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

1,4

Comments

Because for all d|n, d<n, A003415(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, ...

Links

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

Formula

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

Example

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.

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 *)

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
A319685(n) = { my(m=Map(), s, k=0); fordiv(n, d, if((d<n)&&!mapisdefined(m, s=A003415(d)), mapput(m, s, s); k++)); (k); };

Crossrefs

One less than A319686.

Cf. A003415.

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

Sequence in context: A226378 A347460 A033273 * A343652 A034836 A316365

Adjacent sequences: A319682 A319683 A319684 * A319686 A319687 A319688

Keyword

nonn,changed

Author

Antti Karttunen, Oct 02 2018

Status

approved