|
%I #21 May 15 2023 14:27:28
%S 1,3,7,15,31,127,1023,8191,131071,524287,2147483647
%N Numbers k such that sigma(phi(sigma(k))) = k.
%C If n=2^k-1, where either k=1, or n is a Mersenne prime (A000668), or sigma(n)=3*2^(k-1), then n is in the sequence; are there any terms not of these forms? The last form includes the terms 15 and 1023; are there others like this?
%C Is this sequence infinite?
%C It is conjectured that there are infinitely many Mersenne primes. So this conjecture also supports that this sequence is infinite. Additionally, if n=2^k-1, where either k=1, or n is a Mersenne prime (A000668), or sigma(n)=3*2^(k-1), then A000217(n) divides sigma(A000217(n)). - _Altug Alkan_, Jul 25 2016
%e sigma(phi(sigma(31))) = sigma(phi(32)) = sigma(16) = 31.
%t Select[Range[1, 10^6], DivisorSigma[1, EulerPhi[DivisorSigma[1, # ]]]==#&]
%Y Cf. A000217, A000668.
%K nonn,more
%O 1,2
%A _Joseph L. Pe_, Dec 15 2001
%E Edited by _Dean Hickerson_, Feb 20 2002
%E a(11) from _Jud McCranie_, Jun 23 2005; no more terms < 4000000000.
|
