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a(n) = least positive integer k such that (prime(n+k)-prime(n))/n is an integer.
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%I #11 Jan 30 2023 09:28:12

%S 1,1,2,1,6,2,4,6,4,7,5,6,6,6,13,10,14,4,23,12,16,4,42,6,20,5,10,10,10,

%T 10,23,6,24,6,37,12,38,14,40,22,151,6,16,16,46,22,60,10,49,25,65,43,

%U 16,18,18,27,19,38,56,19,144,30,21,21,21,10,42,32,66

%N a(n) = least positive integer k such that (prime(n+k)-prime(n))/n is an integer.

%e a(5) = 6 because 5 divides 20, which is prime(5+6) - prime(5)), and if 0 < k < 6, then 5 does not divide prime(5+k) - prime(5).

%p f:= proc(n) local p,k,q;

%p p:= ithprime(n); q:= p;

%p for k from 1 do

%p q:= nextprime(q);

%p if (q - p) mod n = 0 then return k fi;

%p od

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Jan 26 2023

%t p[n_] := Prime[n];

%t a[n_] := Select[Range[1000], IntegerQ[(p[n + #] - p[n])/n] &, 1]

%t Flatten[Table[a[n], {n, 1, 130}]]

%Y Cf. A000040, A072063.

%K nonn

%O 1,3

%A _Clark Kimberling_, Jan 26 2023