A213079 -id:A213079 - cp's OEIS frontend

A213107 Primes p such that 2p^2-1, 3p^2-2, 4p^2-3, and 5p^2-4 are also prime.

17569, 43781, 70321, 229561, 251231, 426131, 426551, 453289, 635051, 727201, 729791, 741709, 944689, 981091, 1015309, 1078081, 1128761, 1228429, 1231229, 1282961, 1289149, 1302349, 1351099, 1723481, 1763159, 1823779, 2078339, 2260889, 2336519, 2357879

Offset: 1

Zak Seidov, Jun 05 2012

Subsequence of A213079: a(1) = 17569 = A213079(10) =A213078(44)= A106483(389) = A000040(2019).

Vincenzo Librandi, Table of n, a(n) for n = 1..740

Cf. A000040, A106483, A213078, A213079.

[p: p in PrimesUpTo(2500000) | forall{i*p^2-i+1: i in [2..5] | IsPrime(i*p^2-i+1)}]; // Vincenzo Librandi, Apr 08 2013

Select[Prime[Range[200000]], PrimeQ[2 #^2 - 1] && PrimeQ[3 #^2 - 2] && PrimeQ[4 #^2 - 3] && PrimeQ[5 #^2 - 4] &] (* T. D. Noe, Jun 06 2012 *)

A213125 Primes p such that 2p^2-1, 3p^2-2, 4p^2-3, 5p^2-4 and 6p^2-5 are also prime.

Original entry on oeis.org

981091, 2260889, 3553271, 3918251, 6038551, 9499279, 10310761, 12377429, 13670719, 13783139, 14881649, 15480529, 18114461, 18727199, 20418341, 21793829, 24170089, 25276649, 30814321, 36104069, 49319579, 52650599, 63485311, 73614031, 76141591, 77646199, 87177089

Offset: 1

Zak Seidov, Jun 05 2012

nonn

Subsequence of A213107:

a(1)=981091=A213107(14)=A213079(148)=A213078(970)==A106483(10205)=A000040(77144).

Cf. A000040, A106483, A213078, A213079, A213107.

A213159 Primes p such that (k+1)*p^2-k are prime for k=1..6.

Original entry on oeis.org

3918251, 18727199, 395564539, 397687709, 503720279, 873201911, 1088927209, 1329433951, 2108335769, 2376506131, 3684190621, 3728773019, 3934099049, 3971294419, 4272771301, 5047170421, 5091014389, 5380213021, 6187560259, 6219076681, 7243803841, 8309591011, 8448425231

Offset: 1

Zak Seidov, Jun 06 2012

nonn

Subsequence of A213125: a(1)=A213125(4), a(2)=A213125(14),

a(3)=A213125(60).

Cf. A106483, A213078, A213079, A213107, A213125.

A213161 Primes p such that (k+1)*p^2-k are prime for k=1..7.

Original entry on oeis.org

1329433951, 25778112821, 75902670689, 80358496679, 84005465699, 184273377289, 188745495049, 220260667439, 225830918741, 227706130541, 250232659249, 314987199911, 396580371571, 532375084669, 535798256839, 542604984109, 634725913009, 676837365821, 706769028239

Offset: 1

Zak Seidov, Jun 06 2012

nonn

All terms squared are congruent to 1 mod 9240.
More terms: 755943098371, 797370889699, 843028736089, 848657766529, 849148113659, 861950855039, 1035653917759.
Subsequence of A213159: a(1)=A213159(8)=A213125(134).

Cf. A106483, A213078, A213079, A213107, A213125, A213159.

A213334 Primes p such that (k+1)*p^2 - k is prime for k=1..8.

Original entry on oeis.org

75902670689, 84005465699, 188745495049, 220260667439, 314987199911, 1970532645509, 2368000666921, 3702971171899, 3886185845431, 4117783215701, 4166366916251, 6213125459729, 7768065083591, 7946042954849, 8788172678669, 11387502711311, 14643617926211

Offset: 1

Zak Seidov, Jun 08 2012

The square of each term is congruent to 1 (mod 9240).

Note that for each p, (k+1)*p^2 - k (k=1..8) gives 8 primes in arithmetic progression with difference d = p^2 - 1.

Subsequence of A213161. Cf. A106483, A213078, A213079, A213107, A213125, A213159.

a(6)-a(17) from Tyler Busby, Jan 09 2023

A213162 Smallest prime p such that (k+1)*p^2-k are prime for k=1..n.

Original entry on oeis.org

2, 199, 409, 17569, 981091, 3918251, 1329433951, 75902670689, 45048280453021

Offset: 1

Zak Seidov, Jun 06 2012

a(1..8) are the first terms correspondingly in A106483, A213078, A213079, A213107, A213125, A213159, A213161, A213334.

a(n) = {my(p = 2); until (ok, ok = 1; for (k = 1, n, if (! isprime((k+1)*p^2-k), ok = 0; break;);); if (!ok, p = nextprime(p+1);)); return (p);}  \\ Michel Marcus, Apr 19 2013

a(9) from Tyler Busby, Jan 11 2023

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A213107 Primes p such that 2p^2-1, 3p^2-2, 4p^2-3, and 5p^2-4 are also prime.

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A213125 Primes p such that 2p^2-1, 3p^2-2, 4p^2-3, 5p^2-4 and 6p^2-5 are also prime.

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A213159 Primes p such that (k+1)*p^2-k are prime for k=1..6.

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A213161 Primes p such that (k+1)*p^2-k are prime for k=1..7.

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A213334 Primes p such that (k+1)*p^2 - k is prime for k=1..8.

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A213162 Smallest prime p such that (k+1)*p^2-k are prime for k=1..n.

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