A132746 Numbers k such that prime(k) + prime(k+1) is a perfect power.

2, 7, 15, 18, 20, 28, 61, 152, 190, 293, 377, 492, 558, 564, 789, 919, 942, 1332, 1768, 2343, 2429, 2693, 2952, 3136, 3720, 3928, 4837, 5421, 5722, 6870, 7347, 8126, 8193, 9465, 9857, 9927, 10410, 10483, 10653, 12685, 13005, 13763, 13955, 16033, 16342

Offset

1,1

Comments

First terms absent in A064397: 2, 18, 28, 564, 1332, 3928, 12415, 13005, 16886.

Links

Robert Israel, Table of n, a(n) for n = 1..1000

Formula

a(n) >> n^2/log^2 n. - Charles R Greathouse IV, Mar 08 2025

Example

2 is a term because prime(2) + prime(3) = 3 + 5 = 8 = 2^3 (perfect power);
7 is a term because prime(7) + prime(8) = 17 + 19 = 36 = 6^2 (perfect power);
39867 is a term because prime(39867) + prime(39868) = 478241 + 478243 = 956484 = 978^2 (perfect power).

Maple

ispow:= proc(n) local F;
  F:= ifactors(n)[2];
  igcd(F[.., 2]) > 1
end proc:
q:= 2: count:= 0: R:= NULL:
for i from 1 while count < 100 do
  p:= q; q:= nextprime(q);
  if ispow(p+q) then count:= count+1; R:= R, i;
  fi
od:
R; # Robert Israel, Mar 09 2026

Mathematica

Select[Range[16342], ResourceFunction["PerfectPowerQ"][Prime[#]+Prime[#+1]]&] (* James C. McMahon, Mar 08 2025 *)

Prog

(PARI) s=[]; for(n=1, 41530, a=prime(n)+prime(n+1); if(ispower(a), s=concat(s, n))); s

Crossrefs

Cf. A064397 (numbers k such that prime(k) + prime(k+1) is a square).

Cf. A001043, A001597.

Sequence in context: A088824 A034903 A070898 * A252475 A350043 A375748

Adjacent sequences: A132743 A132744 A132745 * A132747 A132748 A132749

Keyword

nonn

Author

Zak Seidov, Nov 17 2007

Status

approved