A129518 -id:A129518 - cp's OEIS frontend

A129517 Odd primes p such that p divides (p-1)!!-1.

11, 19, 23, 31, 43, 67, 71, 131, 151, 163, 167, 179, 223, 227, 239, 271, 311, 347, 359, 419, 439, 443, 463, 487, 491, 523, 563, 571, 619, 647, 683, 691, 719, 739, 751, 787, 911, 919, 947, 967, 983, 1019, 1031, 1039, 1051, 1063, 1091, 1103, 1123, 1187, 1223

Offset: 1

Alexander Adamchuk, Apr 18 2007

nonn

All terms are primes of the form 4k+3 (A002145). A002145 is the union of {a(n)} and A129518 (numbers k > 2 such that k divides (k-2)!!-1 and (k-3)!!-1).

Primes such that ((p-1)/2)! = (-1)^((p^2-1)/8) in the finite field F_p of p elements. For example, 11 is a term since ((11-1)/2)! = 5! = -1 = (-1)^((121-1)/8) in F_11. - Luis H. Gallardo, Dec 30 2021

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A006882, A002145, A129518, A129516.

Select[Prime@Range[2,200],Mod[(#-1)!!-1,#]==0&] (* Giorgos Kalogeropoulos, Jan 09 2022 *)

A260298 Primes p such that hyperfactorial(p-1) == 1 (mod p).

Original entry on oeis.org

2, 3, 7, 47, 59, 79, 83, 103, 107, 127, 139, 191, 199, 211, 251, 263, 283, 307, 331, 367, 379, 383, 431, 467, 479, 499, 503, 547, 587, 599, 607, 631, 643, 659, 727, 743, 811, 823, 827, 839, 859, 863, 883, 887, 907, 971, 991, 1087, 1151, 1163, 1171, 1259, 1283

Offset: 1

Matthew Campbell, Jul 22 2015

nonn

Does this contain 2 and the entries of A129518? - R. J. Mathar, Aug 07 2015

Primes p such that (p-1)!! == -1 (mod p). - Thomas Ordowski, Jul 26 2016

Matthew Campbell and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2516 terms from Campbell)

Cf. A000040, A002109, A129518, A260178, A260297, A260299.

fQ[n_] := Mod[Hyperfactorial[n - 1], n] == 1; Select[ Prime@ Range@ 210, fQ] (* Robert G. Wilson v, Aug 06 2015 *)

is(p)=prod(k=2, p-1, Mod(k, p)^k)==1 && isprime(p) \\ Charles R Greathouse IV, Aug 05 2015

a(n) = prime(A260299(n)).

A129516 Numbers k such that k divides (k-1)!! - 1.

Original entry on oeis.org

1, 2, 8, 11, 16, 19, 23, 31, 32, 43, 64, 67, 71, 128, 131, 151, 163, 167, 179, 223, 227, 239, 256, 271, 311, 347, 359, 419, 439, 443, 463, 487, 491, 512, 523, 563, 571, 619, 647, 683, 691, 719, 739, 751, 787, 911, 919, 947, 967, 983, 1019, 1024, 1031, 1039

Offset: 1

Alexander Adamchuk, Apr 18 2007

nonn

Terms include all powers of 2 except 2^2 and some primes of the form 4m+3 belonging to A002145. Odd primes in this sequence are listed in A129517. A002145 is the union of A129517 and A129518 (numbers m > 2 such that m divides (m-2)!!-1 and (m-3)!!-1).

Cf. A006882 (double factorials).
Cf. A002145 (primes of the form 4k+3).
Cf. A129517 (odd primes p such that p divides (p-1)!!-1).
Cf. A129518 (numbers k > 2 such that k divides (k-2)!!-1 and (k-3)!!-1).

Select[Range[2048],IntegerQ[((#-1)!!-1)/# ]&]

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A129517 Odd primes p such that p divides (p-1)!!-1.

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A260298 Primes p such that hyperfactorial(p-1) == 1 (mod p).

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A129516 Numbers k such that k divides (k-1)!! - 1.

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