A164571 Numbers n such that n and n+3 are prime powers.

1, 2, 4, 5, 8, 13, 16, 29, 61, 64, 125, 128, 509, 1021, 4093, 4096, 16381, 32768, 65536, 262144, 1048573, 4194301, 16777213, 268435456, 536870909, 1073741824, 36028797018963968

Offset: 1

Author

Daniel Forgues, Aug 16 2009

Keywords

• nonn

Comments

Numbers n such that n + (0, 3) is a prime power pair.

n + (0, 2m), m >= 1, being an admissible pattern for prime pairs, since (0, 2m) = (0, 0) (mod 2), has high density.

n + (0, 2m-1), m >= 1, being a non-admissible pattern for prime pairs, since (0, 2m-1) = (0, 1) (mod 2), has low density [the only possible pairs are (2^a - 2m-1, 2^a) or (2^a, 2^a + 2m-1), a >= 0.]

n + (0, 3) being a non-admissible pattern for prime pairs, has only prime power pairs (2^a - 3, 2^a) or (2^a, 2^a + 3), a >= 0.

Numbers n such that n and n+3 are primes would give only 2, for the prime pair (2, 5).

10^18 < a(28) <= 19807040628566084398385987581. - Donovan Johnson, Aug 17 2009

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..50

Crossrefs

Cf. A000961.

Cf. A006549 Numbers n such that n and n+1 are prime powers.

Cf. A120431 Numbers n such that n and n+2 are prime powers.

Cf. A164571 Numbers n such that n and n+3 are prime powers.

Cf. A164572 Numbers n such that n and n+4 are prime powers.

Cf. A164573 Numbers n such that n and n+5 are prime powers.

Cf. A164574 Numbers n such that n and n+6 are prime powers.

Programs

PARI

ispp(n) = (n==1) || isprime(n) || (ispower(n, ,&p) && isprime(p));
isok(n) = ispp(n) && ispp(n+3); \\ Michel Marcus, Aug 31 2013

PARI

v=List(); for(n=0, 1e3, if(isprimepower(2^n-3), listput(v, 2^n-3)); if(isprimepower(2^n+3), listput(v, 2^n))); Set(v) \\ Charles R Greathouse IV, Apr 24 2015

Extensions

Edited by Daniel Forgues, Aug 17 2009

a(20)-a(27) from Donovan Johnson, Aug 17 2009