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Numbers k such that if x = k - phi(k) then k = sigma(x) - x, where phi(k) is the Euler totient function.
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%I #17 Nov 21 2024 06:15:09

%S 36,42,186,222,270,390,396,440,656,2220,4140,5622,9400,20214,94816,

%T 282540,17578122,85046840,125948800,145805120,434435360,11152607958,

%U 11160256626

%N Numbers k such that if x = k - phi(k) then k = sigma(x) - x, where phi(k) is the Euler totient function.

%C Fixed points of the transform k -> sigma(k-phi(k)) - k + phi(k).

%C a(24) > 5*10^10, if it exists. - _Amiram Eldar_, Nov 21 2024

%e phi(222) = 72 and 222 - 72 = 150; sigma(150) = 372 and 372 - 150 = 222.

%p with(numtheory); P:=proc(q) local n; k:=0;

%p for n from 1 to q do if 2*n=sigma(n-phi(n))+phi(n) then print(n);

%p fi; od; end: P(10^9);

%t q[k_] := Module[{e = EulerPhi[k]}, DivisorSigma[1, k - e] - k + e == k]; Select[Range[300000], q] (* _Amiram Eldar_, Nov 21 2024 *)

%o (PARI) isok(n) = (x = n - eulerphi(n)) && (n == sigma(x) - x); \\ _Michel Marcus_, Mar 28 2014

%Y Cf. A000010, A000203, A239801.

%K nonn,more

%O 1,1

%A _Paolo P. Lava_, Mar 27 2014

%E a(17)-a(21) from _Michel Marcus_, Mar 28 2014

%E a(22)-a(23) from _Amiram Eldar_, Nov 21 2024