1,2

Numbers n such that A124239(n) is divisible by n.

A124239[n] = Sum[ Sum[ (2k-1)^m, {m,1,n} ], {k,1,n} ] = n + Sum[ (2k-1)((2k-1)^n-1) / (2(k-1)), {k,2,n} ]. If k is in the sequence then 2k is also in the sequence, but if 2m is in the sequence m is not necessarily a term of the sequence.

It appears that a(n) almost coincides with A068563[n] Numbers n such that 2^n (mod n) = 4^n (mod n). The first term that is different is A068563[27] = 136. The terms of A068563[n] that are not the terms of a(n) are listed in A124241[n] = {136,408,620,680,820,...}.

It appears that this sequence (except for the initial 1) is the same as the sequence of numbers n such that p-1 divides n for all primes p that divide n. - Leroy Quet, Jun 27 2008. Maximilian Hasler and Edwin Clark have confirmed that this is true for n <= 2500. Is there a proof?

For links and Mathematica program see A068563.

T. D. Noe, Table of n, a(n) for n=1..1000

Leroy Quet, Home Page (listed in lieu of email address)

Eric Weisstein's World of Mathematics, Carmichael Function [From T. D. Noe (noe(AT)sspectra.com), Sep 11 2008]

A124239[n] begins {1, 14, 197, 3704, 90309, 2704470, 95856025, 3921108576, ...}.

Thus a(0) = 1 because 1 divides A124239[1] = 1.

a(1) = 2 because 2 divides A124239[2] = 14.

a(3) = 4 because 4 divides A124239[4] = 3704, but 3 does not divide A124239[3] = 197.

Do[f=n + Sum[ (2k-1)((2k-1)^n-1) / (2(k-1)), {k, 2, n} ]; If[IntegerQ[f/n], Print[n]], {n, 1, 900}]

Flatten[Position[Table[n/CarmichaelLambda[n], {n, 440}], _Integer]] [From T. D. Noe (noe(AT)sspectra.com), Sep 11 2008]

Cf. A124239, A124241, A068563.

Sequence in context: A092903 A005153 A068563 this_sequence A068997 A067712 A060765

Adjacent sequences: A124237 A124238 A124239 this_sequence A124241 A124242 A124243

nonn

Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 22 2006

New definition from T. D. Noe, Aug 31 2008