A319686 Number of distinct values obtained when arithmetic derivative (A003415) is applied to the divisors of n.

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 3, 5, 2, 5, 2, 5, 3, 3, 2, 7, 3, 3, 4, 5, 2, 6, 2, 6, 3, 3, 3, 8, 2, 3, 3, 7, 2, 6, 2, 5, 5, 3, 2, 9, 3, 5, 3, 5, 2, 7, 3, 7, 3, 3, 2, 10, 2, 3, 5, 7, 3, 6, 2, 5, 3, 6, 2, 11, 2, 3, 5, 5, 3, 6, 2, 9, 5, 3, 2, 10, 3, 3, 3, 7, 2, 10, 3, 5, 3, 3, 3, 11, 2, 5, 5, 8, 2, 6, 2, 7, 6, 3, 2, 11, 2, 6, 3, 8, 2, 6, 3, 5, 5, 3, 3, 14

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = 1+A319685(n).

Mathematica

d[0] = d[1] = 0; d[n_] := d[n] = n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := CountDistinct[d /@ Divisors[n]]; Array[a, 100] (* Amiram Eldar, Apr 17 2024 *)

Prog

(PARI)
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A319686(n) = { my(m=Map(), s, k=0); fordiv(n, d, if(!mapisdefined(m, s=A003415(d)), mapput(m, s, s); k++)); (k); };
(PARI) a(n) = my(d = divisors(n)); for(i = 1, #d, d[i] = A003415(d[i])); #Set(d) \\ uses A003415 listed at Antti's programs. David A. Corneth, Oct 02 2018

Crossrefs

One more than A319685.

Cf. A003415.

Cf. also A184395, A319696.

Sequence in context: A335516 A335549 A181796 * A326082 A067554 A135981

Adjacent sequences: A319683 A319684 A319685 * A319687 A319688 A319689

Keyword

nonn,changed

Author

Antti Karttunen, Oct 02 2018

Status

approved