

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



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
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OFFSET

0,1


COMMENTS

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


LINKS



EXAMPLE

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.


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 *)


PROG

(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


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



