A259495 Numbers k such that sigma(k) + phi(k) + d(k) = sigma(k+1) + phi(k+1) + d(k+1), where sigma(k) is the sum of the divisors of k, phi(k) the Euler totient function of k and d(k) the number of divisors of k.

4, 285, 902, 2013, 8493, 37406, 61918, 90094, 120001, 184484, 250550, 303853, 352941, 360446, 375565, 501693, 724934, 889285, 940093, 995630, 1079662, 1473565, 1488957, 1517206, 1573045, 1581806, 1692302, 1864285, 2048973, 2693517, 3393934, 3509997, 4083526, 4194406

Offset: 1

Author

Paolo P. Lava, Jun 29 2015

Keywords

• nonn

Examples

			sigma(4) + phi(4) + d(4) = 7 + 2 + 3 = 12 and sigma(5) + phi(5) + d(5) = 6 + 4 + 2 = 12.
sigma(285) + phi(285) + d(285) = 480 + 144 + 8 = 632 and sigma(286) + phi(286) + d(286) = 504 + 120 + 8 = 632.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A000005, A000010, A000203, A233541.

Cf. A054004, A145749, A259496.

Programs

Maple

with(numtheory): P:=proc(q) local n; for n from 1 to q do
if sigma(n)+phi(n)+tau(n)=sigma(n+1)+phi(n+1)+tau(n+1)
then print(n); fi; od; end: P(10^9);

Mathematica

f[n_] := Module[{fct = FactorInteger[n]}, p = fct[[All, 1]]; e = fct[[All, 2]]; Times @@ (e + 1) + Times @@ ((p^(e + 1) - 1)/(p - 1)) + Times @@ ((p - 1)*p^(e - 1))]; f1 = 0; s = {}; Do[f2 = f[n]; If[f2 == f1, AppendTo[s, n - 1]];  f1 = f2, {n, 2, 10^5}]; s (* Amiram Eldar, Jul 12 2019 *)