Inhaltsverzeichnis

Development of
Algorithmic Constructions

11:01:45
Deutsch
28.Mar 2024

Polynom = x^2-112x+23

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) = 23 = 23
f(1) = 11 = 11
f(2) = 197 = 197
f(3) = 19 = 19
f(4) = 409 = 409
f(5) = 1 = 1
f(6) = 613 = 613
f(7) = 89 = 89
f(8) = 809 = 809
f(9) = 113 = 113
f(10) = 997 = 997
f(11) = 17 = 17
f(12) = 1177 = 11*107
f(13) = 79 = 79
f(14) = 1349 = 19*71
f(15) = 179 = 179
f(16) = 1513 = 17*89
f(17) = 199 = 199
f(18) = 1669 = 1669
f(19) = 109 = 109
f(20) = 1817 = 23*79
f(21) = 59 = 59
f(22) = 1957 = 19*103
f(23) = 253 = 11*23
f(24) = 2089 = 2089
f(25) = 269 = 269
f(26) = 2213 = 2213
f(27) = 71 = 71
f(28) = 2329 = 17*137
f(29) = 149 = 149
f(30) = 2437 = 2437
f(31) = 311 = 311
f(32) = 2537 = 43*59
f(33) = 323 = 17*19
f(34) = 2629 = 11*239
f(35) = 167 = 167
f(36) = 2713 = 2713
f(37) = 43 = 43
f(38) = 2789 = 2789
f(39) = 353 = 353
f(40) = 2857 = 2857
f(41) = 361 = 19*19
f(42) = 2917 = 2917
f(43) = 23 = 23
f(44) = 2969 = 2969
f(45) = 187 = 11*17
f(46) = 3013 = 23*131
f(47) = 379 = 379
f(48) = 3049 = 3049
f(49) = 383 = 383
f(50) = 3077 = 17*181
f(51) = 193 = 193
f(52) = 3097 = 19*163
f(53) = 97 = 97
f(54) = 3109 = 3109
f(55) = 389 = 389
f(56) = 3113 = 11*283
f(57) = 389 = 389
f(58) = 3109 = 3109
f(59) = 97 = 97
f(60) = 3097 = 19*163
f(61) = 193 = 193
f(62) = 3077 = 17*181
f(63) = 383 = 383
f(64) = 3049 = 3049
f(65) = 379 = 379
f(66) = 3013 = 23*131
f(67) = 187 = 11*17
f(68) = 2969 = 2969
f(69) = 23 = 23
f(70) = 2917 = 2917
f(71) = 361 = 19*19
f(72) = 2857 = 2857
f(73) = 353 = 353
f(74) = 2789 = 2789
f(75) = 43 = 43
f(76) = 2713 = 2713
f(77) = 167 = 167
f(78) = 2629 = 11*239
f(79) = 323 = 17*19
f(80) = 2537 = 43*59
f(81) = 311 = 311
f(82) = 2437 = 2437
f(83) = 149 = 149
f(84) = 2329 = 17*137
f(85) = 71 = 71
f(86) = 2213 = 2213
f(87) = 269 = 269
f(88) = 2089 = 2089
f(89) = 253 = 11*23
f(90) = 1957 = 19*103
f(91) = 59 = 59
f(92) = 1817 = 23*79
f(93) = 109 = 109
f(94) = 1669 = 1669
f(95) = 199 = 199
f(96) = 1513 = 17*89
f(97) = 179 = 179
f(98) = 1349 = 19*71
f(99) = 79 = 79
f(100) = 1177 = 11*107

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^2-112x+23

f(0)=23
f(1)=11
f(2)=197
f(3)=19
f(4)=409
f(5)=1
f(6)=613
f(7)=89
f(8)=809
f(9)=113
f(10)=997
f(11)=17
f(12)=107
f(13)=79
f(14)=71
f(15)=179
f(16)=1
f(17)=199
f(18)=1669
f(19)=109
f(20)=1
f(21)=59
f(22)=103
f(23)=1
f(24)=2089
f(25)=269
f(26)=2213
f(27)=1
f(28)=137
f(29)=149
f(30)=2437
f(31)=311
f(32)=43
f(33)=1
f(34)=239
f(35)=167
f(36)=2713
f(37)=1
f(38)=2789
f(39)=353
f(40)=2857
f(41)=1
f(42)=2917
f(43)=1
f(44)=2969
f(45)=1
f(46)=131
f(47)=379
f(48)=3049
f(49)=383
f(50)=181
f(51)=193
f(52)=163
f(53)=97
f(54)=3109
f(55)=389
f(56)=283
f(57)=1
f(58)=1
f(59)=1
f(60)=1
f(61)=1
f(62)=1
f(63)=1
f(64)=1
f(65)=1
f(66)=1
f(67)=1
f(68)=1
f(69)=1
f(70)=1
f(71)=1
f(72)=1
f(73)=1
f(74)=1
f(75)=1
f(76)=1
f(77)=1
f(78)=1
f(79)=1
f(80)=1
f(81)=1
f(82)=1
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^2-112x+23 could be written as f(y)= y^2-3113 with x=y+56

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-56
f'(x)>2x-113 with x > 56

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

23, 11, 197, 19, 409, 1, 613, 89, 809, 113, 997, 17, 107, 79, 71, 179, 1, 199, 1669, 109, 1, 59, 103, 1, 2089, 269, 2213, 1, 137, 149, 2437, 311, 43, 1, 239, 167, 2713, 1, 2789, 353, 2857, 1, 2917, 1, 2969, 1, 131, 379, 3049, 383, 181, 193, 163, 97, 3109, 389, 283, 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, 251, 1, 487, 1, 1, 1, 983, 139, 1, 1, 1511, 1, 1787, 241, 1, 277, 1, 157, 2663, 1, 2971, 1, 173, 431, 1, 1, 3943, 257, 4283, 557, 421, 601, 4987, 1, 5351, 1, 1, 739, 359, 787, 6491, 1, 1, 443, 317, 937, 7703, 1, 8123, 521, 503, 1, 1, 1151, 9431, 1, 9883, 1, 10343, 661, 569, 1381, 11287, 1, 1, 751, 12263, 1, 12763, 1627, 577, 1, 811, 439, 1301, 911, 14843, 1889, 15383, 1, 1, 1013, 16487, 1, 1, 1, 17623, 2239, 1, 1, 1, 1193, 19387, 1, 19991, 1, 1873, 1307, 1117, 673, 21851, 1, 1, 2851, 23131, 733, 1399, 1, 24443, 1, 25111, 3181, 1, 1, 1, 419, 1181, 1, 1, 3527, 28571, 1, 29287, 1, 30011, 3797, 433, 3889, 1657, 1, 1, 1019, 32987, 1, 33751, 1, 1, 1091, 821, 1, 1, 4561, 36887, 1, 37691, 2381, 1, 1, 39323, 4967, 40151, 461, 2411, 647, 709, 1, 42683, 1, 43543, 1, 499, 2803, 1, 1429, 46171, 5827, 2477, 5939, 47963, 1, 48871, 3083, 49787, 571, 1, 6397, 1201, 3257, 52583, 829, 1, 1, 1, 6871, 1, 1, 3319, 3557, 1, 7237, 58391, 1, 59387, 1, 1, 1, 61403, 1, 62423, 7867, 593, 1999, 1093, 1, 3449, 1, 6053, 8389, 1, 4261, 68711, 541, 3673, 1, 1, 1, 71963, 1, 73063, 1, 4363, 9341, 953, 1, 1777, 1, 77543, 2441, 1, 9907, 823, 1, 743, 2549, 727, 5171, 859, 617, 84503, 967, 85691, 5393, 1, 1367, 643, 11087, 89303, 11239, 90523, 1, 1, 1, 92987, 11701, 1, 1, 95483, 6007, 1087, 1, 98011, 1, 2309, 12491, 1, 3163, 101863, 1, 1453, 683, 104471, 773, 1, 6653, 4657, 1, 6379, 1, 1, 13807, 5849, 1747, 1, 1, 1, 1, 115223, 14489, 1, 7331, 117991, 3709, 1, 883, 1, 15187, 122203, 1, 1, 1, 5437, 1, 126487, 15901, 127931, 1, 1187, 1, 130843, 16447, 1, 16631, 1, 1051, 135271, 8501, 12433, 17189, 1, 17377, 1, 8783, 8311, 1, 142811, 1, 1, 1, 1291, 4583, 1, 1, 1, 1, 150551, 18917, 152123, 1, 1, 1, 155291, 1, 156887, 857, 9323, 1, 6961, 1, 3761, 1847, 163351, 20521, 164987, 1, 166631, 5233, 1, 21139, 169943, 21347, 15601, 1, 1163, 10883, 9209, 21977, 7681, 22189, 1031, 1, 180071, 1, 181787, 1, 183511, 1213, 185243, 1, 1, 1, 2389, 1, 17317, 1259, 192251, 12071, 1, 6091, 1901, 1069, 1, 1, 8669, 1, 10589, 1, 4721, 1, 1, 25717, 206651, 12973, 10973, 1, 19121, 26407, 212183, 1567, 214043, 3359, 3041, 13553, 1, 1439, 2053, 1, 1487, 13907, 223463, 7013, 225371, 1489, 1361, 1, 1, 7193, 21013, 1, 233083, 1721, 2423, 29501, 12473, 1, 5557, 1, 2707, 2749, 1, 1, 12889, 1, 1033, 15493, 248891, 31237, 14759, 31489, 22993, 1, 2339, 1, 3253, 32251, 1, 32507, 1993, 8191, 1, 1, 2347, 1447, 1951, 1973, 1613, 16901, 271463, 2129, 3853, 34327, 1319, 34591, 277787, 4357, 1, 1, 2153, 1, 14957, 1, 16843, 1, 6709, 9049, 12637, 36467, 1, 36739, 1879, 1, 297191, 1, 1601, 37561, 301591, 1, 303803, 1, 306023, 4799, 1549, 38671, 1237, 3541, 13597, 1, 314983, 1, 1, 2341, 319511, 1, 1, 20183, 1733, 10163, 1, 40939, 2399, 41227, 1, 1, 333287, 20903, 3079, 1, 1, 1, 340283, 21341, 14897, 1, 344987, 43271, 1, 2293, 31793, 5483, 4457, 1, 3313, 1, 356887, 2633, 359291, 22531, 19037, 1, 21419, 45667, 15937, 45971, 5197, 1, 1, 23291, 6337, 46889, 34213, 1, 22283, 23753, 381287, 1, 1, 1, 386263, 1, 9041, 1, 1, 24533, 2833, 1, 396311, 49697, 17341, 1471, 2017, 12583, 36721, 50651, 23911, 50971, 21529, 12823, 1, 1, 414203, 1, 2557, 4751, 22073, 26293, 1, 3307, 18461, 53239, 2621, 1, 429851, 6737, 39317, 1427, 435131, 1, 10181, 1, 440443, 27611, 1, 1, 1, 5081, 5039, 1, 26539, 1, 4679, 1, 456571, 57241, 1, 1, 1, 28961, 7877, 7283, 1, 1, 24749, 1, 1, 1, 475751, 2711, 478523, 1, 481303, 60337, 484091, 1597, 21169, 15259, 489691, 1, 44773, 1, 495323, 1, 29303, 31223, 3191, 62801, 11717, 1, 506683, 2887, 509543, 1, 1, 64231, 1783, 64591, 518171, 1, 521063, 1, 1, 65677, 526871, 66041, 2927, 33203, 9029, 16693, 2719, 67139, 1699, 1, 541531, 1, 544487, 34123, 32203, 1, 4871, 3631, 24061, 1, 50581, 1, 13009, 70111, 1, 70487, 1, 1, 29917, 1, 2063, 1, 4133, 1, 577531, 36191, 30557, 18191, 1, 1, 586711, 1, 53617, 1, 1, 1, 595963, 3931, 1, 1, 1, 1, 1, 1, 608411, 4013, 611543, 1, 614683, 9629, 36343, 38713, 6029, 1, 2467, 1, 1, 1709, 4813, 19753, 14737, 79411, 5843, 79811, 1, 1823, 643303, 2371, 646523, 81017, 649751, 81421, 1, 1, 6133, 1, 1, 3593, 662743, 83047, 1, 1, 669287, 2207, 39563, 1, 675863, 7699, 8597, 2503, 682471, 21379, 1, 85931, 36269, 1, 1, 1, 1, 43591, 1, 5153, 36973, 1, 3583, 44221, 41719, 2777, 712603, 8117, 715991, 1, 1, 1, 722791, 45281, 4057, 90989, 1, 1, 1, 2417, 4691, 1, 739931, 92707, 32321, 93139, 746843, 1, 750311, 4273, 1, 94441, 757271, 5581, 1, 47657, 7879, 11969, 2377, 2237, 70117, 1, 774811, 6067, 1, 48757, 781883, 4259, 785431, 5179, 46411, 4493, 792551, 1, 11213, 5867, 1, 5273, 1, 1, 7541, 50543, 73681, 101537, 2939, 1, 817723, 1, 1, 1, 35869, 1, 1, 9437, 1, 13033, 43997, 52361, 839611, 1, 14293, 1, 19697, 53051, 4549, 26641, 854363, 1, 1, 6323, 1, 1, 865511, 2357, 3637, 1, 4877, 109357, 876731, 1, 880487, 1, 884251, 1,

6. Sequence of the polynom (only primes)

23, 11, 197, 19, 409, 613, 89, 809, 113, 997, 17, 107, 79, 71, 179, 199, 1669, 109, 59, 103, 2089, 269, 2213, 137, 149, 2437, 311, 43, 239, 167, 2713, 2789, 353, 2857, 2917, 2969, 131, 379, 3049, 383, 181, 193, 163, 97, 3109, 389, 283, 251, 487, 983, 139, 1511, 1787, 241, 277, 157, 2663, 2971, 173, 431, 3943, 257, 4283, 557, 421, 601, 4987, 5351, 739, 359, 787, 6491, 443, 317, 937, 7703, 8123, 521, 503, 1151, 9431, 9883, 10343, 661, 569, 1381, 11287, 751, 12263, 12763, 1627, 577, 811, 439, 1301, 911, 14843, 1889, 15383, 1013, 16487, 17623, 2239, 1193, 19387, 19991, 1873, 1307, 1117, 673, 21851, 2851, 23131, 733, 1399, 24443, 25111, 3181, 419, 1181, 3527, 28571, 29287, 30011, 3797, 433, 3889, 1657, 1019, 32987, 33751, 1091, 821, 4561, 36887, 37691, 2381, 39323, 4967, 40151, 461, 2411, 647, 709, 42683, 43543, 499, 2803, 1429, 46171, 5827, 2477, 5939, 47963, 48871, 3083, 49787, 571, 6397, 1201, 3257, 52583, 829, 6871, 3319, 3557, 7237, 58391, 59387, 61403, 62423, 7867, 593, 1999, 1093, 3449, 6053, 8389, 4261, 68711, 541, 3673, 71963, 73063, 4363, 9341, 953, 1777, 77543, 2441, 9907, 823, 743, 2549, 727, 5171, 859, 617, 84503, 967, 85691, 5393, 1367, 643, 11087, 89303, 11239, 90523, 92987, 11701, 95483, 6007, 1087, 98011, 2309, 12491, 3163, 101863, 1453, 683, 104471, 773, 6653, 4657, 6379, 13807, 5849, 1747, 115223, 14489, 7331, 117991, 3709, 883, 15187, 122203, 5437, 126487, 15901, 127931, 1187, 130843, 16447, 16631, 1051, 135271, 8501, 12433, 17189, 17377, 8783, 8311, 142811, 1291, 4583, 150551, 18917, 152123, 155291, 156887, 857, 9323, 6961, 3761, 1847, 163351, 20521, 164987, 166631, 5233, 21139, 169943, 21347, 15601, 1163, 10883, 9209, 21977, 7681, 22189, 1031, 180071, 181787, 183511, 1213, 185243, 2389, 17317, 1259, 192251, 12071, 6091, 1901, 1069, 8669, 10589, 4721, 25717, 206651, 12973, 10973, 19121, 26407, 212183, 1567, 214043, 3359, 3041, 13553, 1439, 2053, 1487, 13907, 223463, 7013, 225371, 1489, 1361, 7193, 21013, 233083, 1721, 2423, 29501, 12473, 5557, 2707, 2749, 12889, 1033, 15493, 248891, 31237, 14759, 31489, 22993, 2339, 3253, 32251, 32507, 1993, 8191, 2347, 1447, 1951, 1973, 1613, 16901, 271463, 2129, 3853, 34327, 1319, 34591, 277787, 4357, 2153, 14957, 16843, 6709, 9049, 12637, 36467, 36739, 1879, 297191, 1601, 37561, 301591, 303803, 306023, 4799, 1549, 38671, 1237, 3541, 13597, 314983, 2341, 319511, 20183, 1733, 10163, 40939, 2399, 41227, 333287, 20903, 3079, 340283, 21341, 14897, 344987, 43271, 2293, 31793, 5483, 4457, 3313, 356887, 2633, 359291, 22531, 19037, 21419, 45667, 15937, 45971, 5197, 23291, 6337, 46889, 34213, 22283, 23753, 381287, 386263, 9041, 24533, 2833, 396311, 49697, 17341, 1471, 2017, 12583, 36721, 50651, 23911, 50971, 21529, 12823, 414203, 2557, 4751, 22073, 26293, 3307, 18461, 53239, 2621, 429851, 6737, 39317, 1427, 435131, 10181, 440443, 27611, 5081, 5039, 26539, 4679, 456571, 57241, 28961, 7877, 7283, 24749, 475751, 2711, 478523, 481303, 60337, 484091, 1597, 21169, 15259, 489691, 44773, 495323, 29303, 31223, 3191, 62801, 11717, 506683, 2887, 509543, 64231, 1783, 64591, 518171, 521063, 65677, 526871, 66041, 2927, 33203, 9029, 16693, 2719, 67139, 1699, 541531, 544487, 34123, 32203, 4871, 3631, 24061, 50581, 13009, 70111, 70487, 29917, 2063, 4133, 577531, 36191, 30557, 18191, 586711, 53617, 595963, 3931, 608411, 4013, 611543, 614683, 9629, 36343, 38713, 6029, 2467, 1709, 4813, 19753, 14737, 79411, 5843, 79811, 1823, 643303, 2371, 646523, 81017, 649751, 81421, 6133, 3593, 662743, 83047, 669287, 2207, 39563, 675863, 7699, 8597, 2503, 682471, 21379, 85931, 36269, 43591, 5153, 36973, 3583, 44221, 41719, 2777, 712603, 8117, 715991, 722791, 45281, 4057, 90989, 2417, 4691, 739931, 92707, 32321, 93139, 746843, 750311, 4273, 94441, 757271, 5581, 47657, 7879, 11969, 2377, 2237, 70117, 774811, 6067, 48757, 781883, 4259, 785431, 5179, 46411, 4493, 792551, 11213, 5867, 5273, 7541, 50543, 73681, 101537, 2939, 817723, 35869, 9437, 13033, 43997, 52361, 839611, 14293, 19697, 53051, 4549, 26641, 854363, 6323, 865511, 2357, 3637, 4877, 109357, 876731, 880487, 884251,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2-112x+23 and
the reducible primes which appear as divisor for the first time
p | x^2-112x+23 and p < x^2-112x+23

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)
11010641.0000000.6000001.0000000.0000000.0000000.000000
21004417270.4400000.1700000.4400004.4000002.8333336.750000
31.0006061734330.6060000.1730000.60600013.77272710.17647116.037037
410.0006.6341.3505.2840.6634000.1350000.66340010.9471957.80346812.203234
5100.00067.84410.22757.6170.6784400.1022700.67844010.2267117.57555510.904050
61.000.000681.99683.315598.6810.6819960.0833150.68199610.0524148.14657310.390700
710.000.0006.834.994704.9756.130.0190.6834990.0704970.68349910.0220448.46156110.239207
8100.000.00068.462.0476.110.49162.351.5560.6846200.0611050.68462010.0164028.66767010.171511
91.000.000.000685.547.17253.931.389631.615.7830.6855470.0539310.68554710.0135368.82603210.129912
1010.000.000.0006.862.901.912482.701.4086.380.200.5040.6862900.0482700.68629010.0108388.95028710.101395


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
245321.2500000.7500000.5000001.6666671.5000002.000000
388531.0000000.6250000.3750001.6000001.6666671.500000
41614680.8750000.3750000.5000001.7500001.2000002.666667
5322610160.8125000.3125000.5000001.8571431.6666672.000000
6644417270.6875000.2656250.4218751.6923081.7000001.687500
71284922270.3828120.1718750.2109381.1136361.2941181.000000
825612553720.4882810.2070310.2812502.5510202.4090912.666667
95122781011770.5429690.1972660.3457032.2240001.9056602.458333
101.0246201784420.6054690.1738280.4316412.2302161.7623762.497175
112.0481.3053329730.6372070.1621090.4750982.1048391.8651692.201357
124.0962.6616042.0570.6496580.1474610.5021972.0390801.8192772.114080
138.1925.4001.1304.2700.6591800.1379390.5212402.0293121.8708612.075839
1416.38410.9382.0598.8790.6676030.1256710.5419312.0255561.8221242.079391
1532.76822.0533.80818.2450.6730040.1162110.5567932.0161821.8494422.054848
1665.53644.3377.02937.3080.6765290.1072540.5692752.0104751.8458512.044834
17131.07288.98613.04475.9420.6789090.0995180.5793912.0070371.8557412.035542
18262.144178.29724.517153.7800.6801490.0935250.5866242.0036521.8795622.024966
19524.288357.17945.953311.2260.6812650.0876480.5936162.0032811.8743322.023839
201.048.576715.02686.953628.0730.6819020.0829250.5989772.0018701.8922162.018061
212.097.1521.431.189164.9481.266.2410.6824440.0786530.6037912.0015901.8969792.016073
224.194.3042.864.446314.0372.550.4090.6829370.0748720.6080652.0014451.9038542.014158
238.388.6085.732.455598.4825.133.9730.6833620.0713450.6120172.0012441.9057692.013000
2416.777.21611.470.8241.143.40610.327.4180.6837140.0681520.6155622.0010321.9105102.011584
2533.554.43222.954.4852.188.57720.765.9080.6840970.0652250.6188722.0011191.9140862.010755
2667.108.86445.932.0194.197.77041.734.2490.6844400.0625520.6218892.0010041.9180362.009748
27134.217.72891.905.3648.065.24683.840.1180.6847480.0600910.6246582.0009001.9213172.008904
28268.435.456183.887.59315.520.579168.367.0140.6850350.0578190.6272162.0008371.9243782.008191
29536.870.912367.927.83229.906.120338.021.7120.6853190.0557040.6296142.0008301.9268692.007648
301.073.741.824736.127.87657.700.298678.427.5780.6855730.0537380.6318352.0007401.9293812.007053
312.147.483.6481.472.770.960111.474.8371.361.296.1230.6858120.0519100.6339032.0007001.9319632.006546
324.294.967.2962.946.510.725215.620.2022.730.890.5230.6860380.0502030.6358352.0006581.9342502.006096
338.589.934.5925.894.805.731417.529.3615.477.276.3700.6862460.0486070.6376392.0006051.9364112.005674
3417.179.869.18411.792.981.275809.289.71710.983.691.5580.6864420.0471070.6393352.0005721.9382822.005320


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
122020011
243121011
385232021
4166332031
53210643061
664171167091
71282212107294
8256532429717920
95121013863741944
101.02417865113781981
112.04833211921371629154
124.09660421339172939295
138.1921.13038274875599555
1416.3842.0596861.37371.02691.017
1532.7683.8081.2542.55471.90291.890
1665.5367.0292.3064.72373.52993.484
17131.07213.0444.3528.69276.54396.485
18262.14424.5178.17316.344712.281912.220
19524.28845.95315.27930.674722.977922.960
201.048.57686.95328.90358.050743.461943.476
212.097.152164.94854.988109.960782.445982.487
224.194.304314.037104.734209.3037157.0779156.944
238.388.608598.482199.757398.7257299.5589298.908
2416.777.2161.143.406381.471761.9357572.1319571.259
2533.554.4322.188.577729.6951.458.88271.094.71691.093.845
2667.108.8644.197.7701.399.7722.797.99872.099.41992.098.335
27134.217.7288.065.2462.689.4975.375.74974.033.36694.031.864
28268.435.45615.520.5795.175.92710.344.65277.762.17697.758.387
29536.870.91229.906.1209.971.12519.934.995714.956.415914.949.689
301.073.741.82457.700.29819.232.67838.467.620728.853.183928.847.099
312.147.483.648111.474.83737.154.49674.320.341755.736.988955.737.833
324.294.967.296215.620.20271.866.766143.753.4367107.813.7039107.806.483
338.589.934.592417.529.361139.170.297278.359.0647208.761.6319208.767.714
3417.179.869.184809.289.717269.763.123539.526.5947404.634.0559404.655.646


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
242110200
383121200
4168262402
532165113535
6642710175958
71282710175958
825672343817181918
9512177908739455241
101.024442224218108100134100
112.048973509464261213273226
124.0962.0571.067990545466562484
138.1924.2702.2112.0591.1091.0001.167994
1416.3848.8794.6084.2712.3442.0662.3702.099
1532.76818.2459.5038.7424.7664.2804.8494.350
1665.53637.30819.22018.0889.5538.8789.9718.906
17131.07275.94239.11736.82519.49418.27019.95118.227
18262.144153.78079.09374.68739.92636.88140.01636.957
19524.288311.226159.452151.77480.46474.89581.05974.808
201.048.576628.073321.263306.810162.287151.405162.754151.627
212.097.1521.266.241646.765619.476326.378306.068327.126306.669
224.194.3042.550.4091.301.3911.249.018656.517617.534657.580618.778
238.388.6085.133.9732.616.2012.517.7721.319.4401.246.2001.321.1531.247.180
2416.777.21610.327.4185.258.0125.069.4062.650.8312.512.4882.652.2412.511.858
2533.554.43220.765.90810.560.98810.204.9205.322.9865.059.1745.325.2425.058.506
2667.108.86441.734.24921.212.48820.521.76110.685.31210.180.06610.687.71310.181.158
27134.217.72883.840.11842.580.94441.259.17421.445.26320.474.04621.445.67220.475.137
28268.435.456168.367.01485.446.58382.920.43143.025.53141.158.77143.025.33741.157.375
29536.870.912338.021.712171.447.745166.573.96786.294.25982.720.84586.297.48882.709.120
301.073.741.824678.427.578343.932.017334.495.561173.061.125166.157.717173.052.299166.156.437
312.147.483.6481.361.296.123689.767.997671.528.126346.971.365333.672.544346.963.745333.688.469
324.294.967.2962.730.890.5231.383.085.3001.347.805.223695.544.877669.890.441695.552.697669.902.508
338.589.934.5925.477.276.3702.772.822.7122.704.453.6581.394.155.6941.344.484.1061.394.114.6411.344.521.929
3417.179.869.18410.983.691.5585.558.051.5965.425.639.9622.793.935.3902.697.922.3622.793.889.0292.697.944.777


8. Check for existing Integer Sequences by OEIS

Found in Database : 23, 11, 197, 19, 409, 1, 613, 89, 809, 113, 997, 17, 107, 79, 71, 179, 1, 199, 1669, 109,
Found in Database : 23, 11, 197, 19, 409, 613, 89, 809, 113, 997, 17, 107, 79, 71, 179, 199, 1669, 109, 59, 103, 2089, 269, 2213, 137, 149, 2437, 311, 43, 239, 167, 2713, 2789, 353,
Found in Database : 11, 17, 19, 23, 43, 59, 71, 79, 89, 97, 103, 107, 109, 113, 131, 137, 139, 149,