1,1

So far the powers have occurred in numerical order. Here is the list of primes and powers: [7, 4], [89, 8], [1831, 16], [5591, 32], [9551, 36], [89689, 64], [396733, 100], [3851459, 128], [11981443, 144], [70396393, 196], [202551667, 216], [1872851947, 256], [10958687879, 324]. I have searched out to the prime p=26689111613.

Thomas R. Nicely, First occurrence prime gaps.

Previous prime before A123996.

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

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), [p, d]]; print(p, d); print(P); print(Q); fi ; p:=q; od od; # let it run with AutoSave enabled.

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[pp, p]; AppendTo[dd, d]]; p = q, {n, 10^7}]; pp (* Robert G. Wilson v *)

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

Adjacent sequences: A123992 A123993 A123994 this_sequence A123996 A123997 A123998

Sequence in context: A119157 A079701 A096208 this_sequence A056161 A076740 A112290

nonn

Walter A. Kehowski (wkehowski(AT)cox.net), Oct 31 2006

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