A359273 a(n) = least positive integer k such that (prime(n+k)-prime(n))/n is an integer.

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, 10, 23, 6, 24, 6, 37, 12, 38, 14, 40, 22, 151, 6, 16, 16, 46, 22, 60, 10, 49, 25, 65, 43, 16, 18, 18, 27, 19, 38, 56, 19, 144, 30, 21, 21, 21, 10, 42, 32, 66

Offset: 1

Author

Clark Kimberling, Jan 26 2023

Keywords

• nonn

Examples

			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).
		

Crossrefs

Cf. A000040, A072063.

Programs

Maple

f:= proc(n) local p,k,q;
  p:= ithprime(n); q:= p;
  for k from 1 do
    q:= nextprime(q);
    if (q - p) mod n = 0 then return k fi;
  od
end proc:
map(f, [$1..100]); # Robert Israel, Jan 26 2023

Mathematica

p[n_] := Prime[n];
a[n_] := Select[Range[1000], IntegerQ[(p[n + #] - p[n])/n] &, 1]
Flatten[Table[a[n], {n, 1, 130}]]