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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.
2
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
Apparently, a(n) = A110968(n-1) - 1 for n >= 3. - Hugo Pfoertner, Jun 17 2024
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
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Aug 04 2017
EXTENSIONS
a(13)-a(34) from Giovanni Resta, Aug 07 2017
STATUS
approved