A190403 Number n for which sigma(n)=sigma(n'), where sigma is the sum of divisors and n' the arithmetic derivative of n.

4, 27, 60, 84, 132, 140, 204, 220, 228, 260, 270, 340, 372, 378, 444, 492, 564, 572, 580, 620, 644, 702, 708, 740, 804, 812, 820, 836, 860, 884, 918, 945, 1026, 1068, 1180, 1242, 1276, 1284, 1292, 1308, 1316, 1364, 1420, 1460, 1484, 1485, 1508, 1564, 1566

Offset: 1

Author

Paolo P. Lava, May 10 2011

Keywords

• nonn
• easy

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A003415, A189057, A190402.

Programs

Maple

with(numtheory);
P:=proc(i)
local f,n,p,pfs;
for n from 1 to i do
    pfs:=ifactors(n)[2];
    f:=n*add(op(2,p)/op(1,p),p=pfs);
    if sigma(n)=sigma(f) then print(n); fi;
od;
end:
P(1000);

Mathematica

d[0] = d[1] = 0; d[n_] := n*Total[f = FactorInteger[n]; f[[All, 2]]/f[[All, 1]] ]; Reap[For[n = 1, n < 2000, n++, If[DivisorSigma[1, n] == DivisorSigma[1, d[n]], Sow[n]]]][[2, 1]] (* Jean-François Alcover, Apr 22 2015 *)

Python

from sympy import factorint, totient
A190402 = [n for n in range(2,10**3) if totient(int(sum([n*e/p for p,e in factorint(n).items()]))) == totient(n)] # Chai Wah Wu, Aug 21 2014