 12251, 599, 1, 1, 1, 1759, 1, 1, 1, 1, 40693, 20549, 2767, 1103, 1693, 419, 3923, 4357, 1, 4441, 1, 1, 9137, 23057, 1, 1, 2789, 1, 1123, 443, 1, 24809, 1, 8419, 10193, 547, 17291, 5233, 4799, 1, 1, 1, 3643, 1621, 11117, 9343, 56533, 5701, 19163, 1, 58453, 1, 2377, 1033, 4027, 30449, 1, 2063, 701, 1, 1, 743, 1171, 1, 2617, 32969, 1303, 1, 67489, 2267, 3607, 1, 4639, 3187, 1, 11863, 71713, 1, 1277, 7333, 2383, 1, 1, 2221, 1, 1321, 1, 2591, 991, 1, 1, 39989, 1, 1229, 1, 3739, 27611, 1, 1787, 2819, 5009, 1, 1151, 43457, 1, 14683, 733, 8929, 1, 9049, 1283, 1, 18461, 46457, 1, 1, 4987, 1, 95989, 9661, 32411, 1, 19697, 16519, 19949, 1619, 3061, 1, 1, 1, 103573, 1, 6991, 52757, 1249, 937, 107473, 983, 36263, 1, 3797, 1, 719, 1933, 1, 1, 6007, 1, 10499, 11617, 3541, 3457, 4729, 19819, 1259, 60149, 1301, 1, 122389, 1, 1567, 5659, 1, 1, 1013, 1117, 128053, 1, 2539, 1, 11903, 21943, 1, 1, 8923, 67289, 2879, 907, 3697, 809, 46091, 1, 27953, 2129, 1487, 4177, 47591, 1, 144289, 967, 3391, 883, 1, 1, 2707, 24943, 8849, 15121, 1747, 15277, 3571, 887, 1, 1, 1, 7159, 158293, 5303, 1, 16069, 53831, 4271, 32621, 1607, 1, 2237, 1, 1, 5419, 1, 169633, 85229, 1, 1831, 34589, 2633, 174613, 1, 58763, 17713, 1, 29803, 35933, 2099, 1, 2939, 1, 6131, 1, 1, 62171, 4931, 37649, 1, 37997, 95429, 1, 1, 1069, 1, 195253, 1, 1, 5821, 39761, 1, 6917, 20149, 6133, 1, 977, 34183, 41201, 103457, 1, 104369, 6763, 7019, 11131, 1931, 1, 9739, 43037, 1, 1, 1493, 1697, 4397, 220789, 1, 1, 6577, 1361, 112757, 45293, 1307, 1, 1, 2647, 1, 3181, 3533, 46829, 10687, 15739, 118529, 2311, 1, 1, 24097, 80651, 121469, 1, 40819, 1, 123449, 4349, 24889, 1, 8363, 251893, 4079, 16927, 11587, 1, 1, 257953, 1, 1097, 1, 1669, 2579, 52817, 1867, 1613, 1, 1, 1, 8719, 1, 1, 136709, 3229, 1, 55313, 12619, 4889, 2543, 1, 1879, 282913, 1327, 1, 4933, 3023, 1, 1, 1, 1, 1721, 2851, 49123, 11833, 148457, 19867, 1, 300193, 1, 4259, 1, 101531, 1, 1427, 1, 1, 155069, 1, 31237, 28499, 10487, 28703, 158429, 1, 5147, 3371, 1847, 322513, 6473, 108263, 2963, 6959, 4973, 3467, 9721, 22111, 166409, 333973, 11171, 1, 1, 1, 169889, 6199, 3001, 1, 1, 115211, 1, 347989, 1, 1, 1, 4703, 16087, 1, 1381, 1237, 35869, 119963, 36109, 362293, 1289, 1, 182957, 1, 4283, 369553, 1, 1, 37321, 6569, 1, 75377, 1, 75869, 17299, 127271, 1, 384289, 1, 13337, 10211, 25951, 6733, 1, 2113, 1, 1, 132263, 1, 1, 66763, 80369, 201557, 26959, 18439, 1777, 1, 1, 41077, 7229, 12157, 1, 1, 1, 209249, 12721, 2477, 1, 1, 3761, 213149, 1, 214457, 17209, 1, 13963, 3947, 8539, 1, 438133, 73243, 1, 1, 1, 222389, 5647, 1, 3709, 45013, 13681, 226409, 90833, 1, 91373, 13477, 153191, 46093, 1, 1, 465013, 1, 31183, 1439, 1, 78643, 24907, 1531, 158663, 47737, 1, 1861, 1, 2713, 1, 5167, 487093, 1, 28817, 1, 164231, 1, 1, 7529, 1, 13151, 167051, 1733, 503989, 1, 17477, 5407, 3089, 3079, 9319, 5039, 27127, 10337, 5573, 51973, 521173, 87103, 1, 23887, 1, 24019, 1, 1, 532789, 2137, 178571, 1, 6337, 90019, 1, 1, 1, 54601, 14797, 1, 1663, 275969, 1, 1, 1, 1, 5431, 5099, 187463, 1, 565393, 94483, 1, 284957, 1, 286469, 52223, 1, 1, 57901, 11383, 4099, 6143, 1, 117329, 10141, 1, 1, 20441, 1801, 595873, 298709, 1, 15803, 120413, 2719, 605173, 1, 18433, 12197, 1, 1, 1, 18121, 41179, 309629, 1, 20747, 623989, 1, 1, 28579, 1, 105319, 1, 3083, 2557, 1, 1, 21383, 1, 322349, 3917, 323957, 25981, 2309, 652753, 65437, 1, 65761, 15331, 1, 3581, 30187, 44383, 2953, 7517, 1, 672289, 3547, 5237, 338609, 12343, 3911, 1, 1889, 7879, 68713, 1, 4603, 1, 4391, 1, 31687, 8221, 10613, 14939, 70381, 12377, 70717, 1, 118423, 1, 1, 4337, 1, 1, 24023, 722389, 1, 241931, 21397, 1, 2833, 5861, 1, 2293, 1, 1901, 1, 8951, 1871, 2927, 8699, 149969, 1, 68483, 15101, 1, 75853, 760273, 1, 152753, 12347, 51151, 384509, 1, 2341, 774289, 1, 259271, 13441, 156269, 2777, 5413, 1, 7103, 4159, 71999, 26459, 2333, 398669, 2131, 400457, 160541, 7057, 1, 80809, 2621, 1, 813493, 1, 9613, 1, 1, 4621, 1, 27539, 827989, 82981, 1, 1, 15187, 1, 8831, 1, 280871, 16889, 11593, 1, 849973, 1, 56911, 1, 171473, 7537, 4327, 86293, 2551, 17333, 1, 145063, 15859, 23003, 5309, 14159, 51749, 29387, 883489, 1, 295751, 444569, 1, 1, 9419, 40759, 6373, 5297, 902389, 1, 2689, 1, 1, 26821, 3323, 152623, 83423, 91957, 307163, 1, 1, 154543, 2239, 1, 3659, 42499, 30223, 1, 32441, 94273, 18523, 1, 189713, 158419, 2683, 477209, 1, 2039, 87299, 1, 964213, 1, 1, 15647, 1, 162343, 976033, 1, 326663, 1, 51787, 2251, 1, 10531, 1, 496949, 1, 1, 5347, 100189, 1, 1, 201581, 1, 2381, 1, 1, 101797,

6. Sequence of the polynom (only primes)

11, 5, 3, 29, 17, 19, 89, 71, 373, 31, 257, 113, 103, 673, 73, 263, 83, 163, 557, 79, 37, 43, 47, 1489, 157, 401, 1049, 229, 2389, 2593, 523, 3253, 337, 1163, 3733, 643, 797, 283, 199, 4513, 4789, 2609, 919, 2909, 181, 613, 331, 389, 3389, 463, 3557, 449, 2663, 8353, 1423, 349, 4457, 607, 4649, 863, 1009, 2141, 107, 5669, 3851, 631, 12433, 6329, 859, 727, 281, 433, 1453, 2503, 7757, 1051, 8009, 16273, 3019, 3677, 491, 6311, 19489, 659, 1823, 10169, 10457, 4241, 3583, 439, 2269, 22993, 353, 4721, 1087, 12269, 857, 839, 359, 13229, 487, 4519, 499, 9371, 569, 28789, 971, 29473, 877, 2011, 6173, 31573, 653, 33013, 5563, 397, 461, 601, 3203, 1187, 3637, 12251, 599, 1759, 40693, 20549, 2767, 1103, 1693, 419, 3923, 4357, 4441, 9137, 23057, 2789, 1123, 443, 24809, 8419, 10193, 547, 17291, 5233, 4799, 3643, 1621, 11117, 9343, 56533, 5701, 19163, 58453, 2377, 1033, 4027, 30449, 2063, 701, 743, 1171, 2617, 32969, 1303, 67489, 2267, 3607, 4639, 3187, 11863, 71713, 1277, 7333, 2383, 2221, 1321, 2591, 991, 39989, 1229, 3739, 27611, 1787, 2819, 5009, 1151, 43457, 14683, 733, 8929, 9049, 1283, 18461, 46457, 4987, 95989, 9661, 32411, 19697, 16519, 19949, 1619, 3061, 103573, 6991, 52757, 1249, 937, 107473, 983, 36263, 3797, 719, 1933, 6007, 10499, 11617, 3541, 3457, 4729, 19819, 1259, 60149, 1301, 122389, 1567, 5659, 1013, 1117, 128053, 2539, 11903, 21943, 8923, 67289, 2879, 907, 3697, 809, 46091, 27953, 2129, 1487, 4177, 47591, 144289, 967, 3391, 883, 2707, 24943, 8849, 15121, 1747, 15277, 3571, 887, 7159, 158293, 5303, 16069, 53831, 4271, 32621, 1607, 2237, 5419, 169633, 85229, 1831, 34589, 2633, 174613, 58763, 17713, 29803, 35933, 2099, 2939, 6131, 62171, 4931, 37649, 37997, 95429, 1069, 195253, 5821, 39761, 6917, 20149, 6133, 977, 34183, 41201, 103457, 104369, 6763, 7019, 11131, 1931, 9739, 43037, 1493, 1697, 4397, 220789, 6577, 1361, 112757, 45293, 1307, 2647, 3181, 3533, 46829, 10687, 15739, 118529, 2311, 24097, 80651, 121469, 40819, 123449, 4349, 24889, 8363, 251893, 4079, 16927, 11587, 257953, 1097, 1669, 2579, 52817, 1867, 1613, 8719, 136709, 3229, 55313, 12619, 4889, 2543, 1879, 282913, 1327, 4933, 3023, 1721, 2851, 49123, 11833, 148457, 19867, 300193, 4259, 101531, 1427, 155069, 31237, 28499, 10487, 28703, 158429, 5147, 3371, 1847, 322513, 6473, 108263, 2963, 6959, 4973, 3467, 9721, 22111, 166409, 333973, 11171, 169889, 6199, 3001, 115211, 347989, 4703, 16087, 1381, 1237, 35869, 119963, 36109, 362293, 1289, 182957, 4283, 369553, 37321, 6569, 75377, 75869, 17299, 127271, 384289, 13337, 10211, 25951, 6733, 2113, 132263, 66763, 80369, 201557, 26959, 18439, 1777, 41077, 7229, 12157, 209249, 12721, 2477, 3761, 213149, 214457, 17209, 13963, 3947, 8539, 438133, 73243, 222389, 5647, 3709, 45013, 13681, 226409, 90833, 91373, 13477, 153191, 46093, 465013, 31183, 1439, 78643, 24907, 1531, 158663, 47737, 1861, 2713, 5167, 487093, 28817, 164231, 7529, 13151, 167051, 1733, 503989, 17477, 5407, 3089, 3079, 9319, 5039, 27127, 10337, 5573, 51973, 521173, 87103, 23887, 24019, 532789, 2137, 178571, 6337, 90019, 54601, 14797, 1663, 275969, 5431, 5099, 187463, 565393, 94483, 284957, 286469, 52223, 57901, 11383, 4099, 6143, 117329, 10141, 20441, 1801, 595873, 298709, 15803, 120413, 2719, 605173, 18433, 12197, 18121, 41179, 309629, 20747, 623989, 28579, 105319, 3083, 2557, 21383, 322349, 3917, 323957, 25981, 2309, 652753, 65437, 65761, 15331, 3581, 30187, 44383, 2953, 7517, 672289, 3547, 5237, 338609, 12343, 3911, 1889, 7879, 68713, 4603, 4391, 31687, 8221, 10613, 14939, 70381, 12377, 70717, 118423, 4337, 24023, 722389, 241931, 21397, 2833, 5861, 2293, 1901, 8951, 1871, 2927, 8699, 149969, 68483, 15101, 75853, 760273, 152753, 12347, 51151, 384509, 2341, 774289, 259271, 13441, 156269, 2777, 5413, 7103, 4159, 71999, 26459, 2333, 398669, 2131, 400457, 160541, 7057, 80809, 2621, 813493, 9613, 4621, 27539, 827989, 82981, 15187, 8831, 280871, 16889, 11593, 849973, 56911, 171473, 7537, 4327, 86293, 2551, 17333, 145063, 15859, 23003, 5309, 14159, 51749, 29387, 883489, 295751, 444569, 9419, 40759, 6373, 5297, 902389, 2689, 26821, 3323, 152623, 83423, 91957, 307163, 154543, 2239, 3659, 42499, 30223, 32441, 94273, 18523, 189713, 158419, 2683, 477209, 2039, 87299, 964213, 15647, 162343, 976033, 326663, 51787, 2251, 10531, 496949, 5347, 100189, 201581, 2381, 101797,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+20x-11 and
the reducible primes which appear as divisor for the first time
p | x^2+20x-11 and p < x^2+20x-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)
1108260.8000000.2000000.8000000.0000000.0000000.000000
21006312510.6300000.1200000.6300007.8750006.0000008.500000
31.000661685930.6610000.0680000.66100010.4920645.66666711.627451
410.0006.6804776.2030.6680000.0477000.66800010.1059007.01470610.460371
5100.00067.3043.71963.5850.6730400.0371900.67304010.0754497.79664610.250685
61.000.000676.92930.641646.2880.6769290.0306410.67692910.0577828.23904210.164158
710.000.0006.790.076259.5726.530.5040.6790080.0259570.67900810.0307068.47139510.104634
8100.000.00068.063.3932.251.22865.812.1650.6806340.0225120.68063410.0239528.67284610.077655
91.000.000.000681.943.88719.865.027662.078.8600.6819440.0198650.68194410.0192468.82408510.060129
1010.000.000.0006.830.094.838177.774.0366.652.320.8020.6830100.0177770.68301010.0156268.94909610.047626


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
245231.2500000.5000000.7500001.6666671.0000003.000000
388261.0000000.2500000.7500001.6000001.0000002.000000
41612390.7500000.1875000.5625001.5000001.5000001.500000
532235180.7187500.1562500.5625001.9166671.6666672.000000
6644011290.6250000.1718750.4531251.7391302.2000001.611111
71287914650.6171880.1093750.5078121.9750001.2727272.241379
8256166241420.6484380.0937500.5546882.1012661.7142862.184615
9512337382990.6582030.0742190.5839842.0301201.5833332.105634
101.024677696080.6611330.0673830.5937502.0089021.8157892.033445
112.0481.3661271.2390.6669920.0620120.6049802.0177251.8405802.037829
124.0962.7382242.5140.6684570.0546880.6137702.0043921.7637802.029056
138.1925.4734015.0720.6680910.0489500.6191411.9989041.7901792.017502
1416.38410.98572210.2630.6704710.0440670.6264042.0071261.8004992.023462
1532.76821.9711.38420.5870.6705020.0422360.6282652.0000911.9168982.005944
1665.53644.0592.57841.4810.6722870.0393370.6329502.0053251.8627172.014912
17131.07288.3094.75283.5570.6737440.0362550.6374892.0043351.8432892.014344
18262.144176.9328.939167.9930.6749420.0341000.6408422.0035561.8811032.010520
19524.288354.39616.795337.6010.6759570.0320340.6439232.0030071.8788452.009614
201.048.576709.85332.051677.8020.6769690.0305660.6464022.0029941.9083662.007701
212.097.1521.421.07960.6561.360.4230.6776230.0289230.6487002.0019341.8924842.007110
224.194.3042.844.625115.4592.729.1660.6782110.0275280.6506842.0017361.9035052.006116
238.388.6085.694.666220.4895.474.1770.6788570.0262840.6525732.0019041.9096742.005806
2416.777.21611.398.362420.93910.977.4230.6793950.0250900.6543052.0015861.9091162.005310
2533.554.43222.813.455806.62222.006.8330.6798940.0240390.6558552.0014681.9162442.004736
2667.108.86445.659.1521.546.88044.112.2720.6803740.0230500.6573242.0014131.9177262.004481
27134.217.72891.377.6762.971.59288.406.0840.6808170.0221400.6586772.0013001.9210232.004115
28268.435.456182.868.6115.715.545177.153.0660.6812390.0212920.6599472.0012391.9233952.003856
29536.870.912365.944.54311.013.467354.931.0760.6816250.0205140.6611112.0011341.9269322.003528
301.073.741.824732.271.60721.253.182711.018.4250.6819810.0197940.6621882.0010451.9297452.003258
312.147.483.6481.465.273.97141.060.2681.424.213.7030.6823210.0191200.6632012.0009981.9319592.003062
324.294.967.2962.931.938.58879.416.9502.852.521.6380.6826450.0184910.6641542.0009491.9341562.002875
338.589.934.5925.866.451.186153.773.6075.712.677.5790.6829450.0179020.6650432.0008781.9362822.002676
3417.179.869.18411.737.778.517298.045.32711.439.733.1900.6832290.0173490.6658802.0008311.9382092.002517


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
122011100
242011100
382011100
4163111110
5325313110
66411915150
7128141218150
825624221121110
951238361171200
101.02469671301380
112.0481271251551710
124.096224222110511180
138.192401399120411960
1416.384722720135613650
1532.7681.3841.382169516880
1665.5362.5782.57611.30511.2720
17131.0724.7524.75012.40412.3470
18262.1448.9398.93714.48614.4520
19524.28816.79516.79318.41318.3810
201.048.57632.05132.049116.019116.0310
212.097.15260.65660.654130.379130.2760
224.194.304115.459115.457157.762157.6960
238.388.608220.489220.4871110.2791110.2090
2416.777.216420.939420.9371210.5521210.3860
2533.554.432806.622806.6201403.3751403.2460
2667.108.8641.546.8801.546.8781774.0751772.8040
27134.217.7282.971.5922.971.59011.486.44711.485.1440
28268.435.4565.715.5455.715.54312.857.84112.857.7030
29536.870.91211.013.46711.013.46515.507.18715.506.2790
301.073.741.82421.253.18221.253.180110.627.035110.626.1460
312.147.483.64841.060.26841.060.266120.529.810120.530.4570
324.294.967.29679.416.95079.416.948139.710.040139.706.9090
338.589.934.592153.773.607153.773.605176.886.890176.886.7160
3417.179.869.184298.045.327298.045.3251149.028.2691149.017.0570


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
243031020
386152121
4169274122
532187115355
6642913169776
712865303517131916
8256142657736353635
951229913616373778069
101.024608281327145152163148
112.0481.239565674292318321308
124.0962.5141.1571.357610653636615
138.1925.0722.3252.7471.2321.2431.3111.286
1416.38410.2634.7225.5412.5582.4942.6002.611
1532.76820.5879.56211.0255.1685.0675.1925.160
1665.53641.48119.35422.12710.41510.20010.61510.251
17131.07283.55739.23544.32221.11120.59021.18820.668
18262.144167.99379.26488.72942.40841.48342.39541.707
19524.288337.601159.952177.64985.16083.43885.11283.891
201.048.576677.802322.671355.131171.050167.822170.904168.026
212.097.1521.360.423649.646710.777343.454337.047342.714337.208
224.194.3042.729.1661.305.9751.423.191688.667676.514687.132676.853
238.388.6085.474.1772.626.2032.847.9741.379.3931.358.2011.377.8841.358.699
2416.777.21610.977.4235.275.8705.701.5532.764.8912.724.5112.764.5192.723.502
2533.554.43222.006.83310.595.58711.411.2465.541.6965.463.7875.541.1705.460.180
2667.108.86444.112.27221.278.61622.833.65611.104.58910.953.15711.104.74410.949.782
27134.217.72888.406.08442.711.46545.694.61922.250.93021.957.61122.245.24621.952.297
28268.435.456177.153.06685.707.78891.445.27844.582.13944.004.31644.564.87344.001.738
29536.870.912354.931.076171.934.302182.996.77489.302.21088.171.80089.281.21188.175.855
301.073.741.824711.018.425344.835.305366.183.120178.850.895176.664.125178.827.833176.675.572
312.147.483.6481.424.213.703691.513.705732.699.998358.175.723353.938.203358.149.238353.950.539
324.294.967.2962.852.521.6381.386.455.6701.466.065.968717.235.686709.027.160717.221.529709.037.263
338.589.934.5925.712.677.5792.779.217.4352.933.460.1441.436.159.3331.420.173.1761.436.137.7901.420.207.280
3417.179.869.18411.439.733.1905.570.342.5685.869.390.6222.875.480.9342.844.394.0862.875.450.9432.844.407.227


8. Check for existing Integer Sequences by OEIS

Found in Database : 11, 5, 3, 29, 17, 19, 1, 89, 71, 1, 1, 1, 373, 1, 31, 257, 113, 103, 673, 73,
Found in Database : 11, 5, 3, 29, 17, 19, 89, 71, 373, 31, 257, 113, 103, 673, 73, 263, 83, 163, 557, 79, 37, 43, 47, 1489, 157, 401, 1049, 229,
Found in Database : 3, 5, 11, 17, 19, 29, 31, 37, 43, 47, 71, 73, 79, 83, 89, 103, 107, 113,