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A165552 a(1) = 1, and then a(n) is sum of k*a(k) where k<n and k divides n. 5
1, 1, 1, 3, 1, 6, 1, 15, 4, 8, 1, 54, 1, 10, 9, 135, 1, 78, 1, 100, 11, 14, 1, 822, 6, 16, 40, 162, 1, 262, 1, 2295, 15, 20, 13, 2142, 1, 22, 17, 2220, 1, 420, 1, 334, 180, 26, 1, 22710, 8, 238, 21, 444, 1, 2562, 17, 4818, 23, 32, 1, 10782, 1, 34, 278, 75735, 19, 856, 1, 712 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

a(v+1) is also the number of n-color perfect partitions of v. An n-color perfect partition of v is a partition into j types of each part j which contains one and only one n-color partition of each smaller number. - Augustine O. Munagi, May 09 2020

REFERENCES

A. K. Agarwal and R. Sachdeva, Combinatorics of n-Color Perfect Partitions, Ars Combinatoria 136 (2018), pp. 29--43.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

FORMULA

a(1) = 1, and for n > 1, a(n) = Sum_{d|n, d<n} d*a(d).

EXAMPLE

a(6)=6 because v=5 has six n-color perfect partitions:

(1,1,1,1,1), (1,2,2), (1,2',2'), (1,1,3), (1,1,3'), and (1,1,3'').

MATHEMATICA

a[1] = 1; a[n_] := a[n] = With[{k = Most[Divisors[n]]}, k . (a /@ k)]; Array[a, 100] (* Jean-François Alcover, Mar 31 2017 *)

PROG

(PARI) A165552(n) = if(1==n, n, sumdiv(n, d, if(d<n, d*A165552(d)))); \\ Antti Karttunen, Oct 30 2017

(Python)

from sympy import divisors

from sympy.core.cache import cacheit

@cacheit

def a(n): return 1 if n==1 else sum(d*a(d) for d in divisors(n)[:-1])

print([a(n) for n in range(1, 121)]) # Indranil Ghosh, Oct 30 2017, after PARI code

CROSSREFS

Cf. A002033, A008578, A176891.

Sequence in context: A007650 A236540 A339384 * A067231 A218971 A145791

Adjacent sequences:  A165549 A165550 A165551 * A165553 A165554 A165555

KEYWORD

easy,nonn,look

AUTHOR

Rémy Sigrist, Sep 21 2009

STATUS

approved

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Last modified October 21 02:29 EDT 2021. Contains 348141 sequences. (Running on oeis4.)