A302859 -id:A302859 - cp's OEIS frontend

A300559 a(n) = n*(n+1)!/2 + 1.

1, 2, 7, 37, 241, 1801, 15121, 141121, 1451521, 16329601, 199584001, 2634508801, 37362124801, 566658892801, 9153720576001, 156920924160001, 2845499424768001, 54420176498688001, 1094805903679488001, 23112569077678080001, 510909421717094400001, 11802007641664880640001

Offset: 0

M. F. Hasler, Apr 10 2018

See A301373 and A302859 for the primes: it is remarkable that all of a(1..10) are primes, and only a(11) is the first composite term.

Maheswara Rao Valluri, Primes of the form p = 1 + n! Sum n, for some n ∈ N*, arXiv:1803.11461 [math.GM], 2018.
Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.

Inspired by A302859.

Cf. A301373.

[n*Factorial(n+1)/2+1: n in [0..25]]; // Vincenzo Librandi, Apr 12 2018

Array[# (# + 1)!/2 + 1 &, 22, 0] (* Michael De Vlieger, Apr 10 2018 *)

apply( A300559=n->(n+1)!\2*n+1, [0..25])

a(n) = A180119(n) + 1 = A001286(n+1) + 1.
D-finite with recurrence n*a(n+1) = (n+1)*(n+2)*(a(n)-1) + n. - Chai Wah Wu, Apr 11 2018
E.g.f.: exp(x)-1/(x-1)^3*x. - Simon Plouffe, Jun 21 2018

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

Original entry on oeis.org

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

A305738 Numbers k such that k!*T(k) - 1 is prime, where T(k) is the k-th triangular number.

Original entry on oeis.org

2, 4, 28, 34, 47, 62, 228, 256, 258, 341, 848, 1362, 1709, 2262, 2692, 7907, 10396, 10501

Offset: 1

Maheswara Rao Valluri, Jun 22 2018

The PFGW program has been used to certify all the terms up to a(18), 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. A302859, A301373.

Do[If[ PrimeQ[n(n +1)!/2 - 1], Print@ n], {n, 3000}]

isok(n) = ispseudoprime(n(n+1)!/ 2 - 1);

a(16)-a(18) from Giovanni Resta, Jun 24 2018

cp's OEIS Frontend

A300559 a(n) = n*(n+1)!/2 + 1.

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

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A305738 Numbers k such that k!*T(k) - 1 is prime, where T(k) is the k-th triangular number.

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