A123995 - OEIS

%I #19 Oct 28 2021 12:37:08

%S 2,7,89,1831,5591,9551,89689,396733,3851459,11981443,70396393,

%T 202551667,1872851947,10958687879,47203303159,767644374817,

%U 1999066711391,8817792098461,78610833115261,497687231721157,2069461000669981

%N First occurrence of prime gaps which are perfect powers.

%C 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.

%C 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

%C A138198 is a subsequence. - _M. F. Hasler_, Oct 18 2018

%H Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html">First occurrence prime gaps</a> [For local copy see A000101].

%F Previous prime before A123996.

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

%p 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.

%t 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 *)

%o (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

%Y Cf. A080370, A113472, A000230, A001597 (perfect powers), A075090, A002386, A138198.

%K nonn

%O 1,1

%A _Walter Kehowski_, Oct 31 2006

%E Edited and extended by _Robert G. Wilson v_, Nov 03 2006 and corrected Nov 04 2006

%E Better definition from _M. F. Hasler_, Oct 18 2018