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

1,2

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

Prog

(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

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.

Sequence in context: A294944 A178656 A085443 * A238589 A327045 A288668

Adjacent sequences: A164568 A164569 A164570 * A164572 A164573 A164574

Keyword

nonn

Author

Daniel Forgues, Aug 16 2009

Extensions

Edited by Daniel Forgues, Aug 17 2009

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

Status

approved