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A248815
Numbers equal to the arithmetic derivative of their Euler totient function.
1
31, 32, 80, 81, 112, 176, 192, 244, 368, 752, 859, 912, 944, 1296, 1328, 1712, 1723, 2672, 2864, 3024, 3632, 4208, 5552, 5744, 6128, 6156, 7472, 7664, 8048, 8748, 9008, 9392, 11504, 13424, 13808, 14192, 15728, 16304, 18992, 20412, 20528, 20912, 21104, 21872
OFFSET
1,1
COMMENTS
Solutions of the equation n = (phi(n))’.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from Paolo P. Lava)
EXAMPLE
Euler totient function of 32 is 16 and the arithmetic derivative of 16 is 32.
MAPLE
with(numtheory): P:= proc(q) local a, n, p; for n from 1 to q do
a:=phi(n)*add(op(2, p)/op(1, p), p=ifactors(phi(n))[2]);
if n=a then print(n); fi; od; end: P(10^9);
MATHEMATICA
deriv[n_] := n*Total[#2/#1 & @@@ FactorInteger[n]]; aQ[1] = 1; aQ[n_] := deriv[EulerPhi[n]] == n; Select[Range[25000], aQ] (* Amiram Eldar, Jul 11 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Paolo P. Lava, Oct 15 2014
STATUS
approved