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A190882 Numbers other than prime powers divisible by the sum of the squares of their prime divisors. 4

%I

%S 46206,72105,73346,92412,96096,97440,98098,99528,113883,117040,127680,

%T 134805,138618,143520,146692,150024,165880,165886,184824,192192,

%U 194880,196196,199056,216315,234080,255360,269192,276640,277236,287040,288288,292320,293384,294216,298584,300048,331760

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

%C 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.

%C Koninck & Luca show that this sequence is infinite. - _Charles R Greathouse IV_, Sep 08 2012

%H Charles R Greathouse IV, <a href="/A190882/b190882.txt">Table of n, a(n) for n = 1..10000</a>

%H Jean-Marie de Koninck and Florian Luca, <a href="http://dx.doi.org/10.1016/j.jnt.2007.01.010">Integers divisible by sums of powers of their prime factors</a>, Journal of Number Theory, Volume 128, Issue 3, March 2008, Pages 557-563.

%e 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.

%p with(numtheory):for n from 1 to 200000 do:x:=factorset(n):n1:=nops(x):s:=0:for

%p 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:

%o (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

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

%Y Cf. A066031.

%K nonn

%O 1,1

%A _Michel Lagneau_, May 23 2011

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Last modified October 22 16:15 EDT 2021. Contains 348174 sequences. (Running on oeis4.)