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

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

1,4

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