A069586 Smallest prime power p^k (k >= 2) such that there is a difference of exactly n between p^k and some other prime power q^m (m >= 2); or 0 if no such q^m exists.

8, 25, 125, 4, 4, 0, 9, 8, 16, 2187, 16, 4, 243, 0, 49, 9, 8, 9, 8, 0, 4, 27, 4, 8, 0, 0, 0, 4, 0, 6859, 0, 32, 16, 0, 0, 0, 27, 1331, 25, 9, 8, 0, 0, 81, 4, 243, 81, 16, 32, 0, 0, 0, 0, 27, 9, 8, 64, 0, 0, 4, 64, 0, 961, 64, 16, 0, 0, 0, 0, 0, 0, 9, 8, 169, 0, 49, 4, 0, 49, 0, 0, 0, 0, 0, 0

Offset

1,1

Comments

Any solution for the terms that are currently zero is > 10^14. Since there are so many 'missing' values, I would recommend leaving the more keyword. - Larry Reeves (larryr(AT)acm.org), Jul 02 2002

The nonzero terms are sure terms, only the zero terms are conjectured. - Michel Marcus, Nov 02 2023

Links

Table of n, a(n) for n=1..85.

Example

a(1) = 8 = 2^3 and 8 + 1 = 9 = 3^2;
a(3) = 125 = 5^3 and 125 + 3 = 128 = 2^7.

Prog

(PARI) lista(nn) = my(vpp=select(x->(isprimepower(x) && !isprime(x)), [2..10^4]), v=vector(nn)); for (i=1, #vpp, for (j=1, i-1, my(d = vpp[i] - vpp[j]); if ((d<=nn) && (!v[d] || (vpp[j]<v[d])), v[d] = vpp[j]); ); ); v; \\ Michel Marcus, Oct 30 2023

Crossrefs

First time a difference of n occurs in A025475 (with its initial 1 deleted).

Sequence in context: A135942 A181207 A068315 * A253237 A275151 A302617

Adjacent sequences: A069583 A069584 A069585 * A069587 A069588 A069589

Keyword

nonn

Author

Amarnath Murthy, Mar 24 2002

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jul 02 2002

Description corrected by Karl W. Heuer, Apr 08 2012

Status

approved