A392304 - OEIS

%I #30 Jan 20 2026 16:17:40

%S 1,4,225,256,324,4225,5929,6084,7569,12100,17424,19881,38416,61009,

%T 78400,90601,99225,103684,112225,112896,140625,142884,160000,161604,

%U 174724,184900,195364,202500,211600,230400,231361,262144,266256,291600,304704,329476,331776,346921

%N Squares whose sum of prime factors (with multiplicity) is also a perfect square.

%C A number x = m^2 is in the sequence if sopf(x) is a perfect square, where sopf(k) is the sum of prime factors of k with multiplicity (A001414).

%C Except for a(1)=1, all values of sopfr(a(n)) are even perfect squares because sopfr(n^2) = 2*sopfr(n).

%C For any prime p and any positive integer k, the number x = p^(p * (2*k)^2) is a term of the sequence.

%e a(1) = 1: 1 is a square (1^2) and its sopf is 0 (empty sum), which is 0^2.

%e a(2) = 4: 4 = 2^2. Prime factors are 2, 2. Sum: 2+2 = 4, which is 2^2.

%e a(3) = 225: 225 = 15^2. Prime factors are 3, 3, 5, 5. Sum: 3+3+5+5 = 16, which is 4^2.

%e a(4) = 256: 256 = 16^2 = 2^8. Prime factors are eight 2's. Sum: 2*8 = 16, which is 4^2.

%e a(5) = 324: 324 = 18^2. Prime factors of 324 are 2, 2, 3, 3, 3, 3. Sum: 2+2+3+3+3+3 = 16, which is 4^2.

%t sopf[n_] := Total[Flatten[Table @@@ FactorInteger[n]]];

%t Select[Range[500]^2, IntegerQ[Sqrt[sopf[#]]] &]

%Y Intersection of A000290 and A051448.

%Y Cf. A001414 (sopf).

%K nonn,easy

%O 1,2

%A _Yunus Emre Yaman_, Jan 06 2026