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A237613 Numbers k such that tau(sigma(tau(k))) = sigma(tau(sigma(k))), where tau is A000005 and sigma is A000203.

%I A237613 #20 Nov 04 2024 14:48:29
%S A237613 1,4,9,25,81,289,1681,3481,5041,7921,10201,17161,27889,29929,85849,
%T A237613 146689,331776,458329,491401,552049,579121,597529,683929,703921,
%U A237613 734449,786432,829921,1190281,1203409,1352569,1394761,1423249,1481089,1885129,2036329,2211169
%N A237613 Numbers k such that tau(sigma(tau(k))) = sigma(tau(sigma(k))), where tau is A000005 and sigma is A000203.
%C A237613 The squares of the terms of A053182 are a subset of this sequence. In fact, in general, if p is prime we have tau(p)=2 and tau(p^2)=3. Therefore tau(p^2)=3 -> sigma(3)=4 -> tau(4)=tau(2^2)=3 and if p belongs to A053182 we also have that sigma(p^2)=p^2+p+1 (prime) -> tau(p^2+p+1)=2 -> sigma(2)=3.
%H A237613 Amiram Eldar, <a href="/A237613/b237613.txt">Table of n, a(n) for n = 1..100</a>
%p A237613 with(numtheory); P:=proc(q) local n;
%p A237613 for n from 1 to q do
%p A237613   if tau(sigma(tau(n)))=sigma(tau(sigma(n))) then print(n); fi;
%p A237613 od; end: P(10^6);
%t A237613 s = {}; Do[If[DivisorSigma[1, DivisorSigma[0, DivisorSigma[1, n]]] == DivisorSigma[0, DivisorSigma[1, DivisorSigma[0, n]]], AppendTo[s, n]], {n, 1, 2500000}]; s (* _Amiram Eldar_, Aug 17 2019 *)
%t A237613 With[{ds=DivisorSigma},Select[Range[2220000],ds[0,ds[1,ds[0,#]]]==ds[1,ds[0,ds[1,#]]]&]] (* _Harvey P. Dale_, Nov 04 2024 *)
%o A237613 (PARI) s=[]; for(n=1, 2500000, if(sigma(sigma(sigma(n, 0)), 0) == sigma(sigma(sigma(n), 0)), s=concat(s, n))); s \\ _Colin Barker_, Feb 10 2014
%o A237613 (Magma) [k:k in [1..2300000]| #Divisors(SumOfDivisors(#Divisors(k))) eq SumOfDivisors(#Divisors(SumOfDivisors(k)))]; // _Marius A. Burtea_, Aug 17 2019
%Y A237613 Cf. A000005, A000203, A053182.
%K A237613 nonn
%O A237613 1,2
%A A237613 _Paolo P. Lava_, Feb 10 2014