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

1,2

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

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.

Example

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.

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

Crossrefs

Cf. A000230, A000961, A023056.

Sequence in context: A265808 A265790 A002540 * A082093 A045455 A160031

Adjacent sequences: A290512 A290513 A290514 * A290516 A290517 A290518

Keyword

nonn

Author

Robert G. Wilson v, Aug 04 2017

Extensions

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

Status

approved