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%I #27 Feb 11 2021 01:25:56
%S 1,399,872,1214,2090,5200,5588,21208,29152,638049,1627676,151732410,
%T 274845607,3224070252,54892040166,69020111756,175288968221
%N Numbers k such that Sum_{j=1..k} (sigma(j) + phi(j) + tau(j)) == 0 (mod k).
%C a(17) > 10^11. - _Donovan Johnson_, Jul 07 2013
%C a(18) > 5*10^11. - _Giovanni Resta_, Jul 11 2013
%e Sum_{j=1..399} sigma(j) = 130973;
%e Sum_{j=1..399} phi(j) = 48518;
%e Sum_{j=1..399} tau(j) = 2453;
%e (130973 + 48518 + 2453) / 399 = 456.
%p with(numtheory); ListA227008:=proc(q,h) local a, n; a:=0;
%p for n from 1 to q do a:=a+sigma(n)+phi(n)+tau(n); if (a mod n)=0 then print(n); fi; od; end: ListA227008(10^9);
%o (PARI) s=0; for(n=1, 274845607, s=s+sigma(n)+eulerphi(n)+numdiv(n); if(s%n==0, print(n " " s))) /* _Donovan Johnson_, Jul 06 2013 */
%Y Cf. A000005, A000010, A000203.
%Y Cf. A006218, A002088, A024916.
%Y Cf. A233541.
%K nonn,more
%O 1,2
%A _Paolo P. Lava_, Jun 27 2013
%E a(11)-a(14) from _Donovan Johnson_, Jul 06 2013
%E a(15)-a(16) from _Donovan Johnson_, Jul 07 2013
%E a(17) from _Giovanni Resta_, Jul 11 2013
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