Inhaltsverzeichnis

Development of
Algorithmic Constructions

17:49:16
Deutsch
28.Mar 2024

Polynom = x^2+25x-31

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) = 31 = 31
f(1) = 5 = 5
f(2) = 23 = 23
f(3) = 53 = 53
f(4) = 85 = 5*17
f(5) = 119 = 7*17
f(6) = 155 = 5*31
f(7) = 193 = 193
f(8) = 233 = 233
f(9) = 275 = 5*5*11
f(10) = 319 = 11*29
f(11) = 365 = 5*73
f(12) = 413 = 7*59
f(13) = 463 = 463
f(14) = 515 = 5*103
f(15) = 569 = 569
f(16) = 625 = 5*5*5*5
f(17) = 683 = 683
f(18) = 743 = 743
f(19) = 805 = 5*7*23
f(20) = 869 = 11*79
f(21) = 935 = 5*11*17
f(22) = 1003 = 17*59
f(23) = 1073 = 29*37
f(24) = 1145 = 5*229
f(25) = 1219 = 23*53
f(26) = 1295 = 5*7*37
f(27) = 1373 = 1373
f(28) = 1453 = 1453
f(29) = 1535 = 5*307
f(30) = 1619 = 1619
f(31) = 1705 = 5*11*31
f(32) = 1793 = 11*163
f(33) = 1883 = 7*269
f(34) = 1975 = 5*5*79
f(35) = 2069 = 2069
f(36) = 2165 = 5*433
f(37) = 2263 = 31*73
f(38) = 2363 = 17*139
f(39) = 2465 = 5*17*29
f(40) = 2569 = 7*367
f(41) = 2675 = 5*5*107
f(42) = 2783 = 11*11*23
f(43) = 2893 = 11*263
f(44) = 3005 = 5*601
f(45) = 3119 = 3119
f(46) = 3235 = 5*647
f(47) = 3353 = 7*479
f(48) = 3473 = 23*151
f(49) = 3595 = 5*719
f(50) = 3719 = 3719
f(51) = 3845 = 5*769
f(52) = 3973 = 29*137
f(53) = 4103 = 11*373
f(54) = 4235 = 5*7*11*11
f(55) = 4369 = 17*257
f(56) = 4505 = 5*17*53
f(57) = 4643 = 4643
f(58) = 4783 = 4783
f(59) = 4925 = 5*5*197
f(60) = 5069 = 37*137
f(61) = 5215 = 5*7*149
f(62) = 5363 = 31*173
f(63) = 5513 = 37*149
f(64) = 5665 = 5*11*103
f(65) = 5819 = 11*23*23
f(66) = 5975 = 5*5*239
f(67) = 6133 = 6133
f(68) = 6293 = 7*29*31
f(69) = 6455 = 5*1291
f(70) = 6619 = 6619
f(71) = 6785 = 5*23*59
f(72) = 6953 = 17*409
f(73) = 7123 = 17*419
f(74) = 7295 = 5*1459
f(75) = 7469 = 7*11*97
f(76) = 7645 = 5*11*139
f(77) = 7823 = 7823
f(78) = 8003 = 53*151
f(79) = 8185 = 5*1637
f(80) = 8369 = 8369
f(81) = 8555 = 5*29*59
f(82) = 8743 = 7*1249
f(83) = 8933 = 8933
f(84) = 9125 = 5*5*5*73
f(85) = 9319 = 9319
f(86) = 9515 = 5*11*173
f(87) = 9713 = 11*883
f(88) = 9913 = 23*431
f(89) = 10115 = 5*7*17*17
f(90) = 10319 = 17*607
f(91) = 10525 = 5*5*421
f(92) = 10733 = 10733
f(93) = 10943 = 31*353
f(94) = 11155 = 5*23*97
f(95) = 11369 = 11369
f(96) = 11585 = 5*7*331
f(97) = 11803 = 11*29*37
f(98) = 12023 = 11*1093
f(99) = 12245 = 5*31*79
f(100) = 12469 = 37*337

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+25x-31

f(0)=31
f(1)=5
f(2)=23
f(3)=53
f(4)=17
f(5)=7
f(6)=1
f(7)=193
f(8)=233
f(9)=11
f(10)=29
f(11)=73
f(12)=59
f(13)=463
f(14)=103
f(15)=569
f(16)=1
f(17)=683
f(18)=743
f(19)=1
f(20)=79
f(21)=1
f(22)=1
f(23)=37
f(24)=229
f(25)=1
f(26)=1
f(27)=1373
f(28)=1453
f(29)=307
f(30)=1619
f(31)=1
f(32)=163
f(33)=269
f(34)=1
f(35)=2069
f(36)=433
f(37)=1
f(38)=139
f(39)=1
f(40)=367
f(41)=107
f(42)=1
f(43)=263
f(44)=601
f(45)=3119
f(46)=647
f(47)=479
f(48)=151
f(49)=719
f(50)=3719
f(51)=769
f(52)=137
f(53)=373
f(54)=1
f(55)=257
f(56)=1
f(57)=4643
f(58)=4783
f(59)=197
f(60)=1
f(61)=149
f(62)=173
f(63)=1
f(64)=1
f(65)=1
f(66)=239
f(67)=6133
f(68)=1
f(69)=1291
f(70)=6619
f(71)=1
f(72)=409
f(73)=419
f(74)=1459
f(75)=97
f(76)=1
f(77)=7823
f(78)=1
f(79)=1637
f(80)=8369
f(81)=1
f(82)=1249
f(83)=8933
f(84)=1
f(85)=9319
f(86)=1
f(87)=883
f(88)=431
f(89)=1
f(90)=607
f(91)=421
f(92)=10733
f(93)=353
f(94)=1
f(95)=11369
f(96)=331
f(97)=1
f(98)=1093
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2+25x-31 could be written as f(y)= y^2-187.25 with x=y-12.5

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+12.5
f'(x)>2x+24

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

31, 5, 23, 53, 17, 7, 1, 193, 233, 11, 29, 73, 59, 463, 103, 569, 1, 683, 743, 1, 79, 1, 1, 37, 229, 1, 1, 1373, 1453, 307, 1619, 1, 163, 269, 1, 2069, 433, 1, 139, 1, 367, 107, 1, 263, 601, 3119, 647, 479, 151, 719, 3719, 769, 137, 373, 1, 257, 1, 4643, 4783, 197, 1, 149, 173, 1, 1, 1, 239, 6133, 1, 1291, 6619, 1, 409, 419, 1459, 97, 1, 7823, 1, 1637, 8369, 1, 1249, 8933, 1, 9319, 1, 883, 431, 1, 607, 421, 10733, 353, 1, 11369, 331, 1, 1093, 1, 337, 2539, 12923, 1879, 2677, 13619, 1, 829, 1303, 1, 1, 131, 15313, 1, 3163, 16069, 653, 1, 16843, 311, 1579, 3527, 17903, 1069, 1, 18719, 1, 19273, 19553, 3967, 1, 1, 20693, 20983, 1, 21569, 4373, 599, 3209, 157, 1, 1, 2153, 23993, 4861, 3517, 4987, 25253, 1, 5179, 167, 5309, 349, 2473, 5507, 1, 5641, 1, 1699, 1, 29569, 1, 571, 1, 563, 31319, 181, 1, 1117, 6551, 33119, 1, 1, 4889, 1, 1, 7069, 1, 457, 7297, 1, 7451, 37643, 521, 1, 3529, 1, 5659, 40013, 1, 40819, 1, 1, 42043, 1213, 1, 787, 1, 44123, 1, 1, 1297, 45823, 2011, 9337, 47119, 9511, 4363, 1, 1, 1, 1, 1, 1747, 10223, 1, 2081, 1693, 4813, 971, 523, 10867, 7829, 1783, 11149, 3307, 1, 57173, 57653, 1, 1, 11821, 2591, 60083, 2423, 1, 1759, 1171, 62563, 12613, 5779, 1, 1, 547, 13121, 1787, 13327, 67153, 1, 593, 9817, 1259, 6343, 1, 1, 1, 1, 1, 1237, 1, 4357, 14923, 6833, 6883, 2179, 76819, 619, 77933, 1481, 1, 773, 1, 3511, 7393, 1489, 82469, 977, 4919, 1, 1, 85369, 17191, 2339, 87133, 1, 1, 17783, 89513, 929, 18143, 1723, 3677, 13219, 5479, 1103, 1, 1, 95603, 96223, 2767, 3361, 853, 1, 1361, 19997, 1, 1, 1, 102533, 4127, 1, 1229, 1, 1, 21283, 107069, 1, 1, 1, 1, 15767, 1, 3019, 1091, 983, 113719, 1, 967, 1, 1, 117119, 23561, 118493, 119183, 1, 120569, 1, 121963, 122663, 2243, 11279, 1, 4327, 2381, 1493, 7507, 25667, 1, 18539, 26099, 1, 2399, 733, 133403, 1, 19267, 1, 136343, 1, 1, 138569, 1, 1, 140813, 1231, 142319, 1, 143833, 144593, 4153, 6353, 1013, 1, 1, 1, 149969, 1, 151523, 1, 1801, 1019, 30931, 155443, 2029, 1, 1627, 31723, 1, 1223, 32203, 23117, 1301, 163433, 1, 3001, 887, 1, 23929, 168323, 33829, 5861, 34159, 2351, 5563, 4951, 1439, 3181, 1, 1021, 1, 7753, 5119, 10589, 10639, 36343, 2311, 1, 1523, 26449, 37201, 186869, 37547, 188603, 189473, 38069, 1, 1, 1, 17623, 1, 1, 39301, 1, 1, 1, 3391, 40193, 1, 18433, 1, 6599, 8219, 206383, 207293, 41641, 7211, 1, 12409, 211873, 1, 19429, 42929, 215573, 1, 43487, 1571, 1907, 220243, 1, 1777, 2897, 4073, 9781, 1, 1, 7349, 9151, 1, 230693, 1, 6287, 1, 21323, 235523, 1, 3253, 1, 239423, 1, 2099, 1, 1, 1, 22303, 1, 10753, 49663, 8597, 35759, 50263, 252319, 10133, 1, 1, 1, 36767, 1667, 15259, 15319, 52289, 262469, 1, 1, 9157, 1, 24329, 53731, 1, 1, 1553, 1831, 54773, 1, 275963, 3259, 1487, 1, 280183, 281243, 1, 283369, 1, 3911, 40939, 57529, 12553, 1, 1, 1613, 1, 42017, 1, 1, 1, 11939, 299569, 60133, 3919, 2503, 60793, 305069, 1, 307283, 5227, 1, 10711, 62347, 1621, 1, 1, 316219, 9067, 318473, 5417, 2789, 6073, 64601, 324143, 1499, 1187, 1, 2267, 329863, 331013, 2143, 2801, 1, 335633, 336793, 1, 30829, 1, 48779, 4337, 2371, 344969, 1, 15101, 9419, 1, 350869, 1, 1889, 20849, 1, 356819, 1, 4547, 1, 2333, 12511, 14561, 33203, 4759, 1, 1, 74017, 371303, 372523, 4397, 1, 75239, 6397, 1, 6907, 2203, 76471, 54799, 1, 15443, 2467, 77713, 6607, 391063, 1, 1, 1, 1, 10739, 2749, 12899, 1, 402403, 17551, 80989, 36929, 1, 408773, 58579, 82267, 412619, 82781, 1, 24499, 1, 1, 7643, 38333, 1847, 1601, 18503, 1, 61169, 1399, 86161, 1, 3769, 3593, 1367, 12497, 1, 1, 1, 442703, 88807, 12037, 12763, 1, 40853, 1, 2707, 1, 454763, 2833, 91493, 458819, 1, 1597, 1, 1, 6047, 1583, 468353, 469723, 94219, 472469, 1, 2341, 476603, 95597, 43579, 8741, 1, 4519, 1, 28607, 4241, 489113, 15823, 2659, 17011, 1, 1, 1, 1, 500369, 100357, 1, 72089, 101209, 1, 5987, 2729, 46523, 1, 73517, 3559, 2221, 518933, 1, 521819, 1, 74959, 1543, 1, 23003, 21221, 531983, 31379, 1, 9091, 1, 539303, 540773, 9859, 49429, 1, 1, 548153, 2971, 9341, 1, 24091, 1, 22283, 1, 1, 4099, 24481, 112913, 1, 1, 569083, 1, 3691, 1, 10457, 7489, 19937, 1, 18749, 116549, 34369, 2027, 1, 25603, 118081, 53813, 1, 23801, 596569, 1, 599663, 16249, 4157, 604319, 1, 1, 1, 1, 36007, 1, 615253, 6359, 1, 2857, 124309, 21487, 1, 1, 5189, 1, 90149, 2647, 1, 6173, 7499, 37589, 640613, 1, 1, 2347, 647033, 22367, 130051, 651869, 18671, 655103, 12391, 1, 17837, 1, 60293, 1, 7841, 1, 1, 29191, 4517, 26987, 1, 135593, 1993, 61933, 1871, 684569, 27449, 98269, 4129, 138241, 1, 8171, 24007, 63443, 1, 701219, 1, 4219, 706253, 141587, 30853, 20323, 2711, 4733, 1, 2251, 143953, 1, 1, 144973, 1, 29131, 1, 23603, 146681, 9547, 13397, 1, 12547, 148399, 2131, 1, 106739, 32561, 8831, 44257, 13711, 68713, 757583, 4339, 761069, 152563, 764563, 766313, 5297, 3221, 4409, 1, 2273, 155371, 1, 9181, 1, 2113, 1, 787469, 6863, 21379, 72073, 14447, 3923, 159631, 5839, 801733, 32141, 1, 1, 1, 1, 1, 1, 6529, 817933, 1, 23473, 4549, 165037, 1, 828823, 1, 75679, 1, 1931, 10607, 3169, 1, 9923, 14327, 121019, 1, 6121, 1, 1, 11731, 4639, 122867, 1, 863783, 865643, 173501, 869369, 1, 1, 4679, 1, 878719, 3323, 5101, 884353, 25321, 28649, 178001, 891893, 1, 3257, 5717, 1, 1, 31147, 1, 1, 1, 910883, 130399, 16631, 3623, 183707, 1, 1, 184859, 1, 1, 930073, 932003, 1, 1, 1, 1, 941683, 7549, 12953, 6113, 41281, 951413, 27239, 955319, 1, 1, 1, 192631, 1, 27631, 5021, 57119, 11447, 974969, 195389, 88993, 12739, 3709, 33961, 197371, 12517, 1, 1, 1, 2731, 998813, 2459, 18233, 59107, 1, 4649, 1010843, 1, 1014869, 7013, 1018903, 32933, 2657,

6. Sequence of the polynom (only primes)

31, 5, 23, 53, 17, 7, 193, 233, 11, 29, 73, 59, 463, 103, 569, 683, 743, 79, 37, 229, 1373, 1453, 307, 1619, 163, 269, 2069, 433, 139, 367, 107, 263, 601, 3119, 647, 479, 151, 719, 3719, 769, 137, 373, 257, 4643, 4783, 197, 149, 173, 239, 6133, 1291, 6619, 409, 419, 1459, 97, 7823, 1637, 8369, 1249, 8933, 9319, 883, 431, 607, 421, 10733, 353, 11369, 331, 1093, 337, 2539, 12923, 1879, 2677, 13619, 829, 1303, 131, 15313, 3163, 16069, 653, 16843, 311, 1579, 3527, 17903, 1069, 18719, 19273, 19553, 3967, 20693, 20983, 21569, 4373, 599, 3209, 157, 2153, 23993, 4861, 3517, 4987, 25253, 5179, 167, 5309, 349, 2473, 5507, 5641, 1699, 29569, 571, 563, 31319, 181, 1117, 6551, 33119, 4889, 7069, 457, 7297, 7451, 37643, 521, 3529, 5659, 40013, 40819, 42043, 1213, 787, 44123, 1297, 45823, 2011, 9337, 47119, 9511, 4363, 1747, 10223, 2081, 1693, 4813, 971, 523, 10867, 7829, 1783, 11149, 3307, 57173, 57653, 11821, 2591, 60083, 2423, 1759, 1171, 62563, 12613, 5779, 547, 13121, 1787, 13327, 67153, 593, 9817, 1259, 6343, 1237, 4357, 14923, 6833, 6883, 2179, 76819, 619, 77933, 1481, 773, 3511, 7393, 1489, 82469, 977, 4919, 85369, 17191, 2339, 87133, 17783, 89513, 929, 18143, 1723, 3677, 13219, 5479, 1103, 95603, 96223, 2767, 3361, 853, 1361, 19997, 102533, 4127, 1229, 21283, 107069, 15767, 3019, 1091, 983, 113719, 967, 117119, 23561, 118493, 119183, 120569, 121963, 122663, 2243, 11279, 4327, 2381, 1493, 7507, 25667, 18539, 26099, 2399, 733, 133403, 19267, 136343, 138569, 140813, 1231, 142319, 143833, 144593, 4153, 6353, 1013, 149969, 151523, 1801, 1019, 30931, 155443, 2029, 1627, 31723, 1223, 32203, 23117, 1301, 163433, 3001, 887, 23929, 168323, 33829, 5861, 34159, 2351, 5563, 4951, 1439, 3181, 1021, 7753, 5119, 10589, 10639, 36343, 2311, 1523, 26449, 37201, 186869, 37547, 188603, 189473, 38069, 17623, 39301, 3391, 40193, 18433, 6599, 8219, 206383, 207293, 41641, 7211, 12409, 211873, 19429, 42929, 215573, 43487, 1571, 1907, 220243, 1777, 2897, 4073, 9781, 7349, 9151, 230693, 6287, 21323, 235523, 3253, 239423, 2099, 22303, 10753, 49663, 8597, 35759, 50263, 252319, 10133, 36767, 1667, 15259, 15319, 52289, 262469, 9157, 24329, 53731, 1553, 1831, 54773, 275963, 3259, 1487, 280183, 281243, 283369, 3911, 40939, 57529, 12553, 1613, 42017, 11939, 299569, 60133, 3919, 2503, 60793, 305069, 307283, 5227, 10711, 62347, 1621, 316219, 9067, 318473, 5417, 2789, 6073, 64601, 324143, 1499, 1187, 2267, 329863, 331013, 2143, 2801, 335633, 336793, 30829, 48779, 4337, 2371, 344969, 15101, 9419, 350869, 1889, 20849, 356819, 4547, 2333, 12511, 14561, 33203, 4759, 74017, 371303, 372523, 4397, 75239, 6397, 6907, 2203, 76471, 54799, 15443, 2467, 77713, 6607, 391063, 10739, 2749, 12899, 402403, 17551, 80989, 36929, 408773, 58579, 82267, 412619, 82781, 24499, 7643, 38333, 1847, 1601, 18503, 61169, 1399, 86161, 3769, 3593, 1367, 12497, 442703, 88807, 12037, 12763, 40853, 2707, 454763, 2833, 91493, 458819, 1597, 6047, 1583, 468353, 469723, 94219, 472469, 2341, 476603, 95597, 43579, 8741, 4519, 28607, 4241, 489113, 15823, 2659, 17011, 500369, 100357, 72089, 101209, 5987, 2729, 46523, 73517, 3559, 2221, 518933, 521819, 74959, 1543, 23003, 21221, 531983, 31379, 9091, 539303, 540773, 9859, 49429, 548153, 2971, 9341, 24091, 22283, 4099, 24481, 112913, 569083, 3691, 10457, 7489, 19937, 18749, 116549, 34369, 2027, 25603, 118081, 53813, 23801, 596569, 599663, 16249, 4157, 604319, 36007, 615253, 6359, 2857, 124309, 21487, 5189, 90149, 2647, 6173, 7499, 37589, 640613, 2347, 647033, 22367, 130051, 651869, 18671, 655103, 12391, 17837, 60293, 7841, 29191, 4517, 26987, 135593, 1993, 61933, 1871, 684569, 27449, 98269, 4129, 138241, 8171, 24007, 63443, 701219, 4219, 706253, 141587, 30853, 20323, 2711, 4733, 2251, 143953, 144973, 29131, 23603, 146681, 9547, 13397, 12547, 148399, 2131, 106739, 32561, 8831, 44257, 13711, 68713, 757583, 4339, 761069, 152563, 764563, 766313, 5297, 3221, 4409, 2273, 155371, 9181, 2113, 787469, 6863, 21379, 72073, 14447, 3923, 159631, 5839, 801733, 32141, 6529, 817933, 23473, 4549, 165037, 828823, 75679, 1931, 10607, 3169, 9923, 14327, 121019, 6121, 11731, 4639, 122867, 863783, 865643, 173501, 869369, 4679, 878719, 3323, 5101, 884353, 25321, 28649, 178001, 891893, 3257, 5717, 31147, 910883, 130399, 16631, 3623, 183707, 184859, 930073, 932003, 941683, 7549, 12953, 6113, 41281, 951413, 27239, 955319, 192631, 27631, 5021, 57119, 11447, 974969, 195389, 88993, 12739, 3709, 33961, 197371, 12517, 2731, 998813, 2459, 18233, 59107, 4649, 1010843, 1014869, 7013, 1018903, 32933, 2657,

7. Distribution of the primes

Legend of the table: I distinguish between primes p= x^2+25x-31 and
the reducible primes which appear as divisor for the first time
p | x^2+25x-31 and p < x^2+25x-31

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)
1109720.9000000.7000000.9000000.0000000.0000000.000000
21006928410.6900000.2800000.6900007.6666674.00000020.500000
31.0006801745060.6800000.1740000.6800009.8550726.21428612.341463
410.0006.9161.2515.6650.6916000.1251000.69160010.1705887.18965511.195652
5100.00069.0209.75059.2700.6902000.0975000.6902009.9797577.79376510.462489
61.000.000690.88878.856612.0320.6908880.0788560.69088810.0099688.08779510.326168
710.000.0006.909.844669.0706.240.7740.6909840.0669070.69098410.0013958.48470610.196810
8100.000.00069.108.2765.799.58663.308.6900.6910830.0579960.69108310.0014238.66813010.144364
91.000.000.000691.202.39351.198.906640.003.4870.6912020.0511990.69120210.0017318.82802810.109252
1010.000.000.0006.913.317.235458.174.7976.455.142.4380.6913320.0458170.69133210.0018728.94891710.086105


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)
123301.5000001.5000000.0000000.0000000.0000000.000000
245411.2500001.0000000.2500001.6666671.333333inf
388711.0000000.8750000.1250001.6000001.7500001.000000
41614950.8750000.5625000.3125001.7500001.2857145.000000
532231490.7187500.4375000.2812501.6428571.5555561.800000
6644620260.7187500.3125000.4062502.0000001.4285712.888889
71288937520.6953120.2890620.4062501.9347831.8500002.000000
8256170571130.6640620.2226560.4414061.9101121.5405412.173077
9512342992430.6679690.1933590.4746092.0117651.7368422.150442
101.0246981785200.6816410.1738280.5078122.0409361.7979802.139918
112.0481.3983281.0700.6826170.1601560.5224612.0028651.8426972.057692
124.0962.8285862.2420.6904300.1430660.5473632.0228901.7865852.095327
138.1925.6551.0654.5900.6903080.1300050.5603031.9996461.8174062.047279
1416.38411.2991.9079.3920.6896360.1163940.5732421.9980551.7906102.046187
1532.76822.6273.56319.0640.6905210.1087340.5817872.0025671.8683802.029813
1665.53645.2656.62338.6420.6906890.1010590.5896302.0004861.8588272.026962
17131.07290.55012.36078.1900.6908420.0942990.5965422.0004421.8662242.023446
18262.144181.05323.120157.9330.6906620.0881960.6024671.9994811.8705502.019862
19524.288362.19743.635318.5620.6908360.0832270.6076092.0005031.8873272.017071
201.048.576724.55482.415642.1390.6909890.0785970.6123912.0004421.8887362.015743
212.097.1521.448.881156.7001.292.1810.6908800.0747200.6161601.9996871.9013532.012307
224.194.3042.898.170297.9132.600.2570.6909780.0710280.6199502.0002821.9011682.012301
238.388.6085.796.470567.9085.228.5620.6909930.0677000.6232932.0000451.9062882.010787
2416.777.21611.593.5961.084.72910.508.8670.6910320.0646550.6263772.0001131.9100432.009896
2533.554.43223.187.0112.077.45921.109.5520.6910270.0619130.6291141.9999851.9151872.008737
2667.108.86446.376.0383.983.84742.392.1910.6910570.0593640.6316932.0000871.9176542.008199
27134.217.72892.757.3237.655.15785.102.1660.6910960.0570350.6340612.0001131.9215492.007496
28268.435.456185.523.13014.730.499170.792.6310.6911280.0548750.6362522.0000921.9242582.006913
29536.870.912371.067.23728.384.869342.682.3680.6911670.0528710.6382962.0001131.9269452.006424
301.073.741.824742.179.87554.774.905687.404.9700.6912090.0510130.6401962.0001231.9297222.005954
312.147.483.6481.484.447.403105.823.0341.378.624.3690.6912500.0492780.6419722.0001181.9319622.005549
324.294.967.2962.969.068.881204.674.0422.764.394.8390.6912900.0476540.6436362.0001171.9341162.005183
338.589.934.5925.938.439.440396.306.3845.542.133.0560.6913250.0461360.6451892.0001021.9362802.004827
3417.179.869.18411.877.483.394768.167.93111.109.315.4630.6913610.0447130.6466472.0001021.9383182.004520


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
123120012
244130022
387252023
4169363024
532144103245
664205153458
712837122588912
825657203712141417
951299326722242627
101.0241785712140464745
112.04832810522379848382
124.096586199387148151147140
138.1921.065363702259279259268
1416.3841.9076591.248472491467477
1532.7683.5631.1912.372906881889887
1665.5366.6232.1904.4331.6621.6501.6551.656
17131.07212.3604.0998.2613.1223.0723.0713.095
18262.14423.1207.66215.4585.7955.7535.7855.787
19524.28843.63514.57229.06310.93010.85311.00310.849
201.048.57682.41527.45554.96020.63120.55620.77920.449
212.097.152156.70052.459104.24139.33439.06439.27539.027
224.194.304297.91399.683198.23074.78174.37474.52374.235
238.388.608567.908189.605378.303142.175142.001142.046141.686
2416.777.2161.084.729361.500723.229271.440271.281271.176270.832
2533.554.4322.077.459692.6991.384.760519.542519.073519.678519.166
2667.108.8643.983.8471.328.5202.655.327995.707995.702996.698995.740
27134.217.7287.655.1572.552.4115.102.7461.913.1801.913.8151.914.0021.914.160
28268.435.45614.730.4994.912.0699.818.4303.681.0013.682.4393.683.0213.684.038
29536.870.91228.384.8699.462.04118.922.8287.094.6487.096.1167.096.3937.097.712
301.073.741.82454.774.90518.258.45536.516.45013.691.15413.693.17013.696.40613.694.175
312.147.483.648105.823.03435.271.73670.551.29826.454.23326.455.18126.456.86726.456.753
324.294.967.296204.674.04268.220.855136.453.18751.165.87351.167.16751.170.42551.170.577
338.589.934.592396.306.384132.100.677264.205.70799.074.04399.073.30799.082.14099.076.894
3417.179.869.184768.167.931256.053.970512.113.961192.040.301192.041.105192.045.648192.040.877


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
241011000
381011000
4165232111
5329632322
6642613137478
712852312111121415
8256113714226332826
951224313710655665765
101.024520275245129148118125
112.0481.070578492261289261259
124.0962.2421.1671.075570577548547
138.1924.5902.3472.2431.1741.1771.1101.129
1416.3849.3924.8664.5262.3892.3302.3622.311
1532.76819.0649.8379.2274.7334.7954.8224.714
1665.53638.64219.85818.7849.6699.6479.7469.580
17131.07278.19040.16638.02419.45619.51019.75119.473
18262.144157.93381.14276.79139.33939.39739.66139.536
19524.288318.562163.449155.11379.48979.40879.89779.768
201.048.576642.139329.180312.959160.426160.447160.697160.569
212.097.1521.292.181661.285630.896322.797322.732323.480323.172
224.194.3042.600.2571.329.1031.271.154649.614649.534650.740650.369
238.388.6085.228.5622.669.2052.559.3571.306.1561.307.2341.307.4601.307.712
2416.777.21610.508.8675.360.7455.148.1222.626.4602.627.1522.627.2042.628.051
2533.554.43221.109.55210.753.86610.355.6865.276.9575.277.1385.277.7485.277.709
2667.108.86442.392.19121.578.55520.813.63610.596.41210.598.97610.600.42410.596.379
27134.217.72885.102.16643.279.70641.822.46021.274.58921.280.60021.271.50821.275.469
28268.435.456170.792.63186.797.73083.994.90142.689.75742.698.54842.699.16842.705.158
29536.870.912342.682.368174.036.760168.645.60885.662.61885.670.55185.670.00885.679.191
301.073.741.824687.404.970348.908.732338.496.238171.844.879171.846.314171.861.292171.852.485
312.147.483.6481.378.624.369699.329.500679.294.869344.628.987344.641.413344.668.468344.685.501
324.294.967.2962.764.394.8391.401.552.1211.362.842.718691.076.657691.082.892691.096.050691.139.240
338.589.934.5925.542.133.0562.808.514.9822.733.618.0741.385.514.1101.385.516.7331.385.523.7021.385.578.511
3417.179.869.18411.109.315.4635.627.199.3505.482.116.1132.777.299.0622.777.332.2662.777.304.7982.777.379.337


8. Check for existing Integer Sequences by OEIS

Found in Database : 31, 5, 23, 53, 17, 7, 1, 193, 233, 11, 29, 73, 59, 463, 103, 569, 1, 683, 743, 1,
Found in Database : 31, 5, 23, 53, 17, 7, 193, 233, 11, 29, 73, 59, 463, 103, 569, 683, 743, 79, 37, 229, 1373, 1453, 307, 1619, 163, 269, 2069, 433, 139,
Found in Database : 5, 7, 11, 17, 23, 29, 31, 37, 53, 59, 73, 79, 97, 103, 107, 131, 137, 139, 149,