A049579 Numbers k such that prime(k)+2 divides (prime(k)-1)!.

4, 6, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 23, 24, 25, 27, 29, 30, 31, 32, 34, 36, 37, 38, 39, 40, 42, 44, 46, 47, 48, 50, 51, 53, 54, 55, 56, 58, 59, 61, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

Numbers k such that prime(k+1) - prime(k) does not divide prime(k+1) + prime(k). These are the numbers k for which prime(k+1) - prime(k) > 2. - Thomas Ordowski, Mar 31 2022

If we prepend 1, the first differences are A251092 (see also A175632). The complement is A029707. - Gus Wiseman, Dec 03 2024

Examples

			prime(4) = 7, 6!+1 = 721 gives residue 1 when divided by prime(4)+2 = 9.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

The first differences are A251092 except first term, run-lengths A373819.

The complement is A029707.

Runs of terms differing by one have lengths A027833, min A107770, max A155752.

A000040 lists the primes, differences A001223 (run-lengths A333254, A373821).

A038664 finds the first prime gap of difference 2n.

A046933 counts composite numbers between primes.

A071148 gives partial sums of odd primes.

For prime runs: A001359, A005381, A006512, A025584, A067774, A373403.

Cf. A006560, A006562, A037201, A122535, A175632, A176246, A373820.

Programs

Mathematica

pnmQ[n_]:=Module[{p=Prime[n]},Mod[(p-1)!+1,p+2]==1]; Select[Range[ 100],pnmQ] (* Harvey P. Dale, Jun 24 2017 *)

PARI

isok(n) = (((prime(n)-1)! + 1) % (prime(n)+2)) == 1; \\ Michel Marcus, Dec 31 2013

Extensions

Definition edited by Thomas Ordowski, Mar 31 2022