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A178916 Triangular array a(n,k) read by rows: nextprime(k*n!)-k*n!. For 1<=k<=n.

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%I A178916 #13 Sep 17 2021 18:28:57
%S A178916 1,1,1,1,1,1,5,5,1,1,7,1,7,7,1,7,7,1,7,7,7,11,11,1,1,19,1,1,23,11,17,
%T A178916 1,11,1,1,11,17,29,1,1,13,1,13,1,29,11,29,1,13,1,11,11,1,1,17,1,13,17,
%U A178916 29,1,47,13,1,13,19,17,29,1,17,59,1,1,29,1,41,29,1,1
%N A178916 Triangular array a(n,k) read by rows: nextprime(k*n!)-k*n!. For 1<=k<=n.
%C A178916 Conjecture: for every n>1 and 1<=k<=n there is a prime in the interval [k*n!+1, k*n!+3*n*log(n)^2]. [_Robert Gerbicz_, Dec 28 2010]
%e A178916 Triangle begins:
%e A178916 1
%e A178916 1,1
%e A178916 1,1,1
%e A178916 5,5,1,1
%e A178916 7,1,7,7,1
%e A178916 7,7,1,7,7,7
%e A178916 11,11,1,1,19,1,1
%e A178916 23,11,17,1,11,1,1,11
%e A178916 17,29,1,1,13,1,13,1,29
%e A178916 11,29,1,13,1,11,11,1,1,17
%e A178916 1,13,17,29,1,47,13,1,13,19,17
%e A178916 29,1,17,59,1,1,29,1,41,29,1,1
%t A178916 Flatten[Table[NextPrime[k*n!] - k*n!, {n, 12}, {k, n}]]
%K A178916 nonn,tabl
%O A178916 1,7
%A A178916 _Dmitry Kamenetsky_, Dec 29 2010 _Robert Gerbicz_, Dec 29 2010