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%I #28 Mar 18 2020 02:02:22
%S 1,3,14,42,76,376,3608,163712,163944,196128,277688,491136,833064,
%T 849120,905814,911008,1080328,1653520,1847898,1935128,2733024,3145216,
%U 3240984,4586240,4734736,4960560,5805384,13758720,16582752,25244956,34961040,38521440,48177990,56240352
%N Numbers k such that k | phi(k)*d(k) - sigma(k), where phi=A000010, d=A000005 and sigma=A000203.
%C From _Farideh Firoozbakht_, Mar 17 2007: (Start)
%C I. If p is an odd prime then m = 2^k*p is in the sequence iff p = (k+3)*2^k - 1. For example, 14, 76, 376, 163712, 3145216, 1073733632, 1443108749312 and 67185481812096157153425363042304 are such terms. The numbers k such that (k+3)*2^k - 1 is prime up to 10000 are 1, 2, 3, 7, 9, 13, 18, 50, 210, 301, 349, 1160, 1796, 2677 and 8823. Thus 2^8823*(8826*2^8823-1) is the largest such term that I have found.
%C II. If m is in the sequence and 3 | phi(m)*d(m) - sigma(m) but 3 doesn't divide m then 3*m is in the sequence. Thus 1, 14, 163712, 277688, 911008, 1080328, 1653520, 1935128 and 4586240 are such terms and 2^2677*(2680*2^2677-1) is the largest such term that I have found. (End)
%D Inspired by David Wells, Curious and Interesting Numbers (Revised), Penguin Books.
%H Giovanni Resta, <a href="/A055650/b055650.txt">Table of n, a(n) for n = 1..89</a> (terms < 2*10^12)
%t Do[If[Mod[EulerPhi[n]*DivisorSigma[0, n]-DivisorSigma[1, n], n]==0, Print[n]], {n, 1, 1.05*10^7}]
%t Select[Range[6000000],Divisible[EulerPhi[#]DivisorSigma[0,#]- DivisorSigma[ 1,#], #]&] (* _Harvey P. Dale_, Mar 10 2012 *)
%o (PARI) isok(k) = {my(f=factor(k)); (eulerphi(f)*numdiv(f)-sigma(f))%k == 0; } \\ _Jinyuan Wang_, Mar 17 2020
%Y Cf. A000005, A000010, A000203, A079536.
%K nonn
%O 1,2
%A _Robert G. Wilson v_, Jun 06 2000
%E More terms from _Jinyuan Wang_, Mar 17 2020
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