OFFSET
1,1
COMMENTS
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.
The observation is false (unless it is just meant to refer to the Data or the b-file): it is conjectured that every positive even number occurs infinitely often as a prime gap. According to the b-file for A001632 the first case where the gap is 6^3 is where the larger prime is 202551883. - Robert Israel, Apr 15 2026
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
LINKS
K. D. Bajpai, Table of n, a(n) for n = 1..10000
EXAMPLE
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.
MAPLE
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);
# Alternative:
N:= 10^4: # for terms <= N
P:= select(isprime, [2, seq(i, i=3..N, 2)]):
G:= P[2..-1]-P[1..-2]:
J:=select(t -> surd(G[t], 3)::integer, [$1..nops(G)]):
P[J +~ 1]; # Robert Israel, Apr 15 2026
MATHEMATICA
n = 0; Do[t = Prime[k] - Prime[k - 1]; If[IntegerQ[t^(1/3)], n++; Print[n, " ", Prime[k]]], {k, 2, 15*10^4}]
PROG
(PARI) s=[]; forprime(p=3, 4000, if(ispower(p-precprime(p-1), 3), s=concat(s, p))); s \\ Colin Barker, Jun 03 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
K. D. Bajpai, May 31 2014
STATUS
approved