cp's OEIS Frontend

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.

%I A290515 #51 Jun 17 2024 07:50:27
%S A290515 1,5,13,19,32,53,1024,89,512,139,536870912,199,144115188075855859,293,
%T A290515 65521,1831,8192,1069,147573952589676412909,887,524288,1129,
%U A290515 549755813888,4177,17179869184,2477,16384,2971,131072,1331,34359738337,5591,18014398509481951,8467,33554432,9551
%N 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.
%C A290515 Conjecture: a(n) always exists.
%C A290515 When n is odd a(n) is equal to 2^k or 2^k-n for a suitable k. - _Giovanni Resta_, Aug 07 2017
%C A290515 Apparently, a(n) = A110968(n-1) - 1 for n >= 3. - _Hugo Pfoertner_, Jun 17 2024
%H A290515 Robert G. Wilson v, <a href="/A290515/b290515.txt">Table of n, a(n) for n = 1..315</a>
%H A290515 Robert G. Wilson v, <a href="/A290515/a290515.txt">Table of n, a(n) for n = 1..1000, or 0 if no such number is known.</a>
%e A290515 a(1) =  1 since  2 -  1 = 1;
%e A290515 a(2) =  5 since  7 -  5 = 2;
%e A290515 a(3) = 13 since 16 - 13 = 3;
%e A290515 a(4) = 19 since 23 - 19 = 4;
%e A290515 a(5) = 32 since 37 - 32 = 5; etc.
%t A290515 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
%Y A290515 Cf. A000230, A000961, A023056, A110968, A373334.
%K A290515 nonn
%O A290515 1,2
%A A290515 _Robert G. Wilson v_, Aug 04 2017
%E A290515 a(13)-a(34) from _Giovanni Resta_, Aug 07 2017