Search: id:A124240
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%I A124240
%S A124240 1,2,4,6,8,12,16,18,20,24,32,36,40,42,48,54,60,64,72,80,84,96,100,108,
%T A124240 120,126,128,144,156,160,162,168,180,192,200,216,220,240,252,256,272,
%U A124240 288,294,300,312,320,324,336,342,360,378,384,400,420,432,440,468,480
%N A124240 Numbers n such that lambda(n) divides n, where lambda is Carmichael's 
               function, A002322.
%C A124240 Numbers n such that A124239(n) is divisible by n.
%C A124240 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.
%C A124240 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,...}.
%C A124240 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?
%C A124240 For links and Mathematica program see A068563.
%H A124240 T. D. Noe, <a href="b124240.txt">Table of n, a(n) for n=1..1000</a>
%H A124240 Leroy Quet, <a href="http://www.prism-of-spirals.net/">Home Page</a> 
               (listed in lieu of email address)
%H A124240 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CarmichaelFunction.html">Carmichael Function</a> [From T. D. Noe 
               (noe(AT)sspectra.com), Sep 11 2008]
%e A124240 A124239[n] begins {1, 14, 197, 3704, 90309, 2704470, 95856025, 3921108576, 
               ...}.
%e A124240 Thus a(0) = 1 because 1 divides A124239[1] = 1.
%e A124240 a(1) = 2 because 2 divides A124239[2] = 14.
%e A124240 a(3) = 4 because 4 divides A124239[4] = 3704, but 3 does not divide A124239[3] 
               = 197.
%t A124240 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}]
%t A124240 Flatten[Position[Table[n/CarmichaelLambda[n], {n, 440}], _Integer]] [From 
               T. D. Noe (noe(AT)sspectra.com), Sep 11 2008]
%Y A124240 Cf. A124239, A124241, A068563.
%Y A124240 Sequence in context: A092903 A005153 A068563 this_sequence A068997 A067712 
               A060765
%Y A124240 Adjacent sequences: A124237 A124238 A124239 this_sequence A124241 A124242 
               A124243
%K A124240 nonn
%O A124240 1,2
%A A124240 Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 22 2006
%E A124240 New definition from T. D. Noe, Aug 31 2008

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