A392304 Squares whose sum of prime factors (with multiplicity) is also a perfect square.

1, 4, 225, 256, 324, 4225, 5929, 6084, 7569, 12100, 17424, 19881, 38416, 61009, 78400, 90601, 99225, 103684, 112225, 112896, 140625, 142884, 160000, 161604, 174724, 184900, 195364, 202500, 211600, 230400, 231361, 262144, 266256, 291600, 304704, 329476, 331776, 346921

Offset

1,2

Comments

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).

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

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

Links

Table of n, a(n) for n=1..38.

Example

a(1) = 1: 1 is a square (1^2) and its sopf is 0 (empty sum), which is 0^2.
a(2) = 4: 4 = 2^2. Prime factors are 2, 2. Sum: 2+2 = 4, which is 2^2.
a(3) = 225: 225 = 15^2. Prime factors are 3, 3, 5, 5. Sum: 3+3+5+5 = 16, which is 4^2.
a(4) = 256: 256 = 16^2 = 2^8. Prime factors are eight 2's. Sum: 2*8 = 16, which is 4^2.
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.

Mathematica

sopf[n_] := Total[Flatten[Table @@@ FactorInteger[n]]];
Select[Range[500]^2, IntegerQ[Sqrt[sopf[#]]] &]

Crossrefs

Intersection of A000290 and A051448.

Cf. A001414 (sopf).

Sequence in context: A052209 A211610 A364481 * A042539 A381723 A182484

Adjacent sequences: A392301 A392302 A392303 * A392305 A392306 A392307

Keyword

nonn,easy

Author

Yunus Emre Yaman, Jan 06 2026

Status

approved