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

Author

Amarnath Murthy, Mar 24 2002

Keywords

• nonn

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

Examples

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

Crossrefs

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

Programs

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]Michel Marcus, Oct 30 2023

Extensions

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

Description corrected by Karl W. Heuer, Apr 08 2012