A290515 a(n) = smallest number that is the start of a gap of size n between successive prime powers (A000961), or 0 if no such number exists.

1, 5, 13, 19, 32, 53, 1024, 89, 512, 139, 536870912, 199, 144115188075855859, 293, 65521, 1831, 8192, 1069, 147573952589676412909, 887, 524288, 1129, 549755813888, 4177, 17179869184, 2477, 16384, 2971, 131072, 1331, 34359738337, 5591, 18014398509481951, 8467, 33554432, 9551

Offset: 1

Author

Robert G. Wilson v, Aug 04 2017

Keywords

• nonn

Comments

Conjecture: a(n) always exists.

When n is odd a(n) is equal to 2^k or 2^k-n for a suitable k. - Giovanni Resta, Aug 07 2017

Apparently, a(n) = A110968(n-1) - 1 for n >= 3. - Hugo Pfoertner, Jun 17 2024

Examples

			a(1) =  1 since  2 -  1 = 1;
a(2) =  5 since  7 -  5 = 2;
a(3) = 13 since 16 - 13 = 3;
a(4) = 19 since 23 - 19 = 4;
a(5) = 32 since 37 - 32 = 5; etc.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..315
• Robert G. Wilson v, Table of n, a(n) for n = 1..1000, or 0 if no such number is known.

Crossrefs

Cf. A000230, A000961, A023056, A110968, A373334.

Programs

Mathematica

nxt[n_] := nxt[n] = Block[{k = n + 1}, While[! PrimePowerQ@k, k++]; k]; prv[n_] := prv[n] = Block[{k = n - 1}, While[! PrimePowerQ@k, k--]; k]; f[n_] := Block[{d = 0, exp = 2, p, q}, While[d == 0, p = prv[2^exp]; q = nxt[2^exp]; If[n == 2^exp - p, d = p]; If[n == q - 2^exp, d = 2^exp]; exp++]; d]; Do[ t[n] = f[n], {n, 3, 99, 2}]; p = 1; q = 2; t[_] = 0; While[p < 1110000, d = q - p; If[t[d] == 0, t[d] = p]; p = q; q = nxt@ q]; t@# & /@ Range@ 100

Extensions

a(13)-a(34) from Giovanni Resta, Aug 07 2017