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%I #26 Jul 02 2020 12:55:08
%S 140,189,378,1375,2750,2775,2997,4524,5550,5661,5994,6375,11253,11322,
%T 12750,13416,13505,22506,25925,27010,27511,30613,32208,32513,32760,
%U 45917,49665,49959,51850,55022,61061,61226,65026,67488,91834,93605
%N Numbers n such that the sum of the Carmichael lambda functions of the divisors is a proper divisor of n.
%C The auxiliary sequence defined by b(n)=sum_{d|n} A002322(d) starts 1,2,3,4,5,6,7,6,9,10,11,10,13,14,11,10,17,18,19,16,...
%C The auxiliary sequence is A141258. [_Reinhard Zumkeller_, Feb 17 2012]
%H Amiram Eldar, <a href="/A131492/b131492.txt">Table of n, a(n) for n = 1..1000</a>
%H W. D. Banks and F. Luca, <a href="https://www.emis.de/journals/AM/06-1/am1278.pdf">On integers with a special divisibility property</a>, Archivum Mathematicum (BRNO) 42 (2006) pp 31-42.
%F n such that (sum_{d|n} A002322(d)) | n.
%t Select[ Range[100000], Divisible[#, s = Total[ CarmichaelLambda /@ Divisors[#]]] && s < # &] (* _Jean-François Alcover_, Jun 24 2013 *)
%o (PARI) lambda(p,alpha)={ if(p>=3 || alpha<=2, return(p^(alpha-1)*(p-1)), return(2^(alpha-2)) ; ) ; } A002322(n)={ local(pf,rmax,resul) ; if(n==1, return(1) ) ; pf=factor(n) ; rmax=matsize(pf)[1] ; resul= lambda(pf[1,1],pf[1,2]) ; for(r=2,rmax, resul=lcm(resul,lambda(pf[r,1],pf[r,2])) ; ) ; return(resul) ; } b(n)={ sumdiv(n,d,A002322(d)) ; } { for(n=1,120000, l=b(n) ; if( l != 1 && l != n && n%l==0, print1(n,",") ) ; ) ; }
%Y Cf. A002322, A039716.
%K nonn
%O 1,1
%A _R. J. Mathar_, Jul 29 2007
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