A390574 - OEIS

A390574

Numbers for which the square of its largest prime factor equals the sum of the squares of its other prime factors.

4, 9, 25, 49, 121, 169, 240, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2240, 2809, 3402, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 21504, 22201, 22801, 24649, 26569, 27889, 29929

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OFFSET

1,1

COMMENTS

All squares of primes (A001248) are terms because they contain exactly and only 2 equal prime factors.

No squares of nonprimes (A062312) are terms because they contain 2 of the same largest prime factor plus additional prime factors.

Subsequence of only nonsquare terms is A391874.

LINKS

Table of n, a(n) for n=1..44.

EXAMPLE

a(1) = 4 is a term because the prime factors of 4 are {2, 2} and 2^2 = 2^2.

a(7) = 240 is a term (the first nonsquare term) because the prime factors of 240 are {2, 2, 2, 2, 3, 5} and 2^2 + 2^2 + 2^2 + 2^2 + 3^2 = 5^2.

1200 is NOT a term because its prime factors are {2, 2, 2, 2, 3, 5, 5} and 2^2 + 2^2 + 2^2 + 2^2 + 3^2 + 5^2 != 5^2.

MATHEMATICA

f[{p_, e_}]:=Table[p^2, e]; lps[k_]:=Last[f/@FactorInteger[k]//Flatten]; sum[k_]:=Total[f/@FactorInteger[k]//Flatten]; Select[Range[30000], !PrimeQ[#]&&sum[#]==2*lps[#]&] (* James C. McMahon, Jan 03 2026 *)

PROG

(PARI) is_a390574(n)={my(f=Vec(factor(n))); f[2][#f[2]]--; return(if(f[1][#f[1]]^2==sum(i=1, #f[1], f[1][i]^2*f[2][i]), 1, 0))}

CROSSREFS

Cf. A163836, A001248, A062312, A391874.

Sequence in context: A390253 A082180 A246131 * A068999 A179707 A247078

Adjacent sequences: A390571 A390572 A390573 * A390575 A390576 A390577

KEYWORD

nonn

AUTHOR

Charles L. Hohn, Dec 17 2025

STATUS

approved