A326236 -id:A326236 - cp's OEIS frontend

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, 400, 404, 420, 422, 449, 462, 481, 483, 499, 509, 537, 551, 568, 587, 590, 609, 651, 652, 667, 684, 730, 755, 761, 806, 817, 825, 827, 848, 867, 870, 882, 916, 931, 980, 982, 992

Offset: 1

M. F. Hasler and Antonie Dinculescu, Jun 14 2019

nonn

Dinculescu notes that if N = m^2 > 1 is a twin rank (i.e., in A002822), then m is always a multiple of 5, and if N = m^3 > 1, then m is a multiple of 7, cf. A326234. He asks whether there are other pairs (a, b) different from (5, 2) and (7, 3) such that all twin ranks m^b > 1 are of the form m = a*n. (Of course (5, 2) and (7, 3) imply that (5, 2k), (7, 3k) and (35, 6k) is such a pair for any k.) This sequence lists the n for (a, b) = (5, 2), see A326232 for the numbers m.

See A326233, A326234 for m^3 and A326235, A326236 for m^6.

M. F. Hasler, Table of n, a(n) for n = 1..10000
A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

Cf. A002822, A326232 ({1} U {5*a(n)}), A326233 (analog for m^3), A326234, A326235 (analog for m^6), A326230 (least twin rank n^k > 1 for given k).

select( is(n)=!for(s=1,2,ispseudoprime(150*n^2+(-1)^s)||return), [1..10^3])

a(n) = A326232(n+1)/5.

A326232 Numbers k such that N = k^2 is a twin rank (cf. A002822: 6N +- 1 are twin primes).

Original entry on oeis.org

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, 1715, 1850, 1890, 1920, 2000, 2020, 2100, 2110, 2245, 2310, 2405, 2415, 2495, 2545, 2685, 2755, 2840, 2935, 2950, 3045, 3255, 3260, 3335, 3420, 3650, 3775, 3805

Offset: 1

M. F. Hasler and Antonie Dinculescu, Jun 14 2019

nonn

Dinculescu notes that when k^2 > 1 is a twin rank (i.e., in A002822), then k is always a multiple of 5, and if k^3 > 1 is a twin rank, it is divisible by 7. See A326231 for the terms > 1 divided by 5.

See A326234 and A326233 for k^3, A326236 and A326235 for k^6.

A. Dinculescu, Table of n, a(n) for n = 1..10001
A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

Cf. A002822, A326231 (a(n)/5, n>1), A326233, A326234 (analog for k^3), A326235, A326236 (analog for k^6), A326230 (least twin rank m^n for given n).

select( is(n)=!for(s=1,2,ispseudoprime(6*n^2+(-1)^s)||return), [1..5000])

A326234 Numbers n such that N = n^3 is a twin rank (A002822: 6N +- 1 are twin primes).

Original entry on oeis.org

1, 28, 42, 168, 203, 287, 308, 518, 1043, 1057, 1512, 1603, 1638, 1680, 1757, 1988, 2905, 3367, 3927, 4018, 4928, 5033, 5145, 5257, 5292, 5432, 5733, 6762, 7182, 7210, 7798, 8715, 10213, 10318, 10668, 10745, 11088, 12243, 13552, 14245, 14588, 14707, 15155, 15323, 15687, 15722, 15757

Offset: 1

M. F. Hasler and Antonie Dinculescu, Jun 14 2019

nonn

Dinculescu notes that when n^2 or n^3 is a twin rank > 1 (i.e., in A002822), then n is a multiple of 5, resp. 7. It is unknown whether there exist other pairs (a, b) different from (5, 2) and (7, 3) such that n^b => a | n. (Of course (5, 2k) and (7, 3k) and (35, 6k) is a solution for any k.) See A326233 for the terms > 1 divided by 7.

See A326232 and A326231 for the case n^2, A326236 and A326235 for n^6.

A. Dinculescu, Table of n, a(n) for n = 1..14709
A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

Cf. A002822, A326233 (a(n)/7, n>1), A326231, A326232 (analog for n^2), A326235, A326236 (analog for n^6), A326230 (least twin rank n^k > 1 for given k).

select( is(n)=!for(s=1,2,ispseudoprime(6*n^3+(-1)^s)||return), [1..10^5])

a(n) = 7*A326233(n-1), n >= 2.

A326230 Least k > 1 such that k^n is a twin rank (cf. A002822: 6*k^n +- 1 are twin primes).

Original entry on oeis.org

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, 38157, 1185, 1473, 12530, 4338, 1540, 1988, 535, 102, 22606, 13773, 18895, 16373, 2635, 20428, 76300, 23037, 29005, 11078

Offset: 1

M. F. Hasler and Antonie Dinculescu, Jun 16 2019

nonn

Dinculescu observes that when k^2 > 1 is a twin rank (i.e., in A002822) then 5 | k (k is divisible by 5), and if k^3 is a twin rank, then 7 | k; cf. A326232 & A326234. It is unknown whether there are other pairs (a, b) such that a | n whenever n^b > 1 is a twin rank. (Of course 2 | b => 5 | a and 3 | b => 7 | a, so we aren't interested in pairs (a, b) which are consequence of this.)

A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

Cf. A326231 .. A326236, A002822.

a(n)=for(k=2,oo,ispseudoprime(6*k^n-1)&&ispseudoprime(6*k^n+1)&&return(k))

A326233 Numbers n such that N = (7n)^3 is a twin rank (A002822: 6N +- 1 are twin primes).

Original entry on oeis.org

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, 1030, 1114, 1245, 1459, 1474, 1524, 1535, 1584, 1749, 1936, 2035, 2084, 2101, 2165, 2189, 2241, 2246, 2251, 2301, 2305, 2384, 2511, 2541, 2710, 2865, 2955, 2990

Offset: 1

M. F. Hasler and Antonie Dinculescu, Jun 14 2019

nonn

Dinculescu notes that if m^3 > 1 is a twin rank (i.e., in A002822), then m is always a multiple of 7. (Indeed, 6m^3 + 1 == 0 (mod 7) if m == 1, 2 or 4 (mod 7), and 6m^3 - 1 == 0 (mod 7) for m == 3, 5 or 6 (mod 7).)
He asks whether there are other pairs (a, b) different from (5, 2) and (7, 3) such that all twin ranks m^b > 1 are of the form m = a*n. (Of course (5, 2) and (7, 3) imply that (5, 2k), (7, 3k) and (35, 6k) is also such a pair for any k >= 1.)
This sequence lists these m/7 for (a, b) = (7, 3), see A326234 for the numbers m.
See A326231, A326232 for m^2 and A326235, A326236 for m^6.

Robert Israel, Table of n, a(n) for n = 1..10000
A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

Cf. A002822, A326234 ({1} U 7*{a(n)}), A326231 (analog for n^2), A326232, A326235 (analog for n^6), A326236, A326230 (least twin rank n^k > 1 for given k).

filter:= proc(n) local m;
  m:= (7*n)^3;
isprime(6*m+1) and isprime(6*m-1)
end proc:
select(filter, [$1..3000]); # Robert Israel, Jun 17 2019

select( is(n)=!for(s=1,2,ispseudoprime(6*(7*n)^3+(-1)^s)||return), [1..10^4])

a(n) = A326234(n+1)/7.

A326235 Numbers k such that N = (35k)^6 is a twin rank (A002822: 6N +- 1 are twin primes).

Original entry on oeis.org

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, 3216, 3263, 3360, 3632, 3692, 3762, 4094, 4364, 4603, 4689, 4858, 5003, 5177, 5287, 5361, 5426, 5999, 6054, 6285, 6347, 6417, 6457, 6639, 6862, 7269, 7500

Offset: 1

M. F. Hasler and Antonie Dinculescu, Jun 14 2019

nonn

Dinculescu notes that if N = m^2 > 1 is a twin rank (i.e., in A002822), then m is a multiple of 5, and if N = m^3 > 1, then m is a multiple of 7, cf. A326231 and A326233. Thus, when N = m^6, then m is a multiple of 35, and here we list these m/35.

See A326236 for the numbers m.

A. Dinculescu and M. F. Hasler, Table of n, a(n) for n = 1..10000
A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

Cf. A002822, A326231 (analog for m^2), A326232, A326233 (analog for m^3), A326234, A326236 ({1} U {35*a(n)}), A326230 (least twin rank m^n for given n).

select( is(n)=!for(s=1,2,ispseudoprime(6*(35*n)^6+(-1)^s)||return), [1..10^4])

a(n) = A326236(n+1)/35.

a(1..10^4) independently computed using Mathematica and PARI/GP, by A. D. and M. F. Hasler, Jun 19 2019

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.

Original entry on oeis.org

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, 425, 4795, 123, 1850, 23, 240, 518, 850, 873, 5565, 192, 3815, 2520, 25, 385, 1043, 970, 1608, 8330, 462, 5840, 5432, 220, 30, 430, 1057, 1255, 1713, 8470, 948, 6270, 6020, 560, 2023, 32

Offset: 1

M. F. Hasler, Jun 16 2019

We assume that all rows have infinite length, in case this should not be the case we would fill the row with 0's after the last term.

From [Dinculescu] we know that whenever 2|n or 3|n, then all terms of row n are multiples of 5 resp. of 7 (where | means "divides"), cf. A326231 - A326234. We do not know other (independent) pairs (a, b) such that (m^b in A002822) implies a|m.

			The array starts:
[   2     3     5     7    10    12    17    18   ...] = A002822 \ {1}
[   5    10    35    60    70    75   210   240   ...] = A326232 \ {1}
[  28    42   168   203   287   308   518  1043   ...] = A326234 \ {1}
[  70    75    80   175   485   850   970  1255   ...]
[   2    40   208   425   873  1608  1713  1718   ...]
[1820  2590  4795  5565  8330  8470 10640 10710   ...] = A326236 \ {1}
[ 110   123   192   462   948  1242  1255  1747   ...]
[1850  3815  5840  6270  8075  8960  9210 10420   ...]
[2520  5432  6020 10535 24017 29092 29295 29967   ...]
(...)
Column 1 is A326230(n): smallest m > 1 such that m^n is in A002822 (twin ranks).

A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

Cf. A002822, A326230, A326231, ..., A326236.

A326229_row(n,LENGTH=20)={my(g=5^!(n%2)*7^!(n%3),m=max(g,2)-g); vector(LENGTH,i,while(m+=g,for(s=1,2,ispseudoprime(6*m^n+(-1)^s)||next(2));break);m)}

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A326231 Numbers n such that N = (5n)^2 is a twin rank (cf. A002822: 6N +- 1 are twin primes).

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A326232 Numbers k such that N = k^2 is a twin rank (cf. A002822: 6N +- 1 are twin primes).

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A326234 Numbers n such that N = n^3 is a twin rank (A002822: 6N +- 1 are twin primes).

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A326230 Least k > 1 such that k^n is a twin rank (cf. A002822: 6*k^n +- 1 are twin primes).

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A326233 Numbers n such that N = (7n)^3 is a twin rank (A002822: 6N +- 1 are twin primes).

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A326235 Numbers k such that N = (35k)^6 is a twin rank (A002822: 6N +- 1 are twin primes).

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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.

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PARI