A273801 Numbers n for which n = (x - phi(x)) * (y - phi(y)), where n = x + y and x - phi(x) is the Euler cototient function of x.

16, 24, 32, 48, 56, 72, 80, 96, 120, 128, 152, 168, 176, 192, 216, 240, 248, 272, 288, 296, 320, 336, 360, 392, 408, 416, 432, 440, 456, 512, 528, 552, 560, 600, 608, 632, 656, 672, 696, 720, 728, 768, 776, 792, 800, 848, 896, 912, 920, 936, 960, 968, 1008, 1032

Offset: 1

Author

Paolo P. Lava, May 31 2016

Keywords

• nonn

Examples

			16 = 4 + 12 = (4 - phi(4)) * (12 - phi(12)) = 2 * 8 = 16 and also
16 = 8 + 8 = (8 - phi(8)) * (8 - phi(8)) = 4 * 4 = 16;
24 = 4 + 20 = (4 - phi(4)) * (20 - phi(20)) = 2 * 12 = 24.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A051953, A273800.

Programs

Maple

with(numtheory): P:=proc(q) local a,b,k,n; for n from 1 to q do
for k from 1 to trunc(n/2) do if (k-phi(k))*(n-k-phi(n-k))=n then print(n); break; fi;
od; od; end: P(10^9);

Mathematica

Select[Range@ 1032, Function[n, Length@ Select[Times @@ Map[(# - EulerPhi@ #) &, {#, n - #}] & /@ Range[0, Floor[n/2]], # == n &] > 0]] (* Michael De Vlieger, Jun 01 2016 *)

Formula

a(n) = 4*(prime(n+1) + 1). - Paolo P. Lava, Sep 06 2017