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

%I A049579 #39 Mar 18 2025 07:33:24
%S A049579 4,6,8,9,11,12,14,15,16,18,19,21,22,23,24,25,27,29,30,31,32,34,36,37,
%T A049579 38,39,40,42,44,46,47,48,50,51,53,54,55,56,58,59,61,62,63,65,66,67,68,
%U A049579 70,71,72,73,74,75,76,77,78,79,80,82,84,85,86,87,88,90,91,92,93,94,95
%N A049579 Numbers k such that prime(k)+2 divides (prime(k)-1)!.
%C A049579 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
%C A049579 If we prepend 1, the first differences are A251092 (see also A175632). The complement is A029707. - _Gus Wiseman_, Dec 03 2024
%H A049579 Amiram Eldar, <a href="/A049579/b049579.txt">Table of n, a(n) for n = 1..10000</a>
%e A049579 prime(4) = 7, 6!+1 = 721 gives residue 1 when divided by prime(4)+2 = 9.
%t A049579 pnmQ[n_]:=Module[{p=Prime[n]},Mod[(p-1)!+1,p+2]==1]; Select[Range[ 100],pnmQ] (* _Harvey P. Dale_, Jun 24 2017 *)
%o A049579 (PARI) isok(n) = (((prime(n)-1)! + 1) % (prime(n)+2)) == 1; \\ _Michel Marcus_, Dec 31 2013
%Y A049579 The first differences are A251092 except first term, run-lengths A373819.
%Y A049579 The complement is A029707.
%Y A049579 Runs of terms differing by one have lengths A027833, min A107770, max A155752.
%Y A049579 A000040 lists the primes, differences A001223 (run-lengths A333254, A373821).
%Y A049579 A038664 finds the first prime gap of difference 2n.
%Y A049579 A046933 counts composite numbers between primes.
%Y A049579 A071148 gives partial sums of odd primes.
%Y A049579 For prime runs: A001359, A005381, A006512, A025584, A067774, A373403.
%Y A049579 Cf. A006560, A006562, A037201, A122535, A175632, A176246, A373820.
%K A049579 nonn
%O A049579 1,1
%A A049579 _Labos Elemer_
%E A049579 Definition edited by _Thomas Ordowski_, Mar 31 2022