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
Keywords
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.
Programs
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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
Comments