A300559 -id:A300559 - cp's OEIS frontend

A301373 Numbers k such that (k+1)!*k/2 + 1 is prime.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 19, 24, 251, 374, 953, 1104, 1507, 3390, 4443, 5762

Offset: 1

Daniel Suteu, Apr 03 2018

The associated primes are A300559(a(n)) = A180119(a(n))+1 = A001286(a(n)+1)+1. - M. F. Hasler, Apr 10 2018
Looking for primes of the form p(n) = 1 + n! f(n) with a simple polynomial function f, it appears that the choice f(n) = n(n+1)/2 = A000217 is one of the most successful choices for getting a maximum of primes for n = 1..20. - M. F. Hasler, Apr 14 2018
The PFGW program has been used to certify all the terms up to a(23), using a deterministic test which exploits the factorization of a(n) - 1. - Giovanni Resta, Jun 24 2018

Maheswara Rao Valluri, Primes of the form p = 1 + n! Sum n, for some n ∈ N*, arXiv:1803.11461 [math.GM], 2018.

Cf. A090703, A300559, A180119, A001286.

See A302859 for the actual primes.

Do[ If[ PrimeQ[n(n +1)!/2 +1], Print@ n], {n, 4000}] (* Robert G. Wilson v, Apr 05 2018 *)

isok(k) = ispseudoprime((k+1)! * k / 2 + 1);

a(21) from Robert G. Wilson v, Apr 05 2018
a(22) from Vaclav Kotesovec, Apr 06 2018
a(23) from Giovanni Resta, Jun 24 2018

A302859 Primes of the form (k+1)!*k/2 + 1.

Original entry on oeis.org

2, 7, 37, 241, 1801, 15121, 141121, 1451521, 16329601, 199584001, 37362124801, 566658892801, 9153720576001, 23112569077678080001, 186134520519971831808000001

Offset: 1

Maheswara Rao Valluri, Apr 03 2018

nonn

The next term, for k = 251 (see A301373), is
2566282033898537172673689833660299199318441\
    47812028978290091772271674111238846647249346322032725585967946013083615\
    44220440938904033673583084158870025082998875790404475054647299641196409\
    72934112662249702715026203933143550590243427364871765801696382591273000\
    77256511620017707120387621962694782616283336623216978502662268159966484\
    36506095391239127788493879085200485514817503469381297494013097308996216\
    58710310236069486145497777789215839354880000000000000000000000000000000\
    0000000000000000000000000000001

Maheswara Rao Valluri, Primes of the form p = 1 + n! Sum n, for some n ∈ N*, arXiv:1803.11461 [math.GM], 2018.

Cf. A300559, A301373.

Reap[For[k = 1, k <= 1000, k++, If[PrimeQ[p = (k+1)! k/2 + 1], Print["k = ", k, " p = ", p]; Sow[p]]]][[2, 1]]

a(n) = A300559(A301373(n)) for all n >= 1; a(n) = A300559(n) for 1 <= n <= 10. - M. F. Hasler, Apr 15 2018

This sequence was originally submitted as A302174, then withdrawn, then reinstated with a new A-number by N. J. A. Sloane, Apr 14 2018

cp's OEIS Frontend

A301373 Numbers k such that (k+1)!*k/2 + 1 is prime.

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