A123996 -id:A123996 - cp's OEIS frontend

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

Walter Kehowski, Oct 31 2006

nonn

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

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

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101].

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

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, Nov 03 2006 *)

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

Previous prime before A123996.

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

A243155 Larger of the two consecutive primes whose positive difference is a cube.

Original entry on oeis.org

3, 97, 367, 397, 409, 457, 487, 499, 691, 709, 727, 751, 769, 919, 937, 991, 1117, 1171, 1201, 1381, 1447, 1531, 1567, 1579, 1741, 1831, 1987, 2011, 2161, 2221, 2251, 2281, 2467, 2539, 2617, 2671, 2707, 2749, 2851, 2887, 2917, 3019, 3049, 3217, 3229, 3457, 3499

Offset: 1

K. D. Bajpai, May 31 2014

nonn

Observation: All the terms in this sequence, after a(1), are the larger of the two consecutive primes which have positive difference either 2^3 or 4^3.

Superset of A031927 as the sequence contains for example numbers like 89753, 107441, 288647,.. (with gaps of 4^3...) that are not in A031927. - R. J. Mathar, Jun 06 2014

			97 is prime and appears in the sequence because 97 - 89 = 8 = 2^3.
397 is prime and appears in the sequence because 397 - 389 = 8 = 2^3.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A031927, A123996, A118590, A001632.

A243155:= proc() local a; a:=evalf((ithprime(n+1)-ithprime(n))^(1/3)); if a=floor(a) then RETURN (ithprime(n+1)); fi; end: seq(A243155 (), n=1..100);

n = 0; Do[t = Prime[k] - Prime[k - 1]; If[IntegerQ[t^(1/3)], n++; Print[n, " ", Prime[k]]], {k, 2, 15*10^4}]

s=[]; forprime(p=3, 4000, if(ispower(p-precprime(p-1), 3), s=concat(s, p))); s \\ Colin Barker, Jun 03 2014

cp's OEIS Frontend

A123995 First occurrence of prime gaps which are perfect powers.

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