A190882 Numbers other than prime powers divisible by the sum of the squares of their prime divisors.

46206, 72105, 73346, 92412, 96096, 97440, 98098, 99528, 113883, 117040, 127680, 134805, 138618, 143520, 146692, 150024, 165880, 165886, 184824, 192192, 194880, 196196, 199056, 216315, 234080, 255360, 269192, 276640, 277236, 287040, 288288, 292320, 293384, 294216, 298584, 300048, 331760

Offset: 1

Author

Michel Lagneau, May 23 2011

Keywords

• nonn

Comments

The number of distinct prime divisors of n is >= 3, because if n = p^a * q^b where p and q are distinct primes, p^2+q^2 | n => p+q == 0 (mod p) or 0 (mod q), but p==0 (mod q), or q==0 (mod p) is impossible.

Koninck & Luca show that this sequence is infinite. - Charles R Greathouse IV, Sep 08 2012

Examples

			46206 is in the sequence because the prime distinct divisors of this number are {2, 3, 17, 151} and 2^2 + 3^2 + 17^2 + 151^2 = 23103, then 46206 = 23103*2.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Jean-Marie de Koninck and Florian Luca, Integers divisible by sums of powers of their prime factors, Journal of Number Theory, Volume 128, Issue 3, March 2008, Pages 557-563.

Crossrefs

Cf. A066031.

Programs

Maple

with(numtheory):for n from 1 to 200000 do:x:=factorset(n):n1:=nops(x):s:=0:for
  p from 1 to n1 do: s:=s+x[p]^2:od:if n1 >= 2 and irem(n,s)=0 then printf(`%d,`,n):else fi:od:

Mathematica

Select[Range[2,332000],!PrimePowerQ[#]&&Divisible[#,Total[Select[ Divisors[#],PrimeQ]^2]]&] (* Harvey P. Dale, May 24 2022 *)

PARI

is(n)=my(f=factor(n)[,1]);#f>2&n%sum(i=1,#f,f[i]^2)==0 \\ Charles R Greathouse IV, May 23 2011

PARI

is(n)=n>9 && !isprimepower(n) && n%norml2(factor(n)[,1])==0 \\ Charles R Greathouse IV, Feb 03 2016