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A123995
First occurrence of prime gaps which are perfect powers.
2
2, 7, 89, 1831, 5591, 9551, 89689, 396733, 3851459, 11981443, 70396393, 202551667, 1872851947, 10958687879, 47203303159, 767644374817, 1999066711391, 8817792098461, 78610833115261, 497687231721157, 2069461000669981
OFFSET
1,1
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
LINKS
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101].
FORMULA
Previous prime before A123996.
EXAMPLE
a(2)=89 since nextprime(89)-89=97-89=8 is the first occurrence of 8 as a difference between successive primes.
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 *)
PROG
(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
CROSSREFS
Cf. A080370, A113472, A000230, A001597 (perfect powers), A075090, A002386, A138198.
Sequence in context: A079701 A327039 A096208 * A350754 A224439 A304722
KEYWORD
nonn
AUTHOR
Walter Kehowski, Oct 31 2006
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
STATUS
approved