A053706 Primes p such that between p and the next prime, 2 prime powers (A025475) occur.

7, 23, 113, 2179, 32749

Offset: 1

Author

Labos Elemer, Feb 14 2000

Keywords

• hard
• more
• nonn

Comments

No other terms < 4290000000. - Jud McCranie, Jun 20 2000

There are no other terms < 2^63. - Donovan Johnson, Mar 11 2013

Examples

			Between 7 and 11 the 2 prime powers are 8 and 9, between 23 and 29 the 2 prime powers are 25 and 27, between 113 and 127 the 2 prime powers are 121 and 125, while between 32749 and 32771 the 2 prime powers are 32761 = 181^2 and 32768 = 2^15.
		

Crossrefs

Cf. A025475, A053607.

Programs

Mathematica

nn = 2^20; Prime /@ Keys@ Select[PositionIndex[PrimePi /@ Union@ Flatten@ Table[Array[p^# &, Floor@ Log[p, nn] - 1, 2], {p, Prime@ Range@ PrimePi@ Sqrt@ nn}]], Length[#] > 1 &] (* Michael De Vlieger, Mar 21 2024 *)

PARI

isok(p) = isprime(p) && (q=nextprime(p+1)) && (v=vector(q-p, x, p+x)) && (#select(x->(isprimepower(x) && !isprime(x)), v) == 2);
lista(nn) = forprime(p=2, nn, if (isok(p), print1(p, ", "))); \\ Michel Marcus, Jul 15 2017

Extensions

Corrected by James Sellers, Feb 22 2000