Inhaltsverzeichnis

Development of
Algorithmic Constructions

18:14:45
Deutsch
28.Mar 2024

Polynom = x^2+134x-3

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) = 3 = 3
f(1) = 33 = 3*11
f(2) = 269 = 269
f(3) = 51 = 3*17
f(4) = 549 = 3*3*61
f(5) = 173 = 173
f(6) = 837 = 3*3*3*31
f(7) = 123 = 3*41
f(8) = 1133 = 11*103
f(9) = 321 = 3*107
f(10) = 1437 = 3*479
f(11) = 199 = 199
f(12) = 1749 = 3*11*53
f(13) = 477 = 3*3*53
f(14) = 2069 = 2069
f(15) = 279 = 3*3*31
f(16) = 2397 = 3*17*47
f(17) = 641 = 641
f(18) = 2733 = 3*911
f(19) = 363 = 3*11*11
f(20) = 3077 = 17*181
f(21) = 813 = 3*271
f(22) = 3429 = 3*3*3*127
f(23) = 451 = 11*41
f(24) = 3789 = 3*3*421
f(25) = 993 = 3*331
f(26) = 4157 = 4157
f(27) = 543 = 3*181
f(28) = 4533 = 3*1511
f(29) = 1181 = 1181
f(30) = 4917 = 3*11*149
f(31) = 639 = 3*3*71
f(32) = 5309 = 5309
f(33) = 1377 = 3*3*3*3*17
f(34) = 5709 = 3*11*173
f(35) = 739 = 739
f(36) = 6117 = 3*2039
f(37) = 1581 = 3*17*31
f(38) = 6533 = 47*139
f(39) = 843 = 3*281
f(40) = 6957 = 3*3*773
f(41) = 1793 = 11*163
f(42) = 7389 = 3*3*821
f(43) = 951 = 3*317
f(44) = 7829 = 7829
f(45) = 2013 = 3*11*61
f(46) = 8277 = 3*31*89
f(47) = 1063 = 1063
f(48) = 8733 = 3*41*71
f(49) = 2241 = 3*3*3*83
f(50) = 9197 = 17*541
f(51) = 1179 = 3*3*131
f(52) = 9669 = 3*11*293
f(53) = 2477 = 2477
f(54) = 10149 = 3*17*199
f(55) = 1299 = 3*433
f(56) = 10637 = 11*967
f(57) = 2721 = 3*907
f(58) = 11133 = 3*3*1237
f(59) = 1423 = 1423
f(60) = 11637 = 3*3*3*431
f(61) = 2973 = 3*991
f(62) = 12149 = 12149
f(63) = 1551 = 3*11*47
f(64) = 12669 = 3*41*103
f(65) = 3233 = 53*61
f(66) = 13197 = 3*53*83
f(67) = 1683 = 3*3*11*17
f(68) = 13733 = 31*443
f(69) = 3501 = 3*3*389
f(70) = 14277 = 3*4759
f(71) = 1819 = 17*107
f(72) = 14829 = 3*4943
f(73) = 3777 = 3*1259
f(74) = 15389 = 11*1399
f(75) = 1959 = 3*653
f(76) = 15957 = 3*3*3*3*197
f(77) = 4061 = 31*131
f(78) = 16533 = 3*3*11*167
f(79) = 2103 = 3*701
f(80) = 17117 = 17117
f(81) = 4353 = 3*1451
f(82) = 17709 = 3*5903
f(83) = 2251 = 2251
f(84) = 18309 = 3*17*359
f(85) = 4653 = 3*3*11*47
f(86) = 18917 = 18917
f(87) = 2403 = 3*3*3*89
f(88) = 19533 = 3*17*383
f(89) = 4961 = 11*11*41
f(90) = 20157 = 3*6719
f(91) = 2559 = 3*853
f(92) = 20789 = 20789
f(93) = 5277 = 3*1759
f(94) = 21429 = 3*3*2381
f(95) = 2719 = 2719
f(96) = 22077 = 3*3*11*223
f(97) = 5601 = 3*1867
f(98) = 22733 = 127*179
f(99) = 2883 = 3*31*31
f(100) = 23397 = 3*11*709

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+134x-3

f(0)=3
f(1)=11
f(2)=269
f(3)=17
f(4)=61
f(5)=173
f(6)=31
f(7)=41
f(8)=103
f(9)=107
f(10)=479
f(11)=199
f(12)=53
f(13)=1
f(14)=2069
f(15)=1
f(16)=47
f(17)=641
f(18)=911
f(19)=1
f(20)=181
f(21)=271
f(22)=127
f(23)=1
f(24)=421
f(25)=331
f(26)=4157
f(27)=1
f(28)=1511
f(29)=1181
f(30)=149
f(31)=71
f(32)=5309
f(33)=1
f(34)=1
f(35)=739
f(36)=2039
f(37)=1
f(38)=139
f(39)=281
f(40)=773
f(41)=163
f(42)=821
f(43)=317
f(44)=7829
f(45)=1
f(46)=89
f(47)=1063
f(48)=1
f(49)=83
f(50)=541
f(51)=131
f(52)=293
f(53)=2477
f(54)=1
f(55)=433
f(56)=967
f(57)=907
f(58)=1237
f(59)=1423
f(60)=431
f(61)=991
f(62)=12149
f(63)=1
f(64)=1
f(65)=1
f(66)=1
f(67)=1
f(68)=443
f(69)=389
f(70)=4759
f(71)=1
f(72)=4943
f(73)=1259
f(74)=1399
f(75)=653
f(76)=197
f(77)=1
f(78)=167
f(79)=701
f(80)=17117
f(81)=1451
f(82)=5903
f(83)=2251
f(84)=359
f(85)=1
f(86)=18917
f(87)=1
f(88)=383
f(89)=1
f(90)=6719
f(91)=853
f(92)=20789
f(93)=1759
f(94)=2381
f(95)=2719
f(96)=223
f(97)=1867
f(98)=179
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+134x-3 could be written as f(y)= y^2-4492 with x=y-67

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+67
f'(x)>2x+133

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

3, 11, 269, 17, 61, 173, 31, 41, 103, 107, 479, 199, 53, 1, 2069, 1, 47, 641, 911, 1, 181, 271, 127, 1, 421, 331, 4157, 1, 1511, 1181, 149, 71, 5309, 1, 1, 739, 2039, 1, 139, 281, 773, 163, 821, 317, 7829, 1, 89, 1063, 1, 83, 541, 131, 293, 2477, 1, 433, 967, 907, 1237, 1423, 431, 991, 12149, 1, 1, 1, 1, 1, 443, 389, 4759, 1, 4943, 1259, 1399, 653, 197, 1, 167, 701, 17117, 1451, 5903, 2251, 359, 1, 18917, 1, 383, 1, 6719, 853, 20789, 1759, 2381, 2719, 223, 1867, 179, 1, 709, 349, 113, 1, 24749, 1, 1, 1, 1, 2207, 571, 1, 3061, 6977, 1, 1193, 1, 2447, 1, 1, 10159, 857, 1, 439, 10663, 8093, 1, 1381, 33533, 257, 1, 4339, 1, 1, 241, 1, 12239, 9281, 12511, 1, 1, 1, 13063, 4951, 1213, 3371, 997, 1721, 4637, 1, 4733, 1, 2557, 3659, 14783, 509, 887, 1, 46133, 647, 15679, 1, 1453, 2017, 48869, 4111, 503, 1, 1879, 251, 1667, 1, 1, 1, 17863, 751, 613, 1, 1, 1, 18839, 4751, 1, 2417, 1, 14753, 1, 1, 5503, 5087, 1, 7759, 673, 1753, 63629, 1, 21559, 16301, 21911, 1, 66797, 1, 7541, 1, 1, 5791, 6367, 1, 1, 17921, 1, 337, 1789, 2053, 1, 1, 1, 577, 76733, 3221, 1, 1783, 2927, 1, 1, 1, 1, 10243, 1, 2309, 83717, 1171, 1, 21377, 28703, 3613, 87317, 1, 1093, 1013, 9973, 1, 90989, 347, 1, 1, 31159, 1307, 1553, 883, 2909, 1, 32423, 1, 1, 4133, 11093, 25121, 661, 4241, 102437, 1, 34583, 1, 35023, 1, 106397, 1487, 35911, 1, 353, 1, 10039, 1, 12421, 827, 1, 1, 2161, 4801, 38639, 29153, 39103, 1, 118709, 1, 40039, 1373, 2383, 1, 122957, 5153, 1, 31277, 1, 5273, 127277, 10667, 1, 16183, 43399, 3637, 373, 1, 44383, 1, 44879, 5641, 136133, 1, 1, 17299, 15461, 1, 863, 1, 1, 1, 47911, 1, 1, 4057, 2879, 18451, 49463, 401, 8821, 1, 1, 1, 1, 1, 2179, 12959, 52103, 1, 52639, 4409, 14503, 1, 53719, 40493, 4933, 1, 164429, 1, 6151, 1, 18637, 1277, 2777, 1, 449, 3907, 1, 2411, 1, 1, 58711, 22123, 1, 14891, 179549, 7517, 1831, 45533, 20333, 1, 184733, 15467, 1019, 2129, 62743, 1, 189989, 1, 1559, 2833, 2081, 8101, 1129, 1, 1, 467, 1, 16651, 1, 1, 67511, 1, 4007, 2851, 1, 523, 4079, 26119, 69959, 1597, 2551, 8861, 1, 53633, 1, 1, 19759, 18191, 73079, 1619, 6701, 1, 223037, 1, 1, 56477, 1609, 1, 228797, 1, 1, 2633, 1, 19471, 234629, 9817, 4639, 59393, 7229, 1109, 14149, 6709, 7349, 499, 547, 1, 246509, 10313, 521, 1, 1031, 1, 8147, 1, 1, 1, 1, 1, 258677, 3607, 7901, 2111, 1, 10993, 1, 22159, 9887, 1, 1, 1, 1, 1, 91079, 1459, 5399, 1, 3343, 2579, 93199, 35083, 93911, 23567, 1, 1, 1, 71777, 1, 1, 2819, 1, 97511, 1, 3169, 1, 296909, 4139, 557, 6823, 2137, 12601, 1, 1, 1, 1, 1, 25759, 310229, 12973, 1, 78401, 104911, 1, 1, 8837, 3433, 1, 107183, 26891, 323837, 1231, 1, 4813, 36493, 1, 330749, 1627, 10093, 1, 111799, 9349, 30703, 1, 113359, 85313, 114143, 1, 1993, 2621, 2269, 43543, 1, 2657, 1, 14713, 1327, 1, 118903, 1657, 32647, 10009, 120511, 45343, 1, 30431, 1, 1, 1, 1303, 13751, 1, 1, 31247, 125399, 4289, 126223, 1, 3373, 1, 3119, 96221, 11701, 16141, 22861, 32491, 1, 1, 1, 32911, 396197, 16561, 132911, 9091, 1, 5591, 403829, 1, 135463, 1, 136319, 2011, 2089, 1, 1, 1, 46301, 17417, 38119, 35051, 140639, 52903, 141511, 3943, 1, 1, 143263, 107777, 1, 1, 8209, 1, 2861, 1, 5437, 1, 1, 18517, 4793, 111773, 1, 6247, 2267, 12569, 151279, 3347, 152183, 3469, 7529, 1, 1, 10531, 51637, 19421, 467477, 1, 3823, 1, 1, 13177, 1, 1, 1, 119981, 1933, 20113, 28477, 40459, 54101, 1, 54413, 40927, 1, 1871, 1, 124193, 166063, 1, 501029, 1, 15269, 761, 3187, 1, 46327, 1, 56941, 128477, 19087, 1, 518237, 1, 1, 65323, 174679, 1, 1, 1, 1, 132833, 1, 1, 1, 1, 1, 1, 5471, 45259, 11587, 1, 182519, 1, 183511, 1, 553517, 1, 1801, 1, 1, 46751, 562517, 1, 2027, 141761, 5743, 23753, 571589, 1, 1583, 72019, 1, 1, 580733, 8087, 11447, 1, 3691, 24517, 14389, 4481, 1, 74323, 22079, 49807, 6733, 25033, 18253, 1, 201823, 8431, 1, 1, 203911, 2473, 4999, 1, 618029, 2347, 7669, 155693, 1, 2371, 1, 1, 1, 79039, 1613, 1, 3407, 2957, 3499, 160481, 19501, 26881, 646757, 1, 72221, 1733, 1, 1, 3931, 1, 219911, 15031, 7129, 1, 666269, 18553, 4211, 1, 20389, 56207, 1, 1, 6863, 4153, 8429, 28517, 40357, 1, 229799, 7853, 1, 19289, 696077, 881, 233143, 175277, 1, 1, 706157, 1, 2389, 88903, 1, 1, 65119, 29917, 1889, 1, 5879, 10067, 726533, 1, 1, 91459, 244463, 5569, 1, 30773, 1, 3041, 4861, 1, 67927, 1, 1, 1, 22853, 2333, 1, 1, 4787, 1, 254879, 2903, 1, 1, 1, 1, 28711, 64747, 778733, 1913, 2437, 6323, 23813, 10939, 789389, 21977, 24029, 2423, 265511, 937, 800117, 33413, 1, 18307, 1, 33713, 47701, 1, 4451, 102043, 272719, 22777, 4591, 1, 25013, 206813, 276359, 1, 75703, 4091, 1, 104779, 3011, 4127, 2203, 3203, 1, 1, 4651, 1, 1, 1, 6089, 1, 1, 1, 1, 36161, 2357, 217901, 1, 36473, 6311, 73259, 9473, 110359, 294919, 2239, 888533, 1, 297439, 20323, 1, 1, 1, 1, 3719, 6659, 1, 75787, 911357, 38053, 27733, 2029, 2711, 12791, 11119, 8563, 1, 1, 1, 77711, 934469, 3547, 6133, 7583, 104693, 39341, 946133, 1, 28789, 1, 317983, 1, 1, 13331, 1, 14173, 321911, 1, 969677, 1, 1, 121951, 1, 1, 1, 1, 328511, 1, 1, 1, 1, 27653, 19559, 1, 1, 1, 59149, 41981, 1, 252893, 112621, 3847, 1, 2741, 340559, 11633, 341911, 28549, 25117, 1, 1, 258977, 1, 2549, 22171, 1, 10567, 131023, 1, 87691, 12703, 44017, 1, 24103, 1, 1, 1, 2699, 356959, 1, 1, 89759, 1079189, 45053, 1, 3049, 1, 1, 1, 91151, 365303, 137251, 366703, 1, 3769, 1, 1, 16333, 370919, 1, 1116989, 1, 1, 140419, 125053, 93967, 3313, 47161,

6. Sequence of the polynom (only primes)

3, 11, 269, 17, 61, 173, 31, 41, 103, 107, 479, 199, 53, 2069, 47, 641, 911, 181, 271, 127, 421, 331, 4157, 1511, 1181, 149, 71, 5309, 739, 2039, 139, 281, 773, 163, 821, 317, 7829, 89, 1063, 83, 541, 131, 293, 2477, 433, 967, 907, 1237, 1423, 431, 991, 12149, 443, 389, 4759, 4943, 1259, 1399, 653, 197, 167, 701, 17117, 1451, 5903, 2251, 359, 18917, 383, 6719, 853, 20789, 1759, 2381, 2719, 223, 1867, 179, 709, 349, 113, 24749, 2207, 571, 3061, 6977, 1193, 2447, 10159, 857, 439, 10663, 8093, 1381, 33533, 257, 4339, 241, 12239, 9281, 12511, 13063, 4951, 1213, 3371, 997, 1721, 4637, 4733, 2557, 3659, 14783, 509, 887, 46133, 647, 15679, 1453, 2017, 48869, 4111, 503, 1879, 251, 1667, 17863, 751, 613, 18839, 4751, 2417, 14753, 5503, 5087, 7759, 673, 1753, 63629, 21559, 16301, 21911, 66797, 7541, 5791, 6367, 17921, 337, 1789, 2053, 577, 76733, 3221, 1783, 2927, 10243, 2309, 83717, 1171, 21377, 28703, 3613, 87317, 1093, 1013, 9973, 90989, 347, 31159, 1307, 1553, 883, 2909, 32423, 4133, 11093, 25121, 661, 4241, 102437, 34583, 35023, 106397, 1487, 35911, 353, 10039, 12421, 827, 2161, 4801, 38639, 29153, 39103, 118709, 40039, 1373, 2383, 122957, 5153, 31277, 5273, 127277, 10667, 16183, 43399, 3637, 373, 44383, 44879, 5641, 136133, 17299, 15461, 863, 47911, 4057, 2879, 18451, 49463, 401, 8821, 2179, 12959, 52103, 52639, 4409, 14503, 53719, 40493, 4933, 164429, 6151, 18637, 1277, 2777, 449, 3907, 2411, 58711, 22123, 14891, 179549, 7517, 1831, 45533, 20333, 184733, 15467, 1019, 2129, 62743, 189989, 1559, 2833, 2081, 8101, 1129, 467, 16651, 67511, 4007, 2851, 523, 4079, 26119, 69959, 1597, 2551, 8861, 53633, 19759, 18191, 73079, 1619, 6701, 223037, 56477, 1609, 228797, 2633, 19471, 234629, 9817, 4639, 59393, 7229, 1109, 14149, 6709, 7349, 499, 547, 246509, 10313, 521, 1031, 8147, 258677, 3607, 7901, 2111, 10993, 22159, 9887, 91079, 1459, 5399, 3343, 2579, 93199, 35083, 93911, 23567, 71777, 2819, 97511, 3169, 296909, 4139, 557, 6823, 2137, 12601, 25759, 310229, 12973, 78401, 104911, 8837, 3433, 107183, 26891, 323837, 1231, 4813, 36493, 330749, 1627, 10093, 111799, 9349, 30703, 113359, 85313, 114143, 1993, 2621, 2269, 43543, 2657, 14713, 1327, 118903, 1657, 32647, 10009, 120511, 45343, 30431, 1303, 13751, 31247, 125399, 4289, 126223, 3373, 3119, 96221, 11701, 16141, 22861, 32491, 32911, 396197, 16561, 132911, 9091, 5591, 403829, 135463, 136319, 2011, 2089, 46301, 17417, 38119, 35051, 140639, 52903, 141511, 3943, 143263, 107777, 8209, 2861, 5437, 18517, 4793, 111773, 6247, 2267, 12569, 151279, 3347, 152183, 3469, 7529, 10531, 51637, 19421, 467477, 3823, 13177, 119981, 1933, 20113, 28477, 40459, 54101, 54413, 40927, 1871, 124193, 166063, 501029, 15269, 761, 3187, 46327, 56941, 128477, 19087, 518237, 65323, 174679, 132833, 5471, 45259, 11587, 182519, 183511, 553517, 1801, 46751, 562517, 2027, 141761, 5743, 23753, 571589, 1583, 72019, 580733, 8087, 11447, 3691, 24517, 14389, 4481, 74323, 22079, 49807, 6733, 25033, 18253, 201823, 8431, 203911, 2473, 4999, 618029, 2347, 7669, 155693, 2371, 79039, 1613, 3407, 2957, 3499, 160481, 19501, 26881, 646757, 72221, 1733, 3931, 219911, 15031, 7129, 666269, 18553, 4211, 20389, 56207, 6863, 4153, 8429, 28517, 40357, 229799, 7853, 19289, 696077, 881, 233143, 175277, 706157, 2389, 88903, 65119, 29917, 1889, 5879, 10067, 726533, 91459, 244463, 5569, 30773, 3041, 4861, 67927, 22853, 2333, 4787, 254879, 2903, 28711, 64747, 778733, 1913, 2437, 6323, 23813, 10939, 789389, 21977, 24029, 2423, 265511, 937, 800117, 33413, 18307, 33713, 47701, 4451, 102043, 272719, 22777, 4591, 25013, 206813, 276359, 75703, 4091, 104779, 3011, 4127, 2203, 3203, 4651, 6089, 36161, 2357, 217901, 36473, 6311, 73259, 9473, 110359, 294919, 2239, 888533, 297439, 20323, 3719, 6659, 75787, 911357, 38053, 27733, 2029, 2711, 12791, 11119, 8563, 77711, 934469, 3547, 6133, 7583, 104693, 39341, 946133, 28789, 317983, 13331, 14173, 321911, 969677, 121951, 328511, 27653, 19559, 59149, 41981, 252893, 112621, 3847, 2741, 340559, 11633, 341911, 28549, 25117, 258977, 2549, 22171, 10567, 131023, 87691, 12703, 44017, 24103, 2699, 356959, 89759, 1079189, 45053, 3049, 91151, 365303, 137251, 366703, 3769, 16333, 370919, 1116989, 140419, 125053, 93967, 3313, 47161,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+134x-3 and
the reducible primes which appear as divisor for the first time
p | x^2+134x-3 and p < x^2+134x-3

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)
11011381.1000000.3000000.8000000.0000000.0000000.000000
21007620560.7600000.2000000.5600006.9090916.6666677.000000
31.0006241334910.6240000.1330000.4910008.2105266.6500008.767858
410.0006.4038975.5060.6403000.0897000.55060010.2612186.74436111.213849
5100.00065.2546.98958.2650.6525400.0698900.58265010.1911607.79152710.582092
61.000.000660.12556.757603.3680.6601250.0567570.60336810.1162388.12090410.355582
710.000.0006.647.285479.1436.168.1420.6647290.0479140.61681410.0697368.44200710.222853
8100.000.00066.833.1694.140.00962.693.1600.6683320.0414000.62693210.0542058.64044610.164026
91.000.000.000671.101.58636.423.587634.677.9990.6711020.0364240.63467810.0414458.79794910.123561
1010.000.000.0006.733.199.340325.135.1946.408.064.1460.6733200.0325140.64080610.0330558.92650110.096559


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
389361.1250000.3750000.7500001.8000001.5000002.000000
416155100.9375000.3125000.6250001.6666671.6666671.666667
532289190.8750000.2812500.5937501.8666671.8000001.900000
6645015350.7812500.2343750.5468751.7857141.6666671.842105
71289124670.7109380.1875000.5234381.8200001.6000001.914286
8256166411250.6484380.1601560.4882811.8241761.7083331.865672
9512318762420.6210940.1484380.4726561.9156631.8536591.936000
101.0246361365000.6210940.1328120.4882812.0000001.7894742.066116
112.0481.2842411.0430.6269530.1176760.5092772.0188681.7720592.086000
124.0962.5934252.1680.6330570.1037600.5292972.0194701.7634852.078619
138.1925.2417604.4810.6397710.0927730.5469972.0212111.7882352.066882
1416.38410.5281.4079.1210.6425780.0858760.5567022.0087771.8513162.035483
1532.76821.2012.58818.6130.6470030.0789790.5680242.0137731.8393752.040675
1665.53642.6114.83537.7760.6501920.0737760.5764162.0098581.8682382.029549
17131.07285.6488.90376.7450.6534420.0679240.5855182.0099971.8413652.031581
18262.144172.00616.815155.1910.6561510.0641440.5920072.0082901.8886892.022164
19524.288345.02531.599313.4260.6580830.0602700.5978132.0058891.8792152.019614
201.048.576692.24659.322632.9240.6601770.0565740.6036032.0063651.8773382.019373
212.097.1521.387.865112.5471.275.3180.6617860.0536670.6081192.0048731.8972222.014962
224.194.3042.781.975213.4272.568.5480.6632750.0508850.6123902.0045001.8963372.014045
238.388.6085.574.130406.8425.167.2880.6644880.0484990.6159892.0036591.9062352.011755
2416.777.21611.167.344776.77210.390.5720.6656260.0462990.6193262.0034241.9092722.010837
2533.554.43222.372.9801.484.97220.888.0080.6667670.0442560.6225112.0034291.9117222.010285
2667.108.86444.814.5732.844.62741.969.9460.6677890.0423880.6254012.0030671.9156102.009284
27134.217.72889.754.9805.459.81684.295.1640.6687270.0406790.6280482.0028081.9193432.008465
28268.435.456179.745.62310.495.814169.249.8090.6696050.0391000.6305052.0026261.9223752.007823
29536.870.912359.925.29320.208.755339.716.5380.6704130.0376420.6327712.0024151.9254112.007190
301.073.741.824720.669.49838.964.928681.704.5700.6711760.0362890.6348872.0022751.9281212.006686
312.147.483.6481.442.878.04775.213.6811.367.664.3660.6718920.0350240.6368682.0021361.9302922.006242
324.294.967.2962.888.623.828145.373.1532.743.250.6750.6725600.0338470.6387132.0019871.9328022.005792
338.589.934.5925.782.620.506281.284.2845.501.336.2220.6731860.0327460.6404402.0018601.9349122.005408
3417.179.869.18411.575.375.904544.859.80111.030.516.1030.6737760.0317150.6420612.0017531.9370432.005061
3534.359.738.36823.169.805.1221.056.471.94022.113.333.1820.6743300.0307470.6435832.0016461.9389802.004742
3668.719.476.73646.375.661.3132.050.363.45544.325.297.8580.6748550.0298370.6450182.0015561.9407652.004460


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
122010110
242010110
383020120
4165130131
5329171161
664154101293
71282461723154
825641103065246
9512761659129478
101.0241363210319178416
112.04824166174313214335
124.096425118306585924860
138.19276020755210098452110
1416.3841.4073841.022175183847202
1532.7682.5886861.9013333231.568364
1665.5364.8351.2453.5896226112.967635
17131.0728.9032.3196.5831.1361.1635.4471.157
18262.14416.8154.36812.4462.1502.19510.2962.174
19524.28831.5998.29023.3084.0174.17619.2914.115
201.048.57659.32215.60643.7157.5207.83136.1957.776
212.097.152112.54729.47883.06814.33814.75168.73014.728
224.194.304213.42755.817157.60927.17227.916130.43727.902
238.388.608406.842106.053300.78851.66452.892249.12453.162
2416.777.216776.772202.229574.54298.577100.676475.965101.554
2533.554.4321.484.972386.1201.098.851188.252192.716910.599193.405
2667.108.8642.844.627738.0092.106.617360.933368.4501.745.684369.560
27134.217.7285.459.8161.414.1394.045.676692.908705.8693.352.768708.271
28268.435.45610.495.8142.715.0837.780.7301.331.4721.356.6196.449.2581.358.465
29536.870.91220.208.7555.221.02214.987.7322.562.5632.610.76912.425.1692.610.254
301.073.741.82438.964.92810.053.66028.911.2674.936.6825.027.59723.974.5855.026.064
312.147.483.64875.213.68119.384.24555.829.4359.523.2179.692.84046.306.2189.691.406
324.294.967.296145.373.15337.426.925107.946.22718.396.32918.714.62789.549.89818.712.299
338.589.934.592281.284.28472.345.213208.939.07035.576.66636.174.549173.362.40436.170.665
3417.179.869.184544.859.801140.018.189404.841.61168.888.37470.011.682335.953.23770.006.508
3534.359.738.3681.056.471.940271.265.695785.206.244133.532.202135.641.794651.674.042135.623.902
3668.719.476.7362.050.363.455526.039.7741.524.323.680259.080.003263.025.3681.265.243.677263.014.407


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
243121110
386332112
41610372224
532198112359
664351619471113
71286730376122128
8256125596613184153
9512242118124353566106
101.0245002592417977139205
112.0481.043547496169182303389
124.0962.1681.0931.075380398583807
138.1924.4812.2672.2148368561.1961.593
1416.3849.1214.6874.4341.7671.8292.4113.114
1532.76818.6139.5729.0413.7743.7774.8546.208
1665.53637.77619.35618.4207.7737.8219.89312.289
17131.07276.74539.41537.33016.02416.18520.00124.535
18262.144155.19179.53975.65232.93233.26940.28248.708
19524.288313.426160.121153.30567.81267.76981.12396.722
201.048.576632.924322.858310.066138.447138.245163.357192.875
212.097.1521.275.318649.845625.473281.650281.427328.150384.091
224.194.3042.568.5481.306.9881.261.560571.439571.087660.214765.808
238.388.6085.167.2882.626.5252.540.7631.157.1821.157.3881.326.4601.526.258
2416.777.21610.390.5725.275.8845.114.6882.341.4112.340.3772.662.5883.046.196
2533.554.43220.888.00810.597.07210.290.9364.733.1474.729.6845.344.0386.081.139
2667.108.86441.969.94621.278.55820.691.3889.557.3149.551.77410.723.31812.137.540
27134.217.72884.295.16442.706.11941.589.04519.280.59619.269.60621.515.66624.229.296
28268.435.456169.249.80985.699.80083.550.00938.866.58038.848.94043.162.92648.371.363
29536.870.912339.716.538171.916.042167.800.49678.303.04978.255.21286.568.68896.589.589
301.073.741.824681.704.570344.824.857336.879.713157.668.087157.571.246173.574.080192.891.157
312.147.483.6481.367.664.366691.493.046676.171.320317.300.423317.120.082347.961.484385.282.377
324.294.967.2962.743.250.6751.386.412.9321.356.837.743638.300.501637.923.365697.453.224769.573.585
338.589.934.5925.501.336.2222.779.206.8762.722.129.3461.283.425.7931.282.758.6261.397.770.4091.537.381.394
3417.179.869.18411.030.516.1035.570.458.4445.460.057.6592.579.721.6542.578.439.3162.800.901.4683.071.453.665
3534.359.738.36822.113.333.18211.163.615.96910.949.717.2135.183.630.5525.181.025.4965.612.025.3926.136.651.742
3668.719.476.73644.325.297.85822.370.043.03821.955.254.82010.412.662.20610.407.700.67211.243.284.57312.261.650.407


8. Check for existing Integer Sequences by OEIS

Found in Database : 3, 11, 269, 17, 61, 173, 31, 41, 103, 107, 479, 199, 53, 1, 2069, 1, 47, 641, 911, 1,
Found in Database : 3, 11, 269, 17, 61, 173, 31, 41, 103, 107, 479, 199, 53, 2069, 47, 641, 911, 181, 271, 127, 421, 331, 4157, 1511, 1181, 149, 71, 5309, 739, 2039, 139, 281,
Found in Database : 3, 11, 17, 31, 41, 47, 53, 61, 71, 83, 89, 103, 107, 113, 127, 131, 139, 149,