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A173325 Numbers k such that sigma(tau(k)) equals the sum of distinct primes dividing k.

%I A173325 #17 Oct 12 2024 09:03:33
%S A173325 3,10,104,105,175,245,276,343,414,484,532,798,1190,1430,1776,1862,
%T A173325 3105,3174,3712,4394,5049,5054,5104,5994,6256,6360,6975,8125,8480,
%U A173325 8625,9472,9648,10600,12408,12789,14310,16544,16625,16728,19908,20295,21056,21708
%N A173325 Numbers k such that sigma(tau(k)) equals the sum of distinct primes dividing k.
%C A173325 sigma(tau(k)) = A000203(A000005(k)) = A062069(k).
%C A173325 From _Robert Israel_, Nov 07 2016: (Start)
%C A173325 If m is in A023194, sigma(m)^(m-1) is in the sequence.
%C A173325 If p and q are distinct primes, and r and s are distinct primes such that r+s = (p+1)(q+1), then r^(p-1)*s^(q-1) is in the sequence.
%C A173325 (End)
%H A173325 Robert Israel, <a href="/A173325/b173325.txt">Table of n, a(n) for n = 1..1000</a>
%H A173325 C. K. Caldwell, <a href="https://t5k.org/glossary/page.php?sort=Tau">The Prime Glossary, Number of divisors</a>
%H A173325 P. A. MacMahon, <a href="http://dx.doi.org/10.1112/plms/s2-19.1.75">Divisors of numbers and their continuations in the theory of partitions</a>, Proc. London Math. Soc., 19 (1921), 75-113.
%H A173325 W. Sierpinski, <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon42/mon4204.pdf">Number Of Divisors And Their Sum</a>
%F A173325 {k: A062069(k) = A008472(k)}.
%e A173325 k=3 with sigma(tau(3)) = sigma(2) = 3 = A008472(3).
%e A173325 k=10 with sigma(tau(10)) = sigma(4) = 7 = A008472(10).
%p A173325 with(numtheory): for n from 1 to 100000 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)):if sigma(tau(n)) = t2 then print (n): else fi : od :
%Y A173325 Cf. A001222, A001414, A062069, A173320.
%K A173325 nonn
%O A173325 1,1
%A A173325 _Michel Lagneau_, Feb 16 2010
%E A173325 "sopf" uses replaced and examples disentangled by _R. J. Mathar_, Feb 24 2010