cp's OEIS Frontend

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

%I A190882 #26 Oct 20 2024 02:32:50
%S A190882 46206,72105,73346,92412,96096,97440,98098,99528,113883,117040,127680,
%T A190882 134805,138618,143520,146692,150024,165880,165886,184824,192192,
%U A190882 194880,196196,199056,216315,234080,255360,269192,276640,277236,287040,288288,292320,293384,294216,298584,300048,331760
%N A190882 Numbers other than prime powers divisible by the sum of the squares of their prime divisors.
%C A190882 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 A190882 Koninck & Luca show that this sequence is infinite. - _Charles R Greathouse IV_, Sep 08 2012
%H A190882 Charles R Greathouse IV, <a href="/A190882/b190882.txt">Table of n, a(n) for n = 1..10000</a>
%H A190882 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 A190882 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 A190882 with(numtheory):for n from 1 to 200000 do:x:=factorset(n):n1:=nops(x):s:=0:for
%p A190882   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:
%t A190882 Select[Range[2,332000],!PrimePowerQ[#]&&Divisible[#,Total[Select[ Divisors[#],PrimeQ]^2]]&] (* _Harvey P. Dale_, May 24 2022 *)
%o A190882 (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 A190882 (PARI) is(n)=n>9 && !isprimepower(n) && n%norml2(factor(n)[,1])==0 \\ _Charles R Greathouse IV_, Feb 03 2016
%Y A190882 Cf. A066031.
%K A190882 nonn
%O A190882 1,1
%A A190882 _Michel Lagneau_, May 23 2011