A165552 a(1) = 1, and then a(n) is sum of k*a(k) where k

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

Offset: 1

Author

Rémy Sigrist, Sep 21 2009

Keywords

• easy
• nonn
• look

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

Examples

			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'').
		

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

Crossrefs

Cf. A002033, A008578, A176891.

Programs

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 *)

PARI

A165552(n) = if(1==n,n,sumdiv(n,d,if(dA165552(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

Formula

a(1) = 1, and for n > 1, a(n) = Sum_{d|n, d

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