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A319685 Number of distinct values obtained when arithmetic derivative (A003415) is applied to proper divisors of n. 8
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 (list; graph; refs; listen; history; text; internal format)
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.

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: A076526 A226378 A033273 * A034836 A316365 A292886

Adjacent sequences:  A319682 A319683 A319684 * A319686 A319687 A319688

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 02 2018

STATUS

approved

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Last modified February 19 22:36 EST 2020. Contains 332061 sequences. (Running on oeis4.)