A123996 Smallest prime q such that the gap between q and the previous prime is a perfect power that has not occurred earlier as a gap.

3, 11, 97, 1847, 5623, 9587, 89753, 396833, 3851587, 11981587, 70396589, 202551883, 1872852203, 10958688203, 47203303559, 767644375301, 8817792099037, 78610833115937, 497687231721941, 2069461000670881

Offset: 1

Author

Walter Kehowski, Oct 31 2006

Keywords

• nonn

Comments

So far the powers have occurred in numerical order. Here is the list of primes and powers: [11, 4], [97, 8], [1847, 16], [5623, 32], [9587, 36], [89753, 64], [396833, 100], [3851587, 128], [11981587, 144], [70396589, 196], [202551883, 216], [1872852203, 256], [10958688203, 324]. I have searched out to the prime p=26689111613.

Examples

			a(2)=97 since 97-prevprime(97)=97-89=8 is the first occurrence of 8 as a difference between successive primes.
		

Links

• Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101].

Crossrefs

Cf. A080370, A113472, A000230, A001597, A075090.

Programs

Maple

with(numtheory); egcd := proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); return igcd(op(L)) else return 1 fi end: P:={}; Q:=[]; p:=2; for w to 1 do for k from 0 do # keep track if k mod 10^6 = 0 then print(k,p) fi; lastprime:=p; q:=nextprime(p); d:=q-p; x:=egcd(d); if x>1 and not d in P then P:=P union {d}; Q:=[op(Q), [q,d]]; print(q,d); print(P); print(Q); fi ; p:=q; od od; # let it run with AutoSave enabled.

Mathematica

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ@k, k++ ]; k]; perfectPowerQ[x_] := GCD @@ Last /@ FactorInteger@x > 1; dd = {1}; pp = {2}; qq = {3}; p = 3; Do[q = NextPrim@p; d = q - p; If[perfectPowerQ@d && !MemberQ[dd, d], Print@q; AppendTo[qq, q]; AppendTo[dd, d]]; p = q, {n, 10^7}]; qq (* Robert G. Wilson v, Nov 03 2006 *)

Formula

Next prime after A123995.

Extensions

Edited and extended by Robert G. Wilson v, Nov 03 2006, corrected Nov 04 2006

Definition corrected by M. F. Hasler, Oct 19 2018