A260624 a(n) = arithmetic derivative of the n-th composite number.

4, 5, 12, 6, 7, 16, 9, 8, 32, 21, 24, 10, 13, 44, 10, 15, 27, 32, 31, 80, 14, 19, 12, 60, 21, 16, 68, 41, 48, 39, 25, 112, 14, 45, 20, 56, 81, 16, 92, 22, 31, 92, 33, 51, 192, 18, 61, 72, 26, 59, 156, 39, 55, 80, 18, 71, 176, 108, 43, 124, 22, 45, 32, 140, 123

Offset

1,1

Links

Matthew Campbell, Table of n, a(n) for n = 1..10000; used code by Altug Alkan.

Formula

a(n) = A003415(A002808(n)).

Example

The second composite number is 6. 6 = 2 * 3. 6' = 2*3' + 3*2' = 3 * 1 + 2 * 1 = 3 + 2 = 5, so a(2) = 5.

Mathematica

lim = 120; f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]]; f /@ Rest@ Complement[Range@ lim, Prime@ Range@ PrimePi@ lim] (* Michael De Vlieger, Oct 07 2015, after Michael Somos at A003415 *)

Prog

(PARI) forcomposite(n=1, 100, if((a(n) = local(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))), print1(a(n)", "))); \\ Altug Alkan, Oct 06 2015

Crossrefs

Cf. A002808 (composite), A003415 (n').

Cf. A001787 ((2^n)'), A068719 ((2*n)'), A068720 ((n^2)'), A068721 ((n^3)').

Sequence in context: A305041 A316731 A253086 * A067371 A068719 A191161

Adjacent sequences: A260621 A260622 A260623 * A260625 A260626 A260627

Keyword

nonn,easy

Author

Matthew Campbell, Oct 06 2015

Extensions

More terms from Altug Alkan, Oct 06 2015

Status

approved