A319685 Number of distinct values obtained when arithmetic derivative (A003415) is applied to proper divisors of n.

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 7, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 8, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 9, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 10, 1, 2, 4, 4, 2, 5, 1, 8, 4, 2, 1, 9, 2, 2, 2, 6, 1, 9, 2, 4, 2, 2, 2, 10, 1, 4, 4, 7, 1, 5, 1, 6, 5, 2, 1, 10, 1, 5, 2, 7, 1, 5, 2, 4, 4, 2, 2, 13

Offset: 1

Author

Antti Karttunen, Oct 02 2018

Keywords

• nonn

Comments

Because for all d|n, dA003415(d) < A003415(n), it follows that the terms here are one less than in A319686.

Differs from A033273(n) = A000005(n) - A001221(n) at n = 1, 112, 156, 224, 280, 312, 336, 342, 380, 448, 468, 525, 560, 608, 624, 660, 672, 684, 756, 760, 780, 784, 840, 870, 896, 936, 984, 1008, ...

Examples

			The proper divisors of 112 are [1, 2, 4, 7, 8, 14, 16, 28, 56]. Applying arithmetic derivative A003415 to these, we obtain values [0, 1, 4, 1, 12, 9, 32, 32, 92], of which only 7 are distinct: 0, 1, 4, 9, 12, 32, and 92, thus a(112) = 7.
		

Links

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

Crossrefs

One less than A319686.

Cf. A003415.

Cf. also A304793, A305611, A316555, A316556, A319695 for similarly constructed sequences.

Programs

Mathematica

d[0] = d[1] = 0; d[n_] := d[n] = n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := CountDistinct[d /@ Most[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
A319685(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((dA003415(d)), mapput(m,s,s); k++)); (k); };

Formula

a(n) = A319686(n)-1.