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A132065 a(n) = Sum_{k=1 to d(n)} C(d(n)-1, k-1) d_k, where d(n) is the number of divisors of n and d_k is the k-th divisor of n.

%I A132065 #32 Aug 16 2025 07:05:43
%S A132065 1,3,4,9,6,22,8,27,16,32,12,123,14,42,40,81,18,164,20,171,52,62,24,
%T A132065 704,36,72,64,219,30,808,32,243,76,92,72,1765,38,102,88,944,42,1016,
%U A132065 44,315,276,122,48,4075,64,336,112,363,54,1224,104,1170,124,152,60,17815,62
%N A132065 a(n) = Sum_{k=1 to d(n)} C(d(n)-1, k-1) d_k, where d(n) is the number of divisors of n and d_k is the k-th divisor of n.
%H A132065 Rémy Sigrist, <a href="/A132065/b132065.txt">Table of n, a(n) for n = 1..10000</a>
%F A132065 a(p) = p+1, for p prime. - _Michel Marcus_, Sep 13 2014
%e A132065 Since the divisors of 12 are 1,2,3,4,6,12 and since row (d(12)-1) of Pascal's triangle is 1,5,10,10,5,1, a(12) = 1*1 + 5*2 + 10*3 + 10*4 + 5*6 + 1*12 = 123.
%e A132065 From _Peter Luschny_, May 18 2016: (Start)
%e A132065 Also the lower vertex of the accumulation triangle of the divisors of n.
%e A132065 For instance a(39) = 88 because the lower vertex of ATD(39) = 88. ATD(39) is:
%e A132065   [ 39  13  3  1]
%e A132065   [ 52  16  4]
%e A132065   [ 68  20]
%e A132065   [ 88]
%e A132065 (End)
%t A132065 f[n_] := Block[{d, l, k},d = Divisors[n];l = Length[d];Sum[ Binomial[l - 1, k - 1]*d[[k]], {k, l}]];Array[f, 100] (* _Ray Chandler_, Oct 31 2007 *)
%t A132065 Table[Sum[Binomial[Length[Divisors[n]] - 1, k - 1]*Divisors[n][[k]], {k, 1, Length[Divisors[n]]}], {n, 1, 70}] (* _Stefan Steinerberger_, Oct 31 2007 *)
%o A132065 (PARI) a(n) = {d = divisors(n); sum(i=1, #d, d[i]*binomial(#d-1, i-1));} \\ _Michel Marcus_, Sep 13 2014
%o A132065 (Sage)
%o A132065 def A132065(n):
%o A132065     D = divisors(n)[::-1]
%o A132065     T = matrix(ZZ, len(D))
%o A132065     for (m, d) in enumerate(D):
%o A132065         T[0, m] = d
%o A132065         for k in range(m-1, -1, -1) :
%o A132065             T[m-k, k] = T[m-k-1, k+1] + T[m-k-1, k]
%o A132065     return T[len(D)-1,0]
%o A132065 print([A132065(n) for n in range(1,62)]) # _Peter Luschny_, May 18 2016
%Y A132065 Cf. A007318 (Pascal's triangle), A027750 (divisors of n).
%K A132065 nonn
%O A132065 1,2
%A A132065 _Leroy Quet_, Oct 30 2007
%E A132065 Extended by _Ray Chandler_ and _Stefan Steinerberger_, Nov 01 2007