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%I #18 Oct 18 2020 22:35:16
%S 2,3,7,73,241,2161,15121,161281,1088641,10886401,39916801,958003201,
%T 18681062401,1133317785601,9153720576001,83691159552001,
%U 1778437140480001,12804747411456001,851515702861824001,41359334139002880001,766364132575641600001,20232013099996938240001
%N Smallest prime of form (n!)*k + 1.
%C This is one possible generalization of "the least prime problem in special arithmetic progressions" when n in nk+1 is replaced by n!.
%C a(n) is the smallest prime p such that the multiplicative group modulo p has a subgroup of order n!. - _Joerg Arndt_, Oct 18 2020
%H <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>
%e a(5)=241 because in arithmetic progression 120k+1=5!k+1 the second term is prime, 241.
%t sp[n_]:=Module[{nf=n!,k=1},While[!PrimeQ[nf*k+1],k++];nf*k+1]; Array[sp,20] (* _Harvey P. Dale_, Jan 27 2013 *)
%o (PARI) a(n) = for(k=1, oo, if(isprime(k*n! + 1), return(k*n! + 1))); \\ _Daniel Suteu_, Oct 18 2020
%Y Analogous case is A034694. Special case for k=1 is A002981.
%Y Cf. A035093 (values of k).
%K nonn
%O 1,1
%A _Labos Elemer_
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