A338093 Composite numbers which are multiples of the sum of the squares of their prime factors (taken with multiplicity).

16, 27, 256, 540, 756, 1200, 1890, 2940, 3060, 3125, 4050, 4200, 4320, 5460, 6000, 6048, 7920, 8232, 10080, 10164, 10368, 10530, 11232, 11286, 12960, 13104, 13524, 13800, 14000, 14157, 14175, 15708, 15960, 17280, 18200, 18480, 19278, 19683, 19992, 20295, 23814

Offset: 1

Author

Giorgos Kalogeropoulos, Oct 09 2020

Keywords

• nonn

Comments

If a(n)=p1*p2*..*pk where p1,p2,..pk primes, then a(n)=m(p1^2+p2^2+..+pk^2) with m a positive integer.

For the special case of m=1, a(n) is equal to the sum of the squares of its prime factors.

There are only 5 known numbers to have this property:

16, 27 and three more numbers with 123, 163 and 179 digits found by Giorgos Kalogeropoulos (see Rivera links).

It is not known if any smaller numbers than those three exist for the case of m=1.

From Robert Israel, Oct 16 2020: (Start)

Suppose n is in the sequence with n = k*A067666(n). Then n^m is in the sequence if m divides k^m (in particular for m=k).

For any prime p, p^(p^j) is in the sequence if j >= 1 (except j>=2 if p=2). (End)

Examples

			16 = 2*2*2*2 = 1*(2^2 + 2^2 + 2^2 + 2^2).
7920 = 2*2*2*2*3*3*5*11 = 44*(2^2 + 2^2 + 2^2 + 2^2 + 3^2 + 3^2 + 5^2 + 11^2).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2000
• Carlos Rivera, Puzzle 625. Sum of squares of prime divisors, The Prime Puzzles and Problems Connection.
• Carlos Rivera, Puzzle 1019. Follow-up to Puzzle 625, The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A067666.

Programs

Maple

filter:= proc(n) local t;
  if isprime(n) then return false fi;
  n mod add(t[1]^2*t[2],t=ifactors(n)[2]) = 0
end proc:
select(filter, [$4..30000]); # Robert Israel, Oct 16 2020

Mathematica

Select[Range@20000,Mod[#,Total[Flatten[Table@@@FactorInteger@#]^2]]==0&]

PARI

isok(m) = if (!isprime(m) && (m>1), my(f=factor(m)); (m % sum(k=1, #f~, f[k,1]^2*f[k,2])) == 0); \\ Michel Marcus, Oct 11 2020