Inhaltsverzeichnis

Development of
Algorithmic Constructions

13:05:42
Deutsch
29.Mar 2024

Polynom = x^2-13

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) = 13 = 13
f(1) = 3 = 3
f(2) = 9 = 3*3
f(3) = 1 = 1
f(4) = 3 = 3
f(5) = 3 = 3
f(6) = 23 = 23
f(7) = 9 = 3*3
f(8) = 51 = 3*17
f(9) = 17 = 17
f(10) = 87 = 3*29
f(11) = 27 = 3*3*3
f(12) = 131 = 131
f(13) = 39 = 3*13
f(14) = 183 = 3*61
f(15) = 53 = 53
f(16) = 243 = 3*3*3*3*3
f(17) = 69 = 3*23
f(18) = 311 = 311
f(19) = 87 = 3*29
f(20) = 387 = 3*3*43
f(21) = 107 = 107
f(22) = 471 = 3*157
f(23) = 129 = 3*43
f(24) = 563 = 563
f(25) = 153 = 3*3*17
f(26) = 663 = 3*13*17
f(27) = 179 = 179
f(28) = 771 = 3*257
f(29) = 207 = 3*3*23
f(30) = 887 = 887
f(31) = 237 = 3*79
f(32) = 1011 = 3*337
f(33) = 269 = 269
f(34) = 1143 = 3*3*127
f(35) = 303 = 3*101
f(36) = 1283 = 1283
f(37) = 339 = 3*113
f(38) = 1431 = 3*3*3*53
f(39) = 377 = 13*29
f(40) = 1587 = 3*23*23
f(41) = 417 = 3*139
f(42) = 1751 = 17*103
f(43) = 459 = 3*3*3*17
f(44) = 1923 = 3*641
f(45) = 503 = 503
f(46) = 2103 = 3*701
f(47) = 549 = 3*3*61
f(48) = 2291 = 29*79
f(49) = 597 = 3*199
f(50) = 2487 = 3*829
f(51) = 647 = 647
f(52) = 2691 = 3*3*13*23
f(53) = 699 = 3*233
f(54) = 2903 = 2903
f(55) = 753 = 3*251
f(56) = 3123 = 3*3*347
f(57) = 809 = 809
f(58) = 3351 = 3*1117
f(59) = 867 = 3*17*17
f(60) = 3587 = 17*211
f(61) = 927 = 3*3*103
f(62) = 3831 = 3*1277
f(63) = 989 = 23*43
f(64) = 4083 = 3*1361
f(65) = 1053 = 3*3*3*3*13
f(66) = 4343 = 43*101
f(67) = 1119 = 3*373
f(68) = 4611 = 3*29*53
f(69) = 1187 = 1187
f(70) = 4887 = 3*3*3*181
f(71) = 1257 = 3*419
f(72) = 5171 = 5171
f(73) = 1329 = 3*443
f(74) = 5463 = 3*3*607
f(75) = 1403 = 23*61
f(76) = 5763 = 3*17*113
f(77) = 1479 = 3*17*29
f(78) = 6071 = 13*467
f(79) = 1557 = 3*3*173
f(80) = 6387 = 3*2129
f(81) = 1637 = 1637
f(82) = 6711 = 3*2237
f(83) = 1719 = 3*3*191
f(84) = 7043 = 7043
f(85) = 1803 = 3*601
f(86) = 7383 = 3*23*107
f(87) = 1889 = 1889
f(88) = 7731 = 3*3*859
f(89) = 1977 = 3*659
f(90) = 8087 = 8087
f(91) = 2067 = 3*13*53
f(92) = 8451 = 3*3*3*313
f(93) = 2159 = 17*127
f(94) = 8823 = 3*17*173
f(95) = 2253 = 3*751
f(96) = 9203 = 9203
f(97) = 2349 = 3*3*3*3*29
f(98) = 9591 = 3*23*139
f(99) = 2447 = 2447
f(100) = 9987 = 3*3329

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^2-13

f(0)=13
f(1)=3
f(2)=1
f(3)=1
f(4)=1
f(5)=1
f(6)=23
f(7)=1
f(8)=17
f(9)=1
f(10)=29
f(11)=1
f(12)=131
f(13)=1
f(14)=61
f(15)=53
f(16)=1
f(17)=1
f(18)=311
f(19)=1
f(20)=43
f(21)=107
f(22)=157
f(23)=1
f(24)=563
f(25)=1
f(26)=1
f(27)=179
f(28)=257
f(29)=1
f(30)=887
f(31)=79
f(32)=337
f(33)=269
f(34)=127
f(35)=101
f(36)=1283
f(37)=113
f(38)=1
f(39)=1
f(40)=1
f(41)=139
f(42)=103
f(43)=1
f(44)=641
f(45)=503
f(46)=701
f(47)=1
f(48)=1
f(49)=199
f(50)=829
f(51)=647
f(52)=1
f(53)=233
f(54)=2903
f(55)=251
f(56)=347
f(57)=809
f(58)=1117
f(59)=1
f(60)=211
f(61)=1
f(62)=1277
f(63)=1
f(64)=1361
f(65)=1
f(66)=1
f(67)=373
f(68)=1
f(69)=1187
f(70)=181
f(71)=419
f(72)=5171
f(73)=443
f(74)=607
f(75)=1
f(76)=1
f(77)=1
f(78)=467
f(79)=173
f(80)=2129
f(81)=1637
f(82)=2237
f(83)=191
f(84)=7043
f(85)=601
f(86)=1
f(87)=1889
f(88)=859
f(89)=659
f(90)=8087
f(91)=1
f(92)=313
f(93)=1
f(94)=1
f(95)=751
f(96)=9203
f(97)=1
f(98)=1
f(99)=2447

b) Substitution of the polynom
The polynom f(x)=x^2-13 could be written as f(y)= y^2-13 with x=y+0

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+0
f'(x)>2x-1 with x > 4

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

13, 3, 1, 1, 1, 1, 23, 1, 17, 1, 29, 1, 131, 1, 61, 53, 1, 1, 311, 1, 43, 107, 157, 1, 563, 1, 1, 179, 257, 1, 887, 79, 337, 269, 127, 101, 1283, 113, 1, 1, 1, 139, 103, 1, 641, 503, 701, 1, 1, 199, 829, 647, 1, 233, 2903, 251, 347, 809, 1117, 1, 211, 1, 1277, 1, 1361, 1, 1, 373, 1, 1187, 181, 419, 5171, 443, 607, 1, 1, 1, 467, 173, 2129, 1637, 2237, 191, 7043, 601, 1, 1889, 859, 659, 8087, 1, 313, 1, 1, 751, 9203, 1, 1, 2447, 3329, 283, 10391, 883, 277, 2753, 1, 953, 1, 1, 1, 1, 4177, 1063, 12983, 367, 4481, 263, 4637, 1, 14387, 1, 4957, 3779, 569, 1301, 547, 1, 1, 4157, 433, 1429, 757, 491, 5981, 1, 1, 521, 19031, 1609, 6529, 4967, 2239, 1, 1, 1, 1, 5399, 7297, 1, 1, 1, 1, 5849, 7901, 1, 1871, 2053, 8317, 6317, 2843, 1, 1543, 2213, 1, 6803, 9181, 1, 28211, 1, 9629, 7307, 9857, 1, 571, 2551, 10321, 7829, 1, 1, 1, 2729, 1, 8369, 389, 2851, 34583, 971, 11777, 1, 523, 1013, 857, 1, 12541, 1, 1, 1, 39191, 3299, 1481, 439, 13597, 3433, 41603, 1, 1, 10709, 1109, 1213, 44087, 3709, 881, 1, 5087, 3851, 46643, 3923, 5279, 11987, 1, 1, 1699, 1381, 727, 12653, 17021, 1, 51971, 1, 1, 13337, 1993, 4523, 4211, 1, 1, 1, 1, 4759, 57587, 1613, 673, 14759, 19841, 1667, 3559, 1, 1, 15497, 1, 1, 1, 5333, 2389, 16253, 21841, 5503, 1091, 1, 1733, 17027, 22877, 1, 4099, 5851, 23581, 1, 1, 6029, 3169, 1, 8219, 1433, 1, 6301, 76163, 2131, 599, 19457, 1, 1, 3457, 6673, 26881, 1, 1, 6863, 653, 6959, 9343, 1, 1, 1, 86423, 2417, 29201, 1297, 1741, 1, 1, 7549, 1, 1, 10267, 1, 1, 7853, 1171, 823, 32029, 8059, 7487, 907, 1933, 1459, 1447, 2791, 101111, 1, 34129, 1, 11519, 8693, 2441, 677, 11807, 26729, 1559, 1, 108887, 1, 2161, 1, 37181, 1039, 719, 9463, 1, 1249, 1427, 9689, 1093, 9803, 13147, 29753, 39901, 1, 1, 1, 40829, 1, 1, 3461, 125303, 10501, 797, 31859, 1, 10739, 129587, 10859, 1, 32939, 1, 1, 7879, 1, 45137, 1, 1, 3823, 138371, 11593, 2027, 35153, 1, 911, 142871, 11969, 1, 1, 2861, 1, 1, 1, 937, 1291, 50177, 1, 11699, 12739, 51217, 38609, 5749, 13001, 5407, 1, 17599, 2341, 3137, 13399, 161591, 1, 54401, 1, 54941, 1, 7237, 1, 1303, 42227, 18859, 1, 171383, 1, 1, 2557, 58237, 14629, 7669, 1, 59357, 44729, 59921, 1, 2297, 15193, 61057, 3539, 20543, 1, 1031, 919, 1231, 47303, 63361, 15913, 191831, 1, 1, 1, 5009, 1, 197123, 1, 66301, 1, 7433, 1, 1, 997, 22699, 1193, 1, 1327, 207923, 5801, 1, 52667, 70529, 5903, 4027, 17863, 71761, 1, 1049, 1069, 991, 18329, 1, 55457, 74257, 1, 1, 2089, 75521, 2473, 76157, 6373, 230387, 1483, 1, 2011, 1531, 1153, 1, 19763, 26459, 1, 1, 20089, 4567, 1, 6257, 61253, 1907, 2287, 247991, 20749, 3623, 3691, 1, 1, 254003, 1, 28447, 4943, 86017, 21589, 1453, 7253, 1, 65789, 1, 1, 1, 22273, 5261, 1, 2311, 22619, 3449, 1, 10169, 68903, 92221, 23143, 2467, 2591, 93629, 70487, 3253, 1, 1, 1, 1, 72089, 32159, 1, 2887, 1, 1, 73709, 98641, 1, 1, 8311, 1, 75347, 100829, 1, 17923, 1499, 102301, 77003, 1, 1, 13537, 2003, 34843, 2713, 105277, 1, 1777, 8867, 106781, 80369, 2029, 1, 1, 1, 8389, 1, 36607, 27551, 3221, 27743, 12373, 1949, 112129, 1223, 7877, 1, 1439, 6581, 6733, 1, 345731, 28909, 4001, 87317, 1693, 29303, 1, 29501, 1, 1, 1, 1, 359987, 1, 1, 5347, 1, 3389, 2339, 30703, 123217, 92717, 13781, 2393, 374531, 1, 41887, 4111, 126481, 31723, 381911, 1, 7537, 1, 2999, 10781, 1, 1, 130621, 4273, 1511, 32969, 396887, 33179, 4931, 100169, 133981, 33601, 404483, 1, 1, 102077, 1, 1, 1, 1, 138241, 104003, 1, 34883, 14479, 35099, 1, 1999, 141697, 35533, 1, 1, 1, 1, 1, 4021, 4229, 1583, 146077, 1, 5443, 1, 443543, 1, 1, 1, 149629, 2207, 1, 1, 2857, 113903, 11717, 1, 459671, 1, 1951, 1, 51679, 38873, 20341, 1, 1, 117989, 9281, 1, 476087, 4421, 159617, 1, 160541, 13417, 21061, 40483, 1, 122147, 54443, 40949, 37907, 41183, 1, 7309, 1, 1811, 11657, 13963, 168029, 2939, 168977, 1, 509783, 1, 170881, 128519, 1, 1873, 518387, 43319, 3407, 7687, 174721, 43801, 527063, 1, 1, 132857, 7723, 14843, 535811, 44773, 179581, 4657, 1, 45263, 544631, 2677, 1, 10559, 1, 1, 19087, 1, 3041, 1, 1, 15583, 1621, 1, 1, 6163, 1, 47501, 33619, 1, 63839, 1823, 192529, 48259, 580631, 1, 6709, 6361, 195581, 1, 589811, 49279, 1847, 1, 22073, 1, 1, 50051, 2909, 150929, 201757, 1, 1, 16943, 1979, 153269, 1, 17117, 3571, 51613, 8999, 155627, 4079, 3067, 627251, 1, 1, 158003, 1663, 1, 4861, 1, 213329, 160397, 7393, 17911, 646403, 54001, 16657, 1, 1, 54539, 12379, 2383, 73259, 1, 220861, 55351, 665843, 18541, 223037, 12899, 224129, 1, 1, 56443, 13313, 10009, 8423, 56993, 6067, 57269, 76543, 5953, 17749, 1, 30241, 1, 232961, 4073, 2963, 19553, 1, 3467, 13901, 1759, 79147, 1, 1, 1, 26633, 1, 240829, 2081, 725891, 6737, 243101, 1, 1, 1, 3331, 3617, 8501, 185327, 82559, 1, 746483, 62351, 1, 1, 1, 62929, 756887, 1, 1, 190529, 2521, 1, 45139, 1, 256957, 1, 1, 64679, 777911, 1, 6679, 1901, 4937, 65563, 788531, 1, 264029, 8629, 15601, 1, 799223, 66751, 267601, 15473, 89599, 67349, 10253, 1, 30133, 8863, 1, 1, 3121, 2539, 274817, 1, 1, 23053, 831731, 69463, 12107, 1, 1, 1, 29059, 70379, 1, 212057, 1, 1, 853763, 23767, 1, 12637, 287057, 1, 864887, 72229, 2083, 217619, 32309, 1, 67391, 3181, 3371, 4159, 294529, 1, 8293, 1, 1, 223253, 1, 1, 898691, 1, 1, 1, 100699, 75683, 910103, 76001, 3761, 228959, 305917, 1, 2357, 1, 1, 231839, 1973, 25867, 21701, 2687, 1, 1, 104543, 78569, 41077, 78893, 105407, 1, 10949, 4679, 56263, 1, 2833, 240587, 2531, 1, 9049, 1, 1, 243539, 1, 81509, 16067, 81839, 2063, 246509, 19373, 1, 1, 27611, 331997, 249497,

6. Sequence of the polynom (only primes)

13, 3, 23, 17, 29, 131, 61, 53, 311, 43, 107, 157, 563, 179, 257, 887, 79, 337, 269, 127, 101, 1283, 113, 139, 103, 641, 503, 701, 199, 829, 647, 233, 2903, 251, 347, 809, 1117, 211, 1277, 1361, 373, 1187, 181, 419, 5171, 443, 607, 467, 173, 2129, 1637, 2237, 191, 7043, 601, 1889, 859, 659, 8087, 313, 751, 9203, 2447, 3329, 283, 10391, 883, 277, 2753, 953, 4177, 1063, 12983, 367, 4481, 263, 4637, 14387, 4957, 3779, 569, 1301, 547, 4157, 433, 1429, 757, 491, 5981, 521, 19031, 1609, 6529, 4967, 2239, 5399, 7297, 5849, 7901, 1871, 2053, 8317, 6317, 2843, 1543, 2213, 6803, 9181, 28211, 9629, 7307, 9857, 571, 2551, 10321, 7829, 2729, 8369, 389, 2851, 34583, 971, 11777, 523, 1013, 857, 12541, 39191, 3299, 1481, 439, 13597, 3433, 41603, 10709, 1109, 1213, 44087, 3709, 881, 5087, 3851, 46643, 3923, 5279, 11987, 1699, 1381, 727, 12653, 17021, 51971, 13337, 1993, 4523, 4211, 4759, 57587, 1613, 673, 14759, 19841, 1667, 3559, 15497, 5333, 2389, 16253, 21841, 5503, 1091, 1733, 17027, 22877, 4099, 5851, 23581, 6029, 3169, 8219, 1433, 6301, 76163, 2131, 599, 19457, 3457, 6673, 26881, 6863, 653, 6959, 9343, 86423, 2417, 29201, 1297, 1741, 7549, 10267, 7853, 1171, 823, 32029, 8059, 7487, 907, 1933, 1459, 1447, 2791, 101111, 34129, 11519, 8693, 2441, 677, 11807, 26729, 1559, 108887, 2161, 37181, 1039, 719, 9463, 1249, 1427, 9689, 1093, 9803, 13147, 29753, 39901, 40829, 3461, 125303, 10501, 797, 31859, 10739, 129587, 10859, 32939, 7879, 45137, 3823, 138371, 11593, 2027, 35153, 911, 142871, 11969, 2861, 937, 1291, 50177, 11699, 12739, 51217, 38609, 5749, 13001, 5407, 17599, 2341, 3137, 13399, 161591, 54401, 54941, 7237, 1303, 42227, 18859, 171383, 2557, 58237, 14629, 7669, 59357, 44729, 59921, 2297, 15193, 61057, 3539, 20543, 1031, 919, 1231, 47303, 63361, 15913, 191831, 5009, 197123, 66301, 7433, 997, 22699, 1193, 1327, 207923, 5801, 52667, 70529, 5903, 4027, 17863, 71761, 1049, 1069, 991, 18329, 55457, 74257, 2089, 75521, 2473, 76157, 6373, 230387, 1483, 2011, 1531, 1153, 19763, 26459, 20089, 4567, 6257, 61253, 1907, 2287, 247991, 20749, 3623, 3691, 254003, 28447, 4943, 86017, 21589, 1453, 7253, 65789, 22273, 5261, 2311, 22619, 3449, 10169, 68903, 92221, 23143, 2467, 2591, 93629, 70487, 3253, 72089, 32159, 2887, 73709, 98641, 8311, 75347, 100829, 17923, 1499, 102301, 77003, 13537, 2003, 34843, 2713, 105277, 1777, 8867, 106781, 80369, 2029, 8389, 36607, 27551, 3221, 27743, 12373, 1949, 112129, 1223, 7877, 1439, 6581, 6733, 345731, 28909, 4001, 87317, 1693, 29303, 29501, 359987, 5347, 3389, 2339, 30703, 123217, 92717, 13781, 2393, 374531, 41887, 4111, 126481, 31723, 381911, 7537, 2999, 10781, 130621, 4273, 1511, 32969, 396887, 33179, 4931, 100169, 133981, 33601, 404483, 102077, 138241, 104003, 34883, 14479, 35099, 1999, 141697, 35533, 4021, 4229, 1583, 146077, 5443, 443543, 149629, 2207, 2857, 113903, 11717, 459671, 1951, 51679, 38873, 20341, 117989, 9281, 476087, 4421, 159617, 160541, 13417, 21061, 40483, 122147, 54443, 40949, 37907, 41183, 7309, 1811, 11657, 13963, 168029, 2939, 168977, 509783, 170881, 128519, 1873, 518387, 43319, 3407, 7687, 174721, 43801, 527063, 132857, 7723, 14843, 535811, 44773, 179581, 4657, 45263, 544631, 2677, 10559, 19087, 3041, 15583, 1621, 6163, 47501, 33619, 63839, 1823, 192529, 48259, 580631, 6709, 6361, 195581, 589811, 49279, 1847, 22073, 50051, 2909, 150929, 201757, 16943, 1979, 153269, 17117, 3571, 51613, 8999, 155627, 4079, 3067, 627251, 158003, 1663, 4861, 213329, 160397, 7393, 17911, 646403, 54001, 16657, 54539, 12379, 2383, 73259, 220861, 55351, 665843, 18541, 223037, 12899, 224129, 56443, 13313, 10009, 8423, 56993, 6067, 57269, 76543, 5953, 17749, 30241, 232961, 4073, 2963, 19553, 3467, 13901, 1759, 79147, 26633, 240829, 2081, 725891, 6737, 243101, 3331, 3617, 8501, 185327, 82559, 746483, 62351, 62929, 756887, 190529, 2521, 45139, 256957, 64679, 777911, 6679, 1901, 4937, 65563, 788531, 264029, 8629, 15601, 799223, 66751, 267601, 15473, 89599, 67349, 10253, 30133, 8863, 3121, 2539, 274817, 23053, 831731, 69463, 12107, 29059, 70379, 212057, 853763, 23767, 12637, 287057, 864887, 72229, 2083, 217619, 32309, 67391, 3181, 3371, 4159, 294529, 8293, 223253, 898691, 100699, 75683, 910103, 76001, 3761, 228959, 305917, 2357, 231839, 1973, 25867, 21701, 2687, 104543, 78569, 41077, 78893, 105407, 10949, 4679, 56263, 2833, 240587, 2531, 9049, 243539, 81509, 16067, 81839, 2063, 246509, 19373, 27611, 331997, 249497,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2x-13 and
the reducible primes which appear as divisor for the first time
p | x^2x-13 and p < x^2x-13

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)
1105320.5000000.3000000.5000000.0000000.0000000.000000
21006424400.6400000.2400000.64000012.8000008.00000020.000000
31.0006751485270.6750000.1480000.67500010.5468756.16666713.175000
410.0006.8261.0635.7630.6826000.1063000.68260010.1125937.18243310.935484
5100.00068.5918.02360.5680.6859100.0802300.68591010.0484917.54750710.509804
61.000.000686.78365.524621.2590.6867830.0655240.68678310.0127288.16702010.257215
710.000.0006.875.551549.1036.326.4480.6875550.0549100.68755510.0112428.38018110.183270
8100.000.00068.801.1334.746.80764.054.3260.6880110.0474680.68801110.0066368.64465710.124848
91.000.000.000688.418.52641.800.087646.618.4390.6884190.0418000.68841910.0059188.80593810.094844
1010.000.000.0006.887.796.115373.338.6206.514.457.4950.6887800.0373340.68878010.0052468.93152710.074655


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)
122201.0000001.0000000.0000000.0000000.0000000.000000
242200.5000000.5000000.0000001.0000001.000000-nan
384310.5000000.3750000.1250002.0000001.500000inf
4168530.5000000.3125000.1875002.0000001.6666673.000000
532181080.5625000.3125000.2500002.2500002.0000002.666667
6644016240.6250000.2500000.3750002.2222221.6000003.000000
71288329540.6484380.2265620.4218752.0750001.8125002.250000
8256169541150.6601560.2109380.4492192.0361441.8620692.129630
9512347842630.6777340.1640620.5136722.0532541.5555562.286957
101.0246921515410.6757810.1474610.5283201.9942361.7976192.057034
112.0481.3912721.1190.6791990.1328120.5463872.0101161.8013242.068392
124.0962.7974982.2990.6828610.1215820.5612792.0107841.8308822.054513
138.1925.5839014.6820.6815190.1099850.5715331.9960671.8092372.036538
1416.38411.1971.6179.5800.6834110.0986940.5847172.0055531.7946732.046134
1532.76822.4682.96119.5070.6856690.0903630.5953062.0066091.8311692.036221
1665.53644.9245.51639.4080.6854860.0841670.6013181.9994661.8628842.020198
17131.07289.94610.25079.6960.6862340.0782010.6080322.0021821.8582312.022331
18262.144179.88819.354160.5340.6862180.0738300.6123891.9999561.8881952.014329
19524.288359.97636.256323.7200.6866000.0691530.6174472.0011121.8733082.016520
201.048.576720.09868.458651.6400.6867390.0652870.6214522.0004061.8881842.012974
212.097.1521.440.902129.1781.311.7240.6870760.0615970.6254792.0009801.8869672.012958
224.194.3042.882.729245.2792.637.4500.6872960.0584790.6288172.0006421.8987682.010674
238.388.6085.767.236466.3685.300.8680.6875080.0555950.6319132.0006171.9013782.009846
2416.777.21611.537.303889.60310.647.7000.6876770.0530240.6346522.0004911.9075132.008671
2533.554.43223.078.5641.702.15021.376.4140.6877950.0507280.6370672.0003431.9133822.007609
2667.108.86446.165.6523.262.19442.903.4580.6879220.0486100.6393112.0003691.9165142.007046
27134.217.72892.350.2756.262.09086.088.1850.6880630.0466560.6414072.0004111.9195952.006556
28268.435.456184.733.73212.041.795172.691.9370.6881870.0448590.6433282.0003591.9229672.005989
29536.870.912369.535.04023.191.327346.343.7130.6883130.0431970.6451152.0003661.9259032.005558
301.073.741.824739.197.35344.718.053694.479.3000.6884310.0416470.6467842.0003451.9282232.005174
312.147.483.6481.478.638.45986.332.5661.392.305.8930.6885450.0402020.6483432.0003301.9305982.004820
324.294.967.2962.957.746.578166.894.6302.790.851.9480.6886540.0388580.6497962.0003181.9331592.004482
338.589.934.5925.916.385.691322.974.6245.593.411.0670.6887580.0375990.6511592.0003021.9352012.004195
3417.179.869.18411.834.477.024625.688.34811.208.788.6760.6888570.0364200.6524372.0002881.9372682.003927
3534.359.738.36823.672.249.2321.213.333.17222.458.916.0600.6889530.0353130.6536412.0002781.9391972.003688
3668.719.476.73647.350.885.5992.355.031.53444.995.854.0650.6890460.0342700.6547762.0002701.9409602.003474


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
122100110
242100110
383110111
4165130221
53210180523
664161141636
712829127312410
8256541527201017
95128418214321127
101.024151114923592148
112.0482721270371053298
124.09649814966618666180
138.1929011899109327132333
1416.3841.61711.615203591212611
1532.7682.96112.9593771.0643801.140
1665.5365.51615.5147182.0027132.083
17131.07210.250110.2481.3433.7981.3193.790
18262.14419.354119.3522.5177.2052.4977.135
19524.28836.256136.2544.65613.4804.64713.473
201.048.57668.458168.4568.79025.3688.75425.546
212.097.152129.1781129.17616.56547.99516.51348.105
224.194.304245.2791245.27731.42991.28231.41991.149
238.388.608466.3681466.36659.527173.74759.677173.417
2416.777.216889.6031889.601113.486331.135113.664331.318
2533.554.4321.702.15011.702.148217.307633.736217.162633.945
2667.108.8643.262.19413.262.192416.3611.215.627416.1491.214.057
27134.217.7286.262.09016.262.088798.3272.333.041798.0412.332.681
28268.435.45612.041.795112.041.7931.533.1124.488.1851.533.5214.486.977
29536.870.91223.191.327123.191.3252.952.0298.643.7432.951.9698.643.586
301.073.741.82444.718.053144.718.0515.689.06116.667.4175.690.02316.671.552
312.147.483.64886.332.566186.332.56410.977.72932.188.75410.976.35132.189.732
324.294.967.296166.894.6301166.894.62821.215.79962.234.92921.208.50562.235.397
338.589.934.592322.974.6241322.974.62241.032.442120.466.61141.021.372120.454.199
3417.179.869.184625.688.3481625.688.34679.456.102233.412.00179.427.580233.392.665
3534.359.738.3681.213.333.17211.213.333.170153.986.933452.700.147153.969.692452.676.400
3668.719.476.7362.355.031.53412.355.031.532298.725.898878.797.172298.713.977878.794.487


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
120000000
240000000
381011000
4163121020
5328533131
6642412127584
712854272715131610
8256115585732263720
951226314212176567655
101.024541281260147114160120
112.0481.119583536315236325243
124.0962.2991.1811.118627494648530
138.1924.6822.3862.2961.2891.0321.2861.075
1416.3849.5804.9104.6702.6022.1672.5962.215
1532.76819.5079.9879.5205.3104.5005.1664.531
1665.53639.40820.15719.25110.5689.14610.4959.199
17131.07279.69640.79238.90421.20518.63621.26818.587
18262.144160.53482.03778.49742.67937.69542.56337.597
19524.288323.720165.480158.24085.79576.19485.27876.453
201.048.576651.640332.499319.141171.528154.037171.619154.456
212.097.1521.311.724668.907642.817344.037311.903344.128311.656
224.194.3042.637.4501.342.4551.294.995689.963628.840689.540629.107
238.388.6085.300.8682.696.3252.604.5431.383.8221.266.5791.383.3751.267.092
2416.777.21610.647.7005.408.4675.239.2332.773.4792.551.3392.772.1392.550.743
2533.554.43221.376.41410.852.77010.523.6445.559.2915.129.3475.557.5115.130.265
2667.108.86442.903.45821.770.82821.132.63011.136.50110.315.60211.135.16910.316.186
27134.217.72886.088.18543.651.37942.436.80622.307.20720.735.77822.307.91020.737.290
28268.435.456172.691.93787.510.26985.181.66844.683.55941.661.33044.687.83441.659.214
29536.870.912346.343.713175.402.078170.941.63589.495.66283.675.23389.500.36983.672.449
301.073.741.824694.479.300351.521.793342.957.507179.239.656167.991.071179.244.712168.003.861
312.147.483.6481.392.305.893704.396.825687.909.068358.926.415337.212.601358.920.703337.246.174
324.294.967.2962.790.851.9481.411.291.6771.379.560.271718.671.760676.732.374718.703.618676.744.196
338.589.934.5925.593.411.0672.827.333.2702.766.077.7971.438.909.9221.357.776.2441.438.946.3921.357.778.509
3417.179.869.18411.208.788.6765.663.637.8815.545.150.7952.880.788.0472.723.581.0962.880.827.6812.723.591.852
3534.359.738.36822.458.916.06011.344.031.03911.114.885.0215.767.214.8165.462.260.5385.767.134.8505.462.305.856
3668.719.476.73644.995.854.06522.719.734.72722.276.119.33811.544.826.35210.953.157.88111.544.723.96310.953.145.869


8. Check for existing Integer Sequences by OEIS

Found in Database : 13, 3, 1, 1, 1, 1, 23, 1, 17, 1, 29, 1, 131, 1, 61, 53, 1, 1, 311, 1,
Found in Database : 13, 3, 23, 17, 29, 131, 61, 53, 311, 43, 107, 157, 563, 179, 257, 887, 79, 337, 269, 127, 101, 1283, 113,
Found in Database : 3, 13, 17, 23, 29, 43, 53, 61, 79, 101, 103, 107, 113, 127, 131, 139,