A002822 -id:A002822 - cp's OEIS frontend

A060212 Primes q such that 6q-1 and 6q+1 are twin primes. Proper subset of A002822.

2, 3, 5, 7, 17, 23, 47, 103, 107, 137, 283, 313, 347, 373, 397, 443, 467, 577, 593, 653, 773, 787, 907, 1033, 1117, 1423, 1433, 1613, 1823, 2027, 2063, 2137, 2153, 2203, 2287, 2293, 2333, 2347, 2677, 2903, 3257, 3307, 3407, 3413, 3593, 3623, 3673, 3923

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Primes in A182521. Also all primes p for which A182481(p)=1. - Vladimir Shevelev, May 03 2012

Conjecture: a(n) ~ n*log(n)*log(n*log(n))*log(log(n)). - Carl R. White, Nov 16 2023

David Radcliffe, Table of n, a(n) for n = 1..10000

Cf. A001359, A002822, A014574, A027856, A058383, A059960, A182481, A182521, A294731.

lst={}; Do[p=Prime[n]; If[PrimeQ[6*p-1] && PrimeQ[6*p+1], AppendTo[lst,p]], {n,100}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 16 2009 *)

forprime(p=2, 9999, if(isprime(6*p+1) & isprime(6*p-1), print(p))) \\ David Radcliffe, Apr 02 2016

from sympy import *; print([p for p in primerange(2,9999) if isprime(6*p-1) and isprime(6*p+1)]) # David Radcliffe, Apr 02 2016

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

Original entry on oeis.org

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.

A326236 Numbers k such that N = k^6 is a twin rank (cf. A002822: 6N +- 1 are twin primes).

Original entry on oeis.org

1, 1820, 2590, 4795, 5565, 8330, 8470, 10640, 10710, 15960, 16730, 19145, 24535, 26460, 34580, 37065, 41510, 42630, 43505, 48230, 59675, 69160, 84910, 90860, 99540, 103320, 112560, 114205, 117600, 127120, 129220, 131670, 143290, 152740, 161105, 164115, 170030, 175105, 181195, 185045

Offset: 1

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

nonn

Dinculescu notes that when N = m^2 (resp. m^3) > 1 is a twin rank (i.e., in A002822), then m is a multiple of 5 (resp. of 7), cf. A326232 and A326234. Thus, when N = m^6, then m is a multiple of 35. See A326235 for a(n)/35, n > 1.

See A326232 and A326231 for m^2, A326234 and A326233 for m^3.

M. F. Hasler, Table of n, a(n) for n = 1..10001 (3667 terms from A. Dinculescu).
A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

Cf. A002822, A326235 (a(n)/35, n>1), A326231, A326232 (analog for n^2), A326233, A326234 (analog for n^3), A326230 (least twin rank n^k for given k).

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

a(n) = 35*A326235(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

A173233 Numbers k such that k and k+3 are in A002822.

Original entry on oeis.org

2, 7, 30, 100, 107, 135, 172, 217, 322, 352, 452, 562, 590, 667, 707, 917, 940, 975, 1092, 1127, 1222, 1470, 1570, 1950, 2282, 2545, 2772, 2865, 2930, 3007, 3087, 3682, 3770, 3840, 3945, 4447, 4452, 4477, 5142, 5555, 5600, 5625, 5635, 6262, 6442, 7520, 8232

Offset: 1

Juri-Stepan Gerasimov, Feb 13 2010

nonn

			2 is a term because 2 and 5 are in A002822.

Cf. A002822, A173232.

isA002822 := proc(n) isprime(6*n-1) and isprime(6*n+1) ; end proc:
isA173233 := proc(n) isA002822( n ) and isA002822(n+3 ) ;end proc:
for n from 1 to 3000 do if isA173233(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, May 02 2010

SequencePosition[Table[If[AllTrue[6n+{1,-1},PrimeQ],1,0],{n,9000}],{1,,,1}][[;;,1]] (* Harvey P. Dale, Jul 05 2023 *)

Definition and sequence corrected (667, 707, 1097 etc inserted) by R. J. Mathar, May 02 2010

More terms from Jinyuan Wang, May 15 2020

A263282 Numbers n such that 6n is in A002822 but n is not.

Original entry on oeis.org

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, 483, 493, 507, 508, 515, 530, 585, 600, 627, 635, 640, 647, 658, 662, 677, 718, 742, 765, 772, 793, 795, 830, 847, 857

Offset: 1

Jason Kimberley, Oct 13 2015

To use Dinculescu's terminology (see links): non-ranks n such that 6n is a twin-rank.

			Take n = 63; then 6n = 378 and 36n = 2268; now 379, 2267, and 2269 are prime, but 377 = 13 x 29.

Jason Kimberley, Table of n, a(n) for n = 1..27455 (equivalently, a(n) < 10^6).
A. Dinculescu, On Some Infinite Series Related to the Twin Primes, The Open Mathematics Journal, 5 (2012), 8-14.
A. Dinculescu, The Twin Primes Seen from a Different Perspective, The British Journal of Mathematics & Computer Science, 3 (2013), Issue 4, 691-698.

Cf. A002822.

IsInA2822:=func;
[n:n in[1..10^3]|not IsInA2822(n)and IsInA2822(6*n)];

s = Select[Range@ 5184, PrimeQ[6 # - 1] && PrimeQ[6 # + 1] &]; Select[s, IntegerQ[#/6] && ! MemberQ[s, #/6] &]/6 (* Michael De Vlieger, Oct 13 2015, after N. J. A. Sloane at A002822 *)

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A060212 Primes q such that 6*q-1 and 6*q+1 are twin primes. Proper subset of A002822.

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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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A326236 Numbers k such that N = k^6 is a twin rank (cf. 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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A060212 Primes q such that 6q-1 and 6q+1 are twin primes. Proper subset of A002822.