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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).

%I A248881 #20 Nov 11 2024 21:19:42
%S A248881 1,3,5,6,9,11,13,17,18,19,25,26,27,29,36,37,38,41,43,45,49,50,53,54,
%T A248881 59,61,63,65,67,68,72,73,74,75,81,82,83,85,86,87,89,90,95,97,98,99,
%U A248881 100,101,103,107,109,113,117,121,122,125,126,130,131,134,137,139
%N 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).
%C A248881 Number n such that A002322(A074400(n))= A002322(A000593(n)).
%C A248881 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).
%C A248881 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.
%H A248881 Amiram Eldar, <a href="/A248881/b248881.txt">Table of n, a(n) for n = 1..10000</a>
%e A248881 18 is in the sequence because A002322(A074400(18))= A002322(78)= 12 and because A002322(A000593(18)) = A002322(13) = 12.
%t A248881 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
%o A248881 (PARI) a002322(n) = lcm(znstar(n)[2]);
%o A248881 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
%Y A248881 Cf. A000593, A002322, A074400, A077591.
%K A248881 nonn
%O A248881 1,2
%A A248881 _Michel Lagneau_, Mar 05 2015