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

Author

Antti Karttunen, Oct 02 2018

Keywords

• nonn

Links

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

Crossrefs

One more than A319685.

Cf. A003415.

Cf. also A184395, A319696.

Programs

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 *)

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

Formula

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