A233541 -id:A233541 - cp's OEIS frontend

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

Paolo P. Lava, Jun 29 2015

nonn

			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.

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

Cf. A000005, A000010, A000203, A233541.

Cf. A054004, A145749, A259496.

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);

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 *)

A227008 Numbers k such that Sum_{j=1..k} (sigma(j) + phi(j) + tau(j)) == 0 (mod k).

Original entry on oeis.org

1, 399, 872, 1214, 2090, 5200, 5588, 21208, 29152, 638049, 1627676, 151732410, 274845607, 3224070252, 54892040166, 69020111756, 175288968221

Offset: 1

Paolo P. Lava, Jun 27 2013

a(17) > 10^11. - Donovan Johnson, Jul 07 2013

a(18) > 5*10^11. - Giovanni Resta, Jul 11 2013

			Sum_{j=1..399} sigma(j) = 130973;
Sum_{j=1..399} phi(j) = 48518;
Sum_{j=1..399} tau(j) = 2453;
(130973 + 48518 + 2453) / 399 = 456.

Cf. A000005, A000010, A000203.
Cf. A006218, A002088, A024916.
Cf. A233541.

with(numtheory); ListA227008:=proc(q,h) local a, n;  a:=0;
for n from 1 to q do  a:=a+sigma(n)+phi(n)+tau(n); if (a mod n)=0 then print(n); fi; od; end: ListA227008(10^9);

s=0; for(n=1, 274845607, s=s+sigma(n)+eulerphi(n)+numdiv(n); if(s%n==0, print(n " " s))) /* Donovan Johnson, Jul 06 2013 */

a(11)-a(14) from Donovan Johnson, Jul 06 2013
a(15)-a(16) from Donovan Johnson, Jul 07 2013
a(17) from Giovanni Resta, Jul 11 2013

cp's OEIS Frontend

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.

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