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

%I A165552 #33 May 15 2021 06:18:02
%S A165552 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,
%T A165552 162,1,262,1,2295,15,20,13,2142,1,22,17,2220,1,420,1,334,180,26,1,
%U A165552 22710,8,238,21,444,1,2562,17,4818,23,32,1,10782,1,34,278,75735,19,856,1,712
%N A165552 a(1) = 1, and then a(n) is sum of k*a(k) where k<n and k divides n.
%C A165552 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
%D A165552 A. K. Agarwal and R. Sachdeva, Combinatorics of n-Color Perfect Partitions, Ars Combinatoria 136 (2018), pp. 29--43.
%H A165552 Antti Karttunen, <a href="/A165552/b165552.txt">Table of n, a(n) for n = 1..16384</a>
%F A165552 a(1) = 1, and for n > 1, a(n) = Sum_{d|n, d<n} d*a(d).
%e A165552 a(6)=6 because v=5 has six n-color perfect partitions:
%e A165552 (1,1,1,1,1), (1,2,2), (1,2',2'), (1,1,3), (1,1,3'), and (1,1,3'').
%t A165552 a[1] = 1; a[n_] := a[n] = With[{k = Most[Divisors[n]]}, k . (a /@ k)]; Array[a, 100] (* _Jean-François Alcover_, Mar 31 2017 *)
%o A165552 (PARI) A165552(n) = if(1==n,n,sumdiv(n,d,if(d<n,d*A165552(d)))); \\ _Antti Karttunen_, Oct 30 2017
%o A165552 (Python)
%o A165552 from sympy import divisors
%o A165552 from sympy.core.cache import cacheit
%o A165552 @cacheit
%o A165552 def a(n): return 1 if n==1 else sum(d*a(d) for d in divisors(n)[:-1])
%o A165552 print([a(n) for n in range(1, 121)]) # _Indranil Ghosh_, Oct 30 2017, after PARI code
%Y A165552 Cf. A002033, A008578, A176891.
%K A165552 easy,nonn,look
%O A165552 1,4
%A A165552 _Rémy Sigrist_, Sep 21 2009