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%I #13 Aug 12 2023 04:51:36
%S 2,6,14,42,62,186,254,434,762,1302,1778,5334,7874,16382,23622,49146,
%T 55118,114674,165354,262142,344022,507842,786426,1048574,1523526,
%U 1834994,2080514,3145722,3554894,5504982,6241542,7340018,8126402,10664682,14563598,22020054
%N Even integers k such that phi(sum of even divisors of k) = sum of odd divisors of k.
%C The number of divisors of a(n) is a power of 2, and sum of even divisors = 2^(m+1), sum of odd divisors = 2^m for some m.
%C a(n) == 2, 6 (mod 8) or a(n) == 2, 6 (mod 12).
%C a(n) is of the form 2*p1*p2*...pk where p1, p2, ..., pk are Mersenne primes = 3, 7, 31, 127, 8191, ... (see A000668).
%e 62 is a term because its divisors are 1, 2, 31 and 62, the sum of the even divisors of 62 = 62 + 2 = 2^6, the sum of odd divisors = 1 + 31 = 2^5, and phi(2^6) = 2^5.
%p with(numtheory):
%p for n from 2 by 2 to 10^6 do:
%p x:=divisors(n):n1:=nops(x):s0:=0:s1:=0:
%p for k from 1 to n1 do:
%p if irem(x[k],2)=0
%p then
%p s0:=s0+ x[k]:
%p else
%p s1:=s1+ x[k]:
%p fi:
%p od:
%p if s1=phi(s0)
%p then
%p print(n):
%p else
%p fi:
%p od:
%t Select[2 * Range[10^6], (sodd = (s = DivisorSigma[1, #])/(2^(IntegerExponent[#, 2]+1) - 1)) == EulerPhi[s - sodd] &] (* _Amiram Eldar_, Aug 12 2023 *)
%o (PARI) isok(n) = eulerphi(sumdiv(n, d, d*((d % 2)==0))) == sumdiv(n, d, d*(d%2)); \\ _Michel Marcus_, Jan 28 2017
%Y Cf. A000010, A000593, A000668, A146076.
%K nonn
%O 1,1
%A _Michel Lagneau_, Jan 28 2017
%E a(1) inserted by _Amiram Eldar_, Aug 12 2023
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