A368749 a(n) is the smallest prime p such that there are n numbers between p and nextprime(p) which are not prime powers.

2, 5, 13, 19, 31, 53, 359, 89, 241, 139, 509, 113, 839, 293, 2803, 1831, 523, 1069, 11447, 887, 3469, 1129, 1669, 4177, 39581, 2477, 24631, 2971, 16381, 4297, 124601, 5591, 1327, 8467, 22193, 9551, 79493, 30593, 62989, 19333, 410857, 16141, 436913, 15683, 1038337, 81463, 157579

Offset: 0

Author

David James Sycamore and Michael De Vlieger, Jan 04 2024

Keywords

• nonn

Comments

Conjecture: the sequence is infinite (number of nonprime powers between consecutive primes is arbitrarily large; see A014963).

Examples

			a(2) = 13 because between 13 and 17 there are three composite numbers, only one of which (16) is a prime power, and no previous prime has this property.
a(5) = 53 because between 53 and 59 there are 5 composite numbers, none of which are prime powers, and no smaller prime has this property.
		

Crossrefs

Cf. A000040, A000230, A000961, A014963, A024619, A080101, A085970, A093555, A368748.

Programs

Mathematica

p = q = 2; r = a[] = 0; Do[q = NextPrime[q]; If[a[#] == 0, a[#] = p; If[# > r, r = #]] &@ Count[Range[p, q - 1], ?(Not@*PrimePowerQ)]; p = q, {i, 2^16}]; {2}~Join~TakeWhile[Array[a, r], # > 0 &] (* Michael De Vlieger, Jan 04 2024 *)

PARI

f(p) = sum(k=p+1, nextprime(p+1)-1, !isprimepower(k));
a(n) = my(p=2); while(f(p) != n, p=nextprime(p+1)); p; \\ Michel Marcus, Jan 04 2024

Extensions

More terms from Michel Marcus, Jan 04 2024