A396408 Numbers k such that A075254(k) is the square of a prime.

2, 108, 936, 1820, 2166, 4437, 5005, 5008, 5278, 10486, 10545, 10556, 10768, 16144, 18582, 22126, 27420, 27840, 37014, 37149, 37170, 38600, 39321, 44421, 49595, 62830, 68697, 69046, 72208, 72700, 78760, 79935, 91024, 97820, 100284, 113145, 113477, 119470, 123777, 128616, 128652, 133494, 138814

Offset

1,1

Comments

Numbers k such that k + sopfr(k) is a square, where sopfr(k) = A001414(k) is the sum of prime factors of k with multiplicity.

Except for 2, a subset of A386246.

Links

Robert Israel, Table of n, a(n) for n = 1..1250

Example

a(3) = 936 is a term because 936 = 2^3 * 3^2 * 13, 936 + 3*2 + 2*3 + 13 = 961 = 31^2, and 31 is prime.

Maple

filter:= proc(n) local t, s;
 s:= n + add(t[1]*t[2], t=ifactors(n)[2]);
 issqr(s) and isprime(sqrt(s))
end proc:
A:=select(filter, [$1..10^6]);

Mathematica

a075254[n_]:=If[n==1, 1, n +Plus@@Times@@@FactorInteger@n]; q[k_]:=PrimeQ[Sqrt[a075254[k]]]; Select[Range[138814], q] (* James C. McMahon, May 29 2026 *)

Prog

(PARI) upto(nn)= my(f, s); [ k |k<-[1..nn], issquare(s=k+(f=factor(k))[, 1]~*f[, 2]) && isprime(sqrtint(s))]; \\ Ruud H.G. van Tol, May 31 2026

Crossrefs

Cf. A001248, A075254, A386246, A386954.

Sequence in context: A224819 A156502 A111456 * A157067 A364616 A375841

Adjacent sequences: A396405 A396406 A396407 * A396409 A396411 A396412

Keyword

nonn

Author

Will Gosnell and Robert Israel, May 24 2026

Status

approved