Inhaltsverzeichnis

Development of
Algorithmic Constructions

10:36:38
Deutsch
29.Mar 2024

Polynom = x^2+24x-19

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) = 19 = 19
f(1) = 3 = 3
f(2) = 33 = 3*11
f(3) = 31 = 31
f(4) = 93 = 3*31
f(5) = 63 = 3*3*7
f(6) = 161 = 7*23
f(7) = 99 = 3*3*11
f(8) = 237 = 3*79
f(9) = 139 = 139
f(10) = 321 = 3*107
f(11) = 183 = 3*61
f(12) = 413 = 7*59
f(13) = 231 = 3*7*11
f(14) = 513 = 3*3*3*19
f(15) = 283 = 283
f(16) = 621 = 3*3*3*23
f(17) = 339 = 3*113
f(18) = 737 = 11*67
f(19) = 399 = 3*7*19
f(20) = 861 = 3*7*41
f(21) = 463 = 463
f(22) = 993 = 3*331
f(23) = 531 = 3*3*59
f(24) = 1133 = 11*103
f(25) = 603 = 3*3*67
f(26) = 1281 = 3*7*61
f(27) = 679 = 7*97
f(28) = 1437 = 3*479
f(29) = 759 = 3*11*23
f(30) = 1601 = 1601
f(31) = 843 = 3*281
f(32) = 1773 = 3*3*197
f(33) = 931 = 7*7*19
f(34) = 1953 = 3*3*7*31
f(35) = 1023 = 3*11*31
f(36) = 2141 = 2141
f(37) = 1119 = 3*373
f(38) = 2337 = 3*19*41
f(39) = 1219 = 23*53
f(40) = 2541 = 3*7*11*11
f(41) = 1323 = 3*3*3*7*7
f(42) = 2753 = 2753
f(43) = 1431 = 3*3*3*53
f(44) = 2973 = 3*991
f(45) = 1543 = 1543
f(46) = 3201 = 3*11*97
f(47) = 1659 = 3*7*79
f(48) = 3437 = 7*491
f(49) = 1779 = 3*593
f(50) = 3681 = 3*3*409
f(51) = 1903 = 11*173
f(52) = 3933 = 3*3*19*23
f(53) = 2031 = 3*677
f(54) = 4193 = 7*599
f(55) = 2163 = 3*7*103
f(56) = 4461 = 3*1487
f(57) = 2299 = 11*11*19
f(58) = 4737 = 3*1579
f(59) = 2439 = 3*3*271
f(60) = 5021 = 5021
f(61) = 2583 = 3*3*7*41
f(62) = 5313 = 3*7*11*23
f(63) = 2731 = 2731
f(64) = 5613 = 3*1871
f(65) = 2883 = 3*31*31
f(66) = 5921 = 31*191
f(67) = 3039 = 3*1013
f(68) = 6237 = 3*3*3*3*7*11
f(69) = 3199 = 7*457
f(70) = 6561 = 3*3*3*3*3*3*3*3
f(71) = 3363 = 3*19*59
f(72) = 6893 = 61*113
f(73) = 3531 = 3*11*107
f(74) = 7233 = 3*2411
f(75) = 3703 = 7*23*23
f(76) = 7581 = 3*7*19*19
f(77) = 3879 = 3*3*431
f(78) = 7937 = 7937
f(79) = 4059 = 3*3*11*41
f(80) = 8301 = 3*2767
f(81) = 4243 = 4243
f(82) = 8673 = 3*7*7*59
f(83) = 4431 = 3*7*211
f(84) = 9053 = 11*823
f(85) = 4623 = 3*23*67
f(86) = 9441 = 3*3*1049
f(87) = 4819 = 61*79
f(88) = 9837 = 3*3*1093
f(89) = 5019 = 3*7*239
f(90) = 10241 = 7*7*11*19
f(91) = 5223 = 3*1741
f(92) = 10653 = 3*53*67
f(93) = 5431 = 5431
f(94) = 11073 = 3*3691
f(95) = 5643 = 3*3*3*11*19
f(96) = 11501 = 7*31*53
f(97) = 5859 = 3*3*3*7*31
f(98) = 11937 = 3*23*173
f(99) = 6079 = 6079
f(100) = 12381 = 3*4127

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+24x-19

f(0)=19
f(1)=3
f(2)=11
f(3)=31
f(4)=1
f(5)=7
f(6)=23
f(7)=1
f(8)=79
f(9)=139
f(10)=107
f(11)=61
f(12)=59
f(13)=1
f(14)=1
f(15)=283
f(16)=1
f(17)=113
f(18)=67
f(19)=1
f(20)=41
f(21)=463
f(22)=331
f(23)=1
f(24)=103
f(25)=1
f(26)=1
f(27)=97
f(28)=479
f(29)=1
f(30)=1601
f(31)=281
f(32)=197
f(33)=1
f(34)=1
f(35)=1
f(36)=2141
f(37)=373
f(38)=1
f(39)=53
f(40)=1
f(41)=1
f(42)=2753
f(43)=1
f(44)=991
f(45)=1543
f(46)=1
f(47)=1
f(48)=491
f(49)=593
f(50)=409
f(51)=173
f(52)=1
f(53)=677
f(54)=599
f(55)=1
f(56)=1487
f(57)=1
f(58)=1579
f(59)=271
f(60)=5021
f(61)=1
f(62)=1
f(63)=2731
f(64)=1871
f(65)=1
f(66)=191
f(67)=1013
f(68)=1
f(69)=457
f(70)=1
f(71)=1
f(72)=1
f(73)=1
f(74)=2411
f(75)=1
f(76)=1
f(77)=431
f(78)=7937
f(79)=1
f(80)=2767
f(81)=4243
f(82)=1
f(83)=211
f(84)=823
f(85)=1
f(86)=1049
f(87)=1
f(88)=1093
f(89)=239
f(90)=1
f(91)=1741
f(92)=1
f(93)=5431
f(94)=3691
f(95)=1
f(96)=1
f(97)=1
f(98)=1
f(99)=6079

b) Substitution of the polynom
The polynom f(x)=x^2+24x-19 could be written as f(y)= y^2-163 with x=y-12

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+12
f'(x)>2x+23

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

19, 3, 11, 31, 1, 7, 23, 1, 79, 139, 107, 61, 59, 1, 1, 283, 1, 113, 67, 1, 41, 463, 331, 1, 103, 1, 1, 97, 479, 1, 1601, 281, 197, 1, 1, 1, 2141, 373, 1, 53, 1, 1, 2753, 1, 991, 1543, 1, 1, 491, 593, 409, 173, 1, 677, 599, 1, 1487, 1, 1579, 271, 5021, 1, 1, 2731, 1871, 1, 191, 1013, 1, 457, 1, 1, 1, 1, 2411, 1, 1, 431, 7937, 1, 2767, 4243, 1, 211, 823, 1, 1049, 1, 1093, 239, 1, 1741, 1, 5431, 3691, 1, 1, 1, 1, 6079, 4127, 1, 313, 311, 1, 6763, 1, 2333, 619, 127, 701, 1069, 461, 859, 827, 887, 5407, 1, 797, 2833, 421, 1, 659, 821, 1, 443, 1, 1, 1, 9859, 1, 1, 2939, 1, 641, 10723, 7247, 3673, 3191, 1, 2549, 1, 2617, 1, 1051, 1, 1181, 661, 1, 1, 2371, 163, 1, 1933, 397, 4621, 2551, 4733, 1, 1, 467, 1, 30113, 5077, 10271, 15583, 1, 1, 32237, 1811, 10987, 877, 1021, 811, 4919, 5801, 1303, 773, 1, 6053, 1, 883, 12491, 1721, 1, 1, 39041, 1, 1, 1, 1, 6841, 41453, 6977, 1, 3049, 4789, 7253, 829, 7393, 1, 3229, 1, 853, 2447, 1, 15787, 23899, 2297, 1, 49121, 751, 5557, 1097, 5657, 1223, 673, 8713, 17579, 1, 577, 1, 709, 1, 1, 683, 1, 863, 3023, 1, 1, 29443, 733, 907, 60353, 10141, 1, 631, 1889, 3491, 653, 3547, 1129, 1, 1, 1, 2887, 587, 1, 3089, 1087, 1, 2243, 11677, 23531, 1, 23887, 1, 10391, 1, 2237, 1, 24971, 12577, 10859, 1823, 1, 38839, 8693, 571, 1301, 1, 3833, 40531, 27211, 4567, 82781, 1, 1, 6037, 1493, 14281, 1, 1, 1, 1, 1, 1, 8167, 15073, 30347, 45823, 1, 1, 1, 5227, 31567, 47659, 1, 1, 13883, 16301, 10937, 49531, 1, 727, 14423, 1, 643, 51439, 3137, 1, 1, 1, 1, 1, 1, 18013, 1, 18233, 1747, 719, 12377, 983, 1427, 1, 3457, 1171, 1, 6451, 1, 1, 3581, 1, 5693, 1, 120941, 1, 1, 61519, 4583, 2963, 941, 1907, 1361, 2053, 42667, 7151, 1, 1033, 43627, 2861, 44111, 1, 12163, 3203, 1, 1283, 15193, 2083, 2063, 1, 6653, 1, 1, 1, 1, 2657, 48079, 10357, 1, 24421, 147293, 24677, 1, 1, 1, 1, 151937, 25453, 2693, 7013, 51691, 1237, 1, 8747, 52747, 7229, 53279, 1, 23063, 3863, 1, 81931, 1, 27581, 166301, 1, 1, 4441, 56527, 9467, 171233, 1, 8233, 12409, 1877, 1, 176237, 2683, 1, 1, 2851, 1583, 1, 1, 61007, 91939, 1, 1, 1, 3469, 62731, 94531, 63311, 1, 1, 1, 21493, 97159, 1, 2971, 1481, 1, 1, 99823, 1, 1, 1, 1613, 9721, 102523, 1, 1499, 1201, 34781, 1109, 1367, 1, 35393, 213281, 1879, 71711, 1, 10333, 12107, 11519, 12211, 6689, 1, 10601, 5323, 224513, 1, 2287, 113683, 1103, 1, 1061, 1, 4073, 2843, 78031, 1451, 4817, 1, 3449, 1, 79979, 40153, 21991, 5783, 1, 1, 1, 41141, 22531, 1, 11897, 1, 1, 1277, 253853, 14159, 1, 1, 12281, 3923, 259937, 1, 1, 1229, 1, 6311, 266093, 1, 89387, 1, 1, 2153, 38903, 15187, 1, 1, 1511, 46261, 1, 6659, 31193, 12809, 1, 1, 12391, 1, 1, 144103, 1, 1, 291437, 5417, 1, 1, 5189, 49477, 297953, 1, 33349, 21517, 4799, 1, 304541, 1, 102251, 2609, 14713, 1, 1489, 17351, 104479, 157279, 105227, 1, 4129, 1297, 1, 160663, 1327, 4903, 2017, 7759, 1787, 1, 3541, 1669, 331613, 1, 15901, 1, 1, 1, 1, 1381, 1, 1, 3467, 1, 345581, 57793, 115979, 2267, 2383, 1, 3121, 6553, 6229, 16193, 17021, 8539, 359837, 3167, 3659, 7901, 1307, 1, 1279, 61381, 11197, 185359, 123979, 20731, 1, 1, 2129, 9949, 1, 63421, 381761, 1, 1, 1987, 14323, 64661, 1, 1, 1, 28069, 1, 1, 36067, 22111, 133087, 1, 1, 1, 13043, 67601, 1, 1, 1, 1, 412001, 68881, 138191, 207931, 12641, 1, 1, 1, 140779, 1, 1, 3739, 1, 10211, 47797, 1783, 2531, 72353, 435437, 10399, 1, 19973, 1373, 1, 14303, 1, 1931, 31957, 149579, 75013, 1, 1, 1, 32533, 2417, 4019, 459521, 6983, 154079, 231799, 3163, 3701, 7927, 1, 1, 4451, 3847, 11299, 1, 1, 53189, 240043, 1, 3499, 1, 1, 5237, 244219, 2437, 1, 492641, 1, 23593, 248431, 4051, 1, 21787, 1, 1, 36097, 5119, 84701, 509633, 4483, 1, 36709, 1, 28711, 518237, 28871, 173707, 23753, 24953, 12511, 526913, 1, 1, 1, 6577, 1, 1, 89521, 1, 270031, 180511, 1, 77783, 1, 1, 11933, 1, 1559, 1, 1, 8831, 6803, 3271, 1, 562337, 8543, 26921, 2131, 8237, 1, 1, 1, 191467, 1, 1, 96493, 1, 97001, 2819, 292531, 9311, 1, 4643, 98533, 1, 2777, 10453, 1, 1, 33359, 200671, 13121, 1667, 101113, 12413, 14519, 22643, 306463, 2069, 102677, 32507, 1, 1, 28289, 6709, 1, 627101, 1, 30013, 1, 211151, 1997, 1, 106373, 1, 45817, 1, 107441, 646253, 5683, 19681, 325543, 31081, 1, 1, 1, 219727, 2377, 220811, 1, 1, 10111, 1, 335299, 1, 3623, 1, 1, 11909, 1, 227371, 37987, 62311, 1, 32797, 5659, 1, 1, 1, 116177, 3697, 1, 26003, 10663, 705437, 1759, 1, 2671, 3083, 39659, 1, 39847, 12613, 4561, 1, 17239, 1, 3911, 81017, 1, 3539, 17483, 105143, 122953, 2393, 1, 13033, 13789, 106619, 1979, 22721, 19777, 251087, 5471, 18457, 18059, 1, 380983, 84857, 1, 2711, 1, 1, 2399, 1, 2269, 777761, 1, 1, 55933, 1, 131101, 71671, 1, 3259, 3853, 1, 1, 72643, 1, 267551, 402223, 268747, 1, 6089, 1, 11789, 7691, 272351, 1, 117239, 19583, 1, 7001, 8363, 3373, 8573, 19843, 1, 1, 2311, 15569, 1, 1, 1, 5507, 1, 141961, 44927, 6199, 95257, 1, 1, 143813, 864737, 1, 26317, 3851, 1, 1, 875933, 48767, 1, 2089, 294479, 1, 126743, 1, 32999, 14401, 1, 149441, 5581, 1949, 300779, 1, 1, 1, 82723, 7237, 1, 24097, 305867, 1, 1, 6691, 1, 1, 103237, 14107, 1, 155821, 1, 67057, 44797, 1, 944621, 5843, 1, 1, 1, 1, 5867, 2707, 106693, 25321, 9739, 23003, 1, 1, 324011, 44273, 2879, 54331, 139991, 7793, 327967, 1, 1, 1, 1, 23663, 1, 8179, 1, 166973, 1003841, 167641, 4363, 1, 337291, 1, 32771, 1, 2677, 72997,

6. Sequence of the polynom (only primes)

19, 3, 11, 31, 7, 23, 79, 139, 107, 61, 59, 283, 113, 67, 41, 463, 331, 103, 97, 479, 1601, 281, 197, 2141, 373, 53, 2753, 991, 1543, 491, 593, 409, 173, 677, 599, 1487, 1579, 271, 5021, 2731, 1871, 191, 1013, 457, 2411, 431, 7937, 2767, 4243, 211, 823, 1049, 1093, 239, 1741, 5431, 3691, 6079, 4127, 313, 311, 6763, 2333, 619, 127, 701, 1069, 461, 859, 827, 887, 5407, 797, 2833, 421, 659, 821, 443, 9859, 2939, 641, 10723, 7247, 3673, 3191, 2549, 2617, 1051, 1181, 661, 2371, 163, 1933, 397, 4621, 2551, 4733, 467, 30113, 5077, 10271, 15583, 32237, 1811, 10987, 877, 1021, 811, 4919, 5801, 1303, 773, 6053, 883, 12491, 1721, 39041, 6841, 41453, 6977, 3049, 4789, 7253, 829, 7393, 3229, 853, 2447, 15787, 23899, 2297, 49121, 751, 5557, 1097, 5657, 1223, 673, 8713, 17579, 577, 709, 683, 863, 3023, 29443, 733, 907, 60353, 10141, 631, 1889, 3491, 653, 3547, 1129, 2887, 587, 3089, 1087, 2243, 11677, 23531, 23887, 10391, 2237, 24971, 12577, 10859, 1823, 38839, 8693, 571, 1301, 3833, 40531, 27211, 4567, 82781, 6037, 1493, 14281, 8167, 15073, 30347, 45823, 5227, 31567, 47659, 13883, 16301, 10937, 49531, 727, 14423, 643, 51439, 3137, 18013, 18233, 1747, 719, 12377, 983, 1427, 3457, 1171, 6451, 3581, 5693, 120941, 61519, 4583, 2963, 941, 1907, 1361, 2053, 42667, 7151, 1033, 43627, 2861, 44111, 12163, 3203, 1283, 15193, 2083, 2063, 6653, 2657, 48079, 10357, 24421, 147293, 24677, 151937, 25453, 2693, 7013, 51691, 1237, 8747, 52747, 7229, 53279, 23063, 3863, 81931, 27581, 166301, 4441, 56527, 9467, 171233, 8233, 12409, 1877, 176237, 2683, 2851, 1583, 61007, 91939, 3469, 62731, 94531, 63311, 21493, 97159, 2971, 1481, 99823, 1613, 9721, 102523, 1499, 1201, 34781, 1109, 1367, 35393, 213281, 1879, 71711, 10333, 12107, 11519, 12211, 6689, 10601, 5323, 224513, 2287, 113683, 1103, 1061, 4073, 2843, 78031, 1451, 4817, 3449, 79979, 40153, 21991, 5783, 41141, 22531, 11897, 1277, 253853, 14159, 12281, 3923, 259937, 1229, 6311, 266093, 89387, 2153, 38903, 15187, 1511, 46261, 6659, 31193, 12809, 12391, 144103, 291437, 5417, 5189, 49477, 297953, 33349, 21517, 4799, 304541, 102251, 2609, 14713, 1489, 17351, 104479, 157279, 105227, 4129, 1297, 160663, 1327, 4903, 2017, 7759, 1787, 3541, 1669, 331613, 15901, 1381, 3467, 345581, 57793, 115979, 2267, 2383, 3121, 6553, 6229, 16193, 17021, 8539, 359837, 3167, 3659, 7901, 1307, 1279, 61381, 11197, 185359, 123979, 20731, 2129, 9949, 63421, 381761, 1987, 14323, 64661, 28069, 36067, 22111, 133087, 13043, 67601, 412001, 68881, 138191, 207931, 12641, 140779, 3739, 10211, 47797, 1783, 2531, 72353, 435437, 10399, 19973, 1373, 14303, 1931, 31957, 149579, 75013, 32533, 2417, 4019, 459521, 6983, 154079, 231799, 3163, 3701, 7927, 4451, 3847, 11299, 53189, 240043, 3499, 5237, 244219, 2437, 492641, 23593, 248431, 4051, 21787, 36097, 5119, 84701, 509633, 4483, 36709, 28711, 518237, 28871, 173707, 23753, 24953, 12511, 526913, 6577, 89521, 270031, 180511, 77783, 11933, 1559, 8831, 6803, 3271, 562337, 8543, 26921, 2131, 8237, 191467, 96493, 97001, 2819, 292531, 9311, 4643, 98533, 2777, 10453, 33359, 200671, 13121, 1667, 101113, 12413, 14519, 22643, 306463, 2069, 102677, 32507, 28289, 6709, 627101, 30013, 211151, 1997, 106373, 45817, 107441, 646253, 5683, 19681, 325543, 31081, 219727, 2377, 220811, 10111, 335299, 3623, 11909, 227371, 37987, 62311, 32797, 5659, 116177, 3697, 26003, 10663, 705437, 1759, 2671, 3083, 39659, 39847, 12613, 4561, 17239, 3911, 81017, 3539, 17483, 105143, 122953, 2393, 13033, 13789, 106619, 1979, 22721, 19777, 251087, 5471, 18457, 18059, 380983, 84857, 2711, 2399, 2269, 777761, 55933, 131101, 71671, 3259, 3853, 72643, 267551, 402223, 268747, 6089, 11789, 7691, 272351, 117239, 19583, 7001, 8363, 3373, 8573, 19843, 2311, 15569, 5507, 141961, 44927, 6199, 95257, 143813, 864737, 26317, 3851, 875933, 48767, 2089, 294479, 126743, 32999, 14401, 149441, 5581, 1949, 300779, 82723, 7237, 24097, 305867, 6691, 103237, 14107, 155821, 67057, 44797, 944621, 5843, 5867, 2707, 106693, 25321, 9739, 23003, 324011, 44273, 2879, 54331, 139991, 7793, 327967, 23663, 8179, 166973, 1003841, 167641, 4363, 337291, 32771, 2677, 72997,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+24x-19 and
the reducible primes which appear as divisor for the first time
p | x^2+24x-19 and p < x^2+24x-19

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)
1108440.8000000.4000000.4000000.0000000.0000000.000000
21005716410.5700000.1600000.4100007.1250004.00000010.250000
31.000629935360.6290000.0930000.53600011.0350885.81250013.073171
410.0006.4916825.8090.6491000.0682000.58090010.3195557.33333310.837687
5100.00066.0325.09560.9370.6603200.0509500.60937010.1728547.47067510.490102
61.000.000666.16841.260624.9080.6661680.0412600.62490810.0885638.09813510.254985
710.000.0006.699.712347.6246.352.0880.6699710.0347620.63520910.0570918.42520610.164837
8100.000.00067.281.9713.007.83964.274.1320.6728200.0300780.64274110.0425178.65256410.118584
91.000.000.000675.019.04426.509.831648.509.2130.6750190.0265100.64850910.0326898.81358110.089739
1010.000.000.0006.767.911.493237.031.4936.530.880.0000.6767910.0237030.65308810.0262528.94126710.070605


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
244311.0000000.7500000.2500001.3333331.5000001.000000
386330.7500000.3750000.3750001.5000001.0000003.000000
41611560.6875000.3125000.3750001.8333331.6666672.000000
532227150.6875000.2187500.4687502.0000001.4000002.500000
6643912270.6093750.1875000.4218751.7727271.7142861.800000
71287517580.5859380.1328120.4531251.9230771.4166672.148148
8256160281320.6250000.1093750.5156252.1333331.6470592.275862
9512320542660.6250000.1054690.5195312.0000001.9285712.015152
101.024646935530.6308590.0908200.5400392.0187501.7222222.078947
112.0481.3071791.1280.6381840.0874020.5507812.0232201.9247312.039783
124.0962.6493202.3290.6467290.0781250.5686042.0267791.7877092.064716
138.1925.3075774.7300.6478270.0704350.5773932.0033971.8031252.030915
1416.38410.6761.0229.6540.6516110.0623780.5892332.0116831.7712302.041015
1532.76821.4791.86719.6120.6554870.0569760.5985112.0118961.8268102.031490
1665.53643.2253.48639.7390.6595610.0531920.6063692.0124311.8671672.026259
17131.07286.6706.45080.2200.6612400.0492100.6120302.0050901.8502582.018672
18262.144173.82512.203161.6220.6630900.0465510.6165392.0055961.8919382.014735
19524.288348.60022.864325.7360.6649020.0436100.6212922.0054651.8736382.015419
201.048.576698.52243.154655.3680.6661620.0411550.6250082.0037921.8874212.011961
212.097.1521.400.10981.5311.318.5780.6676240.0388770.6287472.0043881.8893032.011966
224.194.3042.804.726154.8612.649.8650.6686990.0369220.6317772.0032201.8994132.009638
238.388.6085.618.107295.2085.322.8990.6697310.0351920.6345392.0030861.9062772.008744
2416.777.21611.252.797562.99810.689.7990.6707190.0335570.6371622.0029521.9071232.008266
2533.554.43222.534.1791.077.62521.456.5540.6715710.0321160.6394552.0025401.9140832.007199
2667.108.86445.121.9912.066.61243.055.3790.6723700.0307950.6415752.0023801.9177472.006631
27134.217.72890.345.7373.968.92286.376.8150.6731280.0295710.6435572.0022551.9204972.006179
28268.435.456180.877.3037.633.466173.243.8370.6738200.0284370.6453842.0020571.9233102.005675
29536.870.912362.103.18514.703.419347.399.7660.6744700.0273870.6470832.0019271.9261792.005265
301.073.741.824724.859.20528.361.236696.497.9690.6750780.0264130.6486642.0018031.9288872.004889
312.147.483.6481.450.939.65354.772.1221.396.167.5310.6756460.0255050.6501412.0016851.9312322.004554
324.294.967.2962.904.161.904105.918.7942.798.243.1100.6761780.0246610.6515172.0015731.9338092.004231
338.589.934.5925.812.669.917205.042.8245.607.627.0930.6766840.0238700.6528142.0014971.9358492.003981
3417.179.869.18411.633.502.821397.332.77911.236.170.0420.6771590.0231280.6540312.0014041.9378042.003730
3534.359.738.36823.282.389.770770.716.64422.511.673.1260.6776070.0224310.6551762.0013221.9397262.003501
3668.719.476.73646.593.769.4161.496.331.94745.097.437.4690.6780290.0217740.6562542.0012451.9414812.003291


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
122100200
243200201
383200201
4165400401
5327511402
66412742523
7128171153725
825628161171146
951254302312191112
101.02493464623242323
112.048179869250414246
124.09632015916084777683
138.192577289287147140140150
1416.3841.022520501261261240260
1532.7681.867966900474482426485
1665.5363.4861.7911.694857892837900
17131.0726.4503.2933.1561.6061.6321.5501.662
18262.14412.2036.1986.0043.0133.0942.9913.105
19524.28822.86411.56111.3025.6285.7635.6745.799
201.048.57643.15421.81221.34110.62310.87910.71810.934
212.097.15281.53141.20040.33020.11220.61220.21820.589
224.194.304154.86178.47376.38738.15939.25438.22839.220
238.388.608295.208149.367145.84072.91974.64972.92174.719
2416.777.216562.998284.744278.253138.963142.463139.290142.282
2533.554.4321.077.625544.861532.763266.399272.261266.364272.601
2667.108.8642.066.6121.043.8781.022.733511.573521.328511.160522.551
27134.217.7283.968.9222.003.6441.965.277983.0061.001.277982.2711.002.368
28268.435.4567.633.4663.852.1453.781.3201.891.1981.925.4431.890.1221.926.703
29536.870.91214.703.4197.419.0947.284.3243.642.0273.708.8463.642.2973.710.249
301.073.741.82428.361.23614.303.70114.057.5347.029.1747.150.7807.028.3607.152.922
312.147.483.64854.772.12227.615.97827.156.14313.577.02813.805.70813.579.11513.810.271
324.294.967.296105.918.79453.394.20252.524.59126.265.78026.693.95826.258.81126.700.245
338.589.934.592205.042.824103.337.308101.705.51550.856.44451.665.33350.849.07151.671.976
3417.179.869.184397.332.779200.207.451197.125.32798.565.884100.097.78998.559.443100.109.663
3534.359.738.368770.716.644388.257.603382.459.040191.223.747194.124.169191.235.293194.133.435
3668.719.476.7361.496.331.947753.649.590742.682.356371.335.386376.819.443371.346.970376.830.148


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
121010100
241010100
383120102
4166240312
53215694524
6642711166759
712858253310151518
8256132656728333833
951226611814857727166
101.024553266287123149142139
112.0481.128539589274277298279
124.0962.3291.1351.194559554621595
138.1924.7302.3612.3691.1781.1361.2271.189
1416.3849.6544.7994.8552.3902.3862.4832.395
1532.76819.6129.8139.7994.8694.9195.0244.800
1665.53639.73919.92819.8119.8459.96610.1029.826
17131.07280.22040.14740.07319.93520.11020.21619.959
18262.144161.62280.79680.82640.33040.44040.56840.284
19524.288325.736162.529163.20781.30181.40781.55481.474
201.048.576655.368327.230328.138164.155163.710164.240163.263
212.097.1521.318.578659.088659.490329.937329.566329.879329.196
224.194.3042.649.8651.325.3551.324.510662.583661.973662.751662.558
238.388.6085.322.8992.661.5522.661.3471.330.7301.330.8771.330.6171.330.675
2416.777.21610.689.7995.347.5685.342.2312.672.3282.672.2412.673.5292.671.701
2533.554.43221.456.55410.730.03910.726.5155.365.9025.361.2905.369.8545.359.508
2667.108.86443.055.37921.531.69721.523.68210.767.20410.759.20110.772.22310.756.751
27134.217.72886.376.81543.194.67543.182.14021.602.99821.587.34821.606.61821.579.851
28268.435.456173.243.83786.632.27186.611.56643.327.67443.296.01743.329.75943.290.387
29536.870.912347.399.766173.726.496173.673.27086.884.15786.802.55886.893.33086.819.721
301.073.741.824696.497.969348.294.334348.203.635174.191.297174.046.902174.198.958174.060.812
312.147.483.6481.396.167.531698.166.301698.001.230349.163.042348.918.190349.169.860348.916.439
324.294.967.2962.798.243.1101.399.240.1591.399.002.951699.773.069699.338.205699.792.153699.339.683
338.589.934.5925.607.627.0932.803.980.5302.803.646.5631.402.332.9541.401.518.1801.402.319.5471.401.456.412
3417.179.869.18411.236.170.0425.618.445.6085.617.724.4342.809.841.5962.808.284.7482.809.832.9402.808.210.758
3534.359.738.36822.511.673.12611.256.468.46711.255.204.6595.629.390.4875.626.546.5315.629.363.6495.626.372.459
3668.719.476.73645.097.437.46922.549.970.74422.547.466.72511.277.105.80011.271.711.33711.277.133.50611.271.486.826


8. Check for existing Integer Sequences by OEIS

Found in Database : 19, 3, 11, 31, 1, 7, 23, 1, 79, 139, 107, 61, 59, 1, 1, 283, 1, 113, 67, 1,
Found in Database : 19, 3, 11, 31, 7, 23, 79, 139, 107, 61, 59, 283, 113, 67, 41, 463, 331, 103, 97, 479, 1601, 281, 197, 2141, 373, 53,
Found in Database : 3, 7, 11, 19, 23, 31, 41, 53, 59, 61, 67, 79, 97, 103, 107, 113, 127, 139,