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

1, 4, 9, 25, 81, 289, 1681, 3481, 5041, 7921, 10201, 17161, 27889, 29929, 85849, 146689, 331776, 458329, 491401, 552049, 579121, 597529, 683929, 703921, 734449, 786432, 829921, 1190281, 1203409, 1352569, 1394761, 1423249, 1481089, 1885129, 2036329, 2211169

Offset: 1

Author

Paolo P. Lava, Feb 10 2014

Keywords

• nonn

Comments

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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A000005, A000203, A053182.

Programs

Magma

[k:k in [1..2300000]| #Divisors(SumOfDivisors(#Divisors(k))) eq SumOfDivisors(#Divisors(SumOfDivisors(k)))]; // Marius A. Burtea, Aug 17 2019

Maple

with(numtheory); P:=proc(q) local n;
for n from 1 to q do
  if tau(sigma(tau(n)))=sigma(tau(sigma(n))) then print(n); fi;
od; end: P(10^6);

Mathematica

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 *)
With[{ds=DivisorSigma},Select[Range[2220000],ds[0,ds[1,ds[0,#]]]==ds[1,ds[0,ds[1,#]]]&]] (* Harvey P. Dale, Nov 04 2024 *)

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