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A212127 Numbers n whose arithmetic derivative equals the sum of its proper divisors. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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,...

LINKS

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

EXAMPLE

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.

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 *)

CROSSREFS

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

Sequence in context: A293511 A325337 A324515 * A307895 A028835 A028834

Adjacent sequences:  A212124 A212125 A212126 * A212128 A212129 A212130

KEYWORD

nonn

AUTHOR

Omar E. Pol, Dec 18 2012

STATUS

approved

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Last modified July 20 22:20 EDT 2019. Contains 325189 sequences. (Running on oeis4.)