A227419 - OEIS

%I #35 Aug 16 2013 18:41:54

%S 4,9,90,25,300,49,735,1770,7644,121,2541,169,5187,6710,8463,289,10982,

%T 361,11913,13202,24339,529,18515,19513,37851,20723,43239,841,35322,

%U 961,43215,20705,146595,270470,110823,1369,62835,46535,632316,1681,106074,1849

%N Least k such that the sum of the semiprime divisors equals n times the sum of the prime divisors, or 0 if no such k exists.

%C Least k such that A076290(k) = n*A008472(k), or 0 if no such k exists. a(n) = n^2 if n is a prime number => A001248 is a subsequence.

%C Conjecture: a(n) > 0.

%H Donovan Johnson, <a href="/A227419/b227419.txt">Table of n, a(n) for n = 2..1000</a>

%e a(12) = 2541: The divisors of 2541 are {1, 3, 7, 11, 21, 33, 77, 121, 231, 363, 847, 2541}, so the sum of the semiprime divisors is 21 + 33 + 77 + 121 = 252, which is 12 times the sum of prime divisors 3 + 7 + 11 = 21.

%p with(numtheory):for n from 2 to 43 do:ii:=0:for k from 2 to 700000 while(ii=0) do:x:=divisors(k):n1:=nops(x): y:=factorset(k):n2:=nops(y):s1:=0:s2:=0:for i from 1 to n1 do: if bigomega(x[i])=2 then s1:=s1+x[i]:else fi:od: s2:=sum('y[i]', 'i'=1..n2):if s1=n*s2 then ii:=1: printf ( "%d %d \n",n,k):else fi:od:od:

%Y Cf. A001248, A076290, A008472.

%K nonn

%O 2,1

%A _Michel Lagneau_, Jul 18 2013