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

1,1

Links

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

Example

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.

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 *)

Crossrefs

Cf. A000005, A000010, A000203, A233541.

Cf. A054004, A145749, A259496.

Sequence in context: A202031 A074309 A113256 * A090088 A253233 A242997

Adjacent sequences: A259492 A259493 A259494 * A259496 A259497 A259498

Keyword

nonn

Author

Paolo P. Lava, Jun 29 2015

Status

approved