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a(n) is the smallest prime p such that there are n numbers between p and nextprime(p) which are not prime powers.
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%I #18 Jan 20 2024 09:38:04

%S 2,5,13,19,31,53,359,89,241,139,509,113,839,293,2803,1831,523,1069,

%T 11447,887,3469,1129,1669,4177,39581,2477,24631,2971,16381,4297,

%U 124601,5591,1327,8467,22193,9551,79493,30593,62989,19333,410857,16141,436913,15683,1038337,81463,157579

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

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

%e 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.

%e 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.

%t 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 *)

%o (PARI) f(p) = sum(k=p+1, nextprime(p+1)-1, !isprimepower(k));

%o a(n) = my(p=2); while(f(p) != n, p=nextprime(p+1)); p; \\ _Michel Marcus_, Jan 04 2024

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

%K nonn

%O 0,1

%A _David James Sycamore_ and _Michael De Vlieger_, Jan 04 2024

%E More terms from _Michel Marcus_, Jan 04 2024