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User:M. F. Hasler/Work in progress/Twin primes

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This is about some recent work on twin primes, mainly inspired by previous work by A. Dinculescu ([[#References|]] [D12]-[D18]) and personal communications with the author. The "Holy Grail" is the Twin prime conjecture, but for the moment we just try to establish some useful intermediate results.

Related sequences

A002822 Numbers n such that 6n-1, 6n+1 are twin primes.
1, 2, 3, 5, 7, 10, 12, 17, 18, 23, 25, 30, 32, 33, 38, 40, 45, 47, 52, 58, 70, 72, 77, 87, 95, 100, 103, 107, 110, 135, 137, 138, 143, ...
A057767 Number of twin prime pairs between P(n)^2 and P(n+1)^2 where P(n) is the n-th prime.
1, 2, 2, 4, 2, 7, 2, 4, 8, 2, 11, 7, 3, 11, 13, 13, 5, 19, 11, 3, 15, 14, 14, 21, 15, 7, 10, 6, 11, 42, 12, 27, 6, 45, 10, 20, 17, 21,...
A263282 Numbers n such that 6n is in A002822 but n is not.
63, 65, 88, 98, 102, 133, 157, 163, 185, 193, 198, 203, 208, 210, 233, 245, 250, 262, 310, 340, 380, 387, 413, 437, 457, 462, 473, 478,...

Twin Goldbach conjecture related

A326228 Primes p such that the distance from p#/6 to the next larger or smaller twin rank is not in A002822. (# = primorial, A034386)
41, 227, 307, 311, 349, 457, 613

Perfect power twin ranks related

A326229 Square array T(n,k) where row n >= 1 lists numbers m > 1 such that 6*m^n +- 1 are twin primes; read by falling antidiagonals.
2, 3, 5, 5, 10, 28, 7, 35, 42, 70, 10, 60, 168, 75, 2, 12, 70, 203, 80, 40, 1820, 17, 75, 287, 175, 208, 2590, 110, 18, 210, 308, 485...
A326230 Least k > 1 such that k^n is a twin rank (cf. A002822: 6*k^n +- 1 are twin primes).
2, 5, 28, 70, 2, 1820, 110, 1850, 2520, 220, 2023, 9415, 647, 2880, 2562, 3895, 2, 51240, 525, 3750, 147, 2350, 355, 4480, 2588, 3370,...
A326231 Numbers n such that N = (5n)^2 is a twin rank (cf. A002822: 6N +- 1 are twin primes).
1, 2, 7, 12, 14, 15, 42, 48, 77, 86, 89, 99, 118, 131, 146, 161, 163, 167, 201, 208, 209, 246, 278, 286, 306, 334, 343, 370, 378, 384,...
A326232 Numbers n such that N = n^2 is a twin rank (cf. A002822: 6N +- 1 are twin primes).
1, 5, 10, 35, 60, 70, 75, 210, 240, 385, 430, 445, 495, 590, 655, 730, 805, 815, 835, 1005, 1040, 1045, 1230, 1390, 1430, 1530, 1670,...
A326233 Numbers n such that N = (7n)^3 is a twin rank (A002822: 6N +- 1 are twin primes).
4, 6, 24, 29, 41, 44, 74, 149, 151, 216, 229, 234, 240, 251, 284, 415, 481, 561, 574, 704, 719, 735, 751, 756, 776, 819, 966, 1026, ...
A326234 Numbers n such that N = n^3 is a twin rank (A002822: 6N +- 1 are twin primes).
1, 28, 42, 168, 203, 287, 308, 518, 1043, 1057, 1512, 1603, 1638, 1680, 1757, 1988, 2905, 3367, 3927, 4018, 4928, 5033, 5145, 5257, ...
A326235 Numbers n such that N = (35n)^6 is a twin rank (A002822: 6N +- 1 are twin primes).
52, 74, 137, 159, 238, 242, 304, 306, 456, 478, 547, 701, 756, 988, 1059, 1186, 1218, 1243, 1378, 1705, 1976, 2426, 2596, 2844, 2952, ...
A326236 Numbers n such that N = n^6 is a twin rank (cf. A002822: 6N +- 1 are twin primes).
1, 1820, 2590, 4795, 5565, 8330, 8470, 10640, 10710, 15960, 16730, 19145, 24535, 26460, 34580, 37065, 41510, 42630, 43505, 48230, ...

Others

A308344(n) = (A001359(n+1)^2 - 1)/24, where A001359 = lesser of twin primes
equivalently: pentagonal numbers (A000326) whose indices are twin ranks (A002822).
1, 5, 12, 35, 70, 145, 210, 425, 477, 782, 925, 1335, 1520, 1617, 2147, 2380, 3015, 3290, 4030, 5017, 7315, 7740, 8855, 11310, 13490, 14950, 15862, 17120, 18095, 27270, 28085, 28497, 30602, 32340, 43265, 44290, 45850, 46905, 49595, 55200, 62935, 67947, 69230, 7052

References

[D12] A. Dinculescu: On Some Infinite Series Related to the Twin Primes.
The Open Mathematics Journal, 5 (2012), 8--14. doi: 10.2174/1874117701205010008
[D13] A. Dinculescu: The Twin Primes Seen from a Different Perspective.
British Journal of Mathematics & Computer Science, 3 no 4 (2013) 691--698. doi: 10.9734/BJMCS/2013/4358
[D16] A. Dinculescu: A Twin Prime Analog of Goldbach’s Conjecture.
British Journal of Mathematics & Computer Science, 12 no 5 (2016) 1--4. doi: 10.9734/BJMCS/2016/22064 - Article no.BJMCS.22064 - ISSN: 2231-0851
[D18] A. Dinculescu: On the Numbers that Determine the Distribution of Twin Primes.
Surveys in Mathematics and its Applications, 13 (2018), 171-181. URL: http://www.utgjiu.ro/math/sma/v13/a13_11.html