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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.
2
3, 11, 97, 1847, 5623, 9587, 89753, 396833, 3851587, 11981587, 70396589, 202551883, 1872852203, 10958688203, 47203303559, 767644375301, 8817792099037, 78610833115937, 497687231721941, 2069461000670881
OFFSET
1,1
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.
LINKS
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101].
FORMULA
Next prime after A123995.
EXAMPLE
a(2)=97 since 97-prevprime(97)=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), [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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Walter Kehowski, Oct 31 2006
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
STATUS
approved