A212127 Numbers n whose arithmetic derivative equals the sum of its proper divisors.

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 220, 223, 227, 229, 233, 239, 241, 251, 257

Offset: 1

Author

Omar E. Pol, Dec 18 2012

Keywords

• nonn

Comments

Numbers n such that A003415(n) = A001065(n). Also numbers n such that A211991(n) = 0. By definition, all prime numbers are in the sequence. Nonprime numbers in the sequence are 1, 12, 18, 220,...

Examples

			The arithmetic derivative of 12 is equal to 16 (see A003415). On the other hand the sum of proper divisors of 12 is equal to 16 since 1+2+3+4+6 = 16, so 12 is in the sequence.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A000203, A001065, A003415, A211991.

Programs

Maple

with(numtheory);
A212127:=proc(i)
local n, p;
for n from 1 to i do
  if sigma(n)/n-1=add(op(2,p)/op(1,p),p=ifactors(n)[2]) then print(n);
fi; od; end:
A212127(1000);  # Paolo P. Lava, Jan 04 2012

Mathematica

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; Select[Range[300], dn[#] == DivisorSigma[1, #] - # &] (* T. D. Noe, Dec 27 2012 *)