A259495 - OEIS

%I #13 Jul 12 2019 14:54:59

%S 4,285,902,2013,8493,37406,61918,90094,120001,184484,250550,303853,

%T 352941,360446,375565,501693,724934,889285,940093,995630,1079662,

%U 1473565,1488957,1517206,1573045,1581806,1692302,1864285,2048973,2693517,3393934,3509997,4083526,4194406

%N 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.

%H Amiram Eldar, <a href="/A259495/b259495.txt">Table of n, a(n) for n = 1..100</a>

%e sigma(4) + phi(4) + d(4) = 7 + 2 + 3 = 12 and sigma(5) + phi(5) + d(5) = 6 + 4 + 2 = 12.

%e sigma(285) + phi(285) + d(285) = 480 + 144 + 8 = 632 and sigma(286) + phi(286) + d(286) = 504 + 120 + 8 = 632.

%p with(numtheory): P:=proc(q) local n; for n from 1 to q do

%p if sigma(n)+phi(n)+tau(n)=sigma(n+1)+phi(n+1)+tau(n+1)

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

%t 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 *)

%Y Cf. A000005, A000010, A000203, A233541.

%Y Cf. A054004, A145749, A259496.

%K nonn

%O 1,1

%A _Paolo P. Lava_, Jun 29 2015