A018784 Numbers n such that sigma(phi(n)) = n.

1, 3, 15, 28, 255, 744, 2418, 20440, 65535, 548856, 2835756, 4059264, 4451832, 10890040, 13192608, 23001132, 54949482, 110771178, 220174080, 445701354, 4294967295, 16331433888, 18377794080, 94951936080, 204721968000, 386940247200, 601662398400, 1433565580920

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

The numbers 2^2^n-1 for n=0,1,...,5 are in the sequence because 2^2^n-1=(2^2^0+1)*(2^2^1+1)*(2^2^2+1)*...*(2^2^(n-1)+1); 2^2^k+1 for k=0,1,2,3 & 4 are primes (Fermat primes); sigma(2^k)=2^(k+1)-1 and phi is a multiplicative function. Hence if p is a known Fermat prime (p=2^2^n+1 for n=0,1,2,3 & 4) then p-2 is in the sequence, note that this is not true for unknown Fermat primes if they exist. - Farideh Firoozbakht, Aug 27 2004

Links

• Graeme L. Cohen, On a conjecture of Makowski and Schinzel, Colloquium Mathematicae, Vol. 74, No. 1 (1997), pp. 1-8. See Notes p. 7.

Crossrefs

Cf. A097645, A097646, A019434.

Programs

Mathematica

Select[Range[10^6], DivisorSigma[1, EulerPhi[#]] == # &] (* Amiram Eldar, Dec 10 2020 *)

PARI

is(n)=sigma(eulerphi(n))==n \\ Charles R Greathouse IV, Nov 27 2013

Formula

sigma(A001229), sorted.

Extensions

Wilson's search was complete only through a(19) = 50319360. Jud McCranie reports Jun 15 1998 that the terms through a(24) are certain.

a(26)-a(28) added. Verified sequence is complete through a(28) by Donovan Johnson, Jun 30 2012