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
Keywords
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
Programs
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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);
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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 *)