A123995 First occurrence of prime gaps which are perfect powers.

2, 7, 89, 1831, 5591, 9551, 89689, 396733, 3851459, 11981443, 70396393, 202551667, 1872851947, 10958687879, 47203303159, 767644374817, 1999066711391, 8817792098461, 78610833115261, 497687231721157, 2069461000669981

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: [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.

The old definition was confusing. What is meant was: primes p such that nextprime(p)-p is an element of A001597 (or A075090: even perfect powers, for n > 1), and p is the smallest prime followed by this gap. - M. F. Hasler, Oct 18 2018

A138198 is a subsequence. - M. F. Hasler, Oct 18 2018

Examples

			a(2)=89 since nextprime(89)-89=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 (perfect powers), A075090, A002386, A138198.

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), [p,d]]; print(p,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[pp, p]; AppendTo[dd, d]]; p = q, {n, 10^7}]; pp (* Robert G. Wilson v, Nov 03 2006 *)

PARI

S=[];print1(p=2);forprime(q=1+p,,ispower(q-p)&& !setsearch(S,q-p)&& !print1(","p)&& S=setunion(S,[q-p]);p=q) \\ M. F. Hasler, Oct 18 2018

Formula

Previous prime before A123996.

Extensions

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

Better definition from M. F. Hasler, Oct 18 2018