A248881 Numbers n such that lambda(sum of even divisors of 2n) = lambda(sum of odd divisors of 2n) where lambda is the Carmichael function (A002322).

1, 3, 5, 6, 9, 11, 13, 17, 18, 19, 25, 26, 27, 29, 36, 37, 38, 41, 43, 45, 49, 50, 53, 54, 59, 61, 63, 65, 67, 68, 72, 73, 74, 75, 81, 82, 83, 85, 86, 87, 89, 90, 95, 97, 98, 99, 100, 101, 103, 107, 109, 113, 117, 121, 122, 125, 126, 130, 131, 134, 137, 139

Offset: 1

Author

Michel Lagneau, Mar 05 2015

Keywords

• nonn

Comments

Number n such that A002322(A074400(n))= A002322(A000593(n)).

The squares of the form p^2 with p prime are in the sequence because the divisors of 2p^2 are {1,2,p,2p,p^2,2p^2} => sum of even divisors s0 = 2+2p+2p^2 = 2(p^2+p+p^2) and sum of odd divisors s1 = 1+p+p^2 and lambda(s0) = lambda(s1) = lambda(2*s0).

A majority of primes are in the sequence: 3, 5, 11, 13, 17, 19, 29, 37, 41, 43, 53, 59, 61, 67, 73, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 157, 163, 173, 179, 181, 193, 197, ... but the primes 7, 23, 31, 47, 71, 79, 127, 151, 167, 191, 223, 239, 263, 367, 383, 431, ... are not in the sequence.

Examples

			18 is in the sequence because A002322(A074400(18))= A002322(78)= 12 and because A002322(A000593(18)) = A002322(13) = 12.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000593, A002322, A074400, A077591.

Programs

Mathematica

lst={};f[x_] := Plus @@ Select[Divisors[x], OddQ[#] &]; g[x_] := Plus @@ Select[Divisors[x], EvenQ[#]&]; Do[If[CarmichaelLambda[f[n]]== CarmichaelLambda[g[n]], AppendTo[lst,n/2]], {n, 1, 500}];lst

PARI

a002322(n) = lcm(znstar(n)[2]);
isok(n) = my(sod = sumdiv(2*n, d, d*(d%2))); my(sed = sigma(2*n) - sod); sod && sed && (a002322(sod) == a002322(sed)); \\ Michel Marcus, Mar 07 2015