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%I #14 May 28 2016 09:22:16
%S 80,400,693,2000,3290,7030,10000,24150,50000,191961,250000,610718,
%T 1214425,1250000,2194778,6250000,31250000,75369362,156250000,
%U 234392726,572760397,588270806,590434574,595208594,781250000,1547001099,2889682738,3906250000,7627258546,10614420142
%N Numbers n such that n = Sum_{i=1..j} (phi(n) mod d(i)), where phi(n) is the Euler totient function of n and d(i) are the divisors of n.
%C This sequence is infinite, since it contains all the numbers of the form 2^4*5^k, for k>0. The key fact is that phi(2^4*5^k) = 2^5*5^(k-1), so only 5 moduli do not vanish. - _Giovanni Resta_, May 25 2016
%e Divisors of 693 are 1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 231, 693 and its Euler totient function is phi(693) = 360. Then 360 mod 1 = 0, 360 mod 3 = 0, 360 mod 7 = 3, 360 mod 9 = 0, 360 mod 11 = 8, 360 mod 21 = 3, 360 mod 33 = 30, 360 mod 63 = 45, 360 mod 77 = 52, 360 mod 99 = 63, 360 mod 231 = 129, 360 mod 693 = 360 and 0 + 0 + 3 + 0 + 8 + 3 + 30 + 45 + 52 + 63 + 129 + 360 = 693.
%p with(numtheory): P:=proc(q) local a,k,n; for n from 1 to q do a:=divisors(n);
%p if add(phi(n) mod a[k],k=1..nops(a))=n then print(n); fi; od; end: P(10^9);
%t Select[Range[10^5], # == Plus @@ Mod[EulerPhi@ #, Divisors@ #] &] (* _Giovanni Resta_, May 25 2016 *)
%Y Cf. A000010.
%K nonn
%O 1,1
%A _Paolo P. Lava_, May 19 2016
%E a(17)-a(30) from _Giovanni Resta_, May 25 2016
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