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

Author

Zak Seidov, Nov 17 2007

Keywords

• nonn

Comments

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

Examples

			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).
		

Crossrefs

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

Cf. A001043, A001597.

Programs

Mathematica

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

PARI

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

Formula

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