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

1,1

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

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.

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A066031.

Sequence in context: A115939 A168630 A361035 * A251278 A202897 A185518

Adjacent sequences: A190879 A190880 A190881 * A190883 A190884 A190885

Keyword

nonn

Author

Michel Lagneau, May 23 2011

Status

approved