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A212127
Numbers n whose arithmetic derivative equals the sum of its proper divisors.
3
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
1,2
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
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
KEYWORD
nonn
AUTHOR
Omar E. Pol, Dec 18 2012
STATUS
approved