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A047968 a(n) = Sum_{d|n} p(d), where p(d) = A000041 = number of partitions of d. 57
1, 3, 4, 8, 8, 17, 16, 30, 34, 52, 57, 99, 102, 153, 187, 261, 298, 432, 491, 684, 811, 1061, 1256, 1696, 1966, 2540, 3044, 3876, 4566, 5846, 6843, 8610, 10203, 12610, 14906, 18491, 21638, 26508, 31290, 38044, 44584, 54133, 63262, 76241 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Inverse Moebius transform of A000041.
Row sums of triangle A137587. - Gary W. Adamson, Jan 27 2008
Row sums of triangle A168021. - Omar E. Pol, Nov 20 2009
Row sums of triangle A168017. Row sums of triangle A168018. - Omar E. Pol, Nov 25 2009
Sum of the partition numbers of the divisors of n. - Omar E. Pol, Feb 25 2014
Conjecture: for n > 6, a(n) is strictly increasing. - Franklin T. Adams-Watters, Apr 19 2014
Number of constant multiset partitions of multisets spanning an initial interval of positive integers with multiplicities an integer partition of n. - Gus Wiseman, Sep 16 2018
LINKS
N. J. A. Sloane, Transforms
FORMULA
G.f.: Sum_{k>0} (-1+1/Product_{i>0} (1-z^(k*i))). - Vladeta Jovovic, Jun 22 2003
G.f.: sum(n>0,A000041(n)*x^n/(1-x^n)). - Mircea Merca, Feb 24 2014.
a(n) = A168111(n) + A000041(n). - Omar E. Pol, Feb 26 2014
a(n) = Sum_{y is a partition of n} A000005(GCD(y)). - Gus Wiseman, Sep 16 2018
EXAMPLE
For n = 10 the divisors of 10 are 1, 2, 5, 10, hence the partition numbers of the divisors of 10 are 1, 2, 7, 42, so a(10) = 1 + 2 + 7 + 42 = 52. - Omar E. Pol, Feb 26 2014
From Gus Wiseman, Sep 16 2018: (Start)
The a(6) = 17 constant multiset partitions:
(111111) (111)(111) (11)(11)(11) (1)(1)(1)(1)(1)(1)
(111222) (12)(12)(12)
(111122) (112)(112)
(112233) (123)(123)
(111112)
(111123)
(111223)
(111234)
(112234)
(112345)
(123456)
(End)
MAPLE
with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l) do c := c+numbpart(l[i]) od: RETURN(c): end: for j from 1 to 60 do printf(`%d, `, a(j)) od: # Zerinvary Lajos, Apr 14 2007
MATHEMATICA
a[n_] := Sum[ PartitionsP[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 44}] (* Jean-François Alcover, Oct 03 2013 *)
CROSSREFS
Cf. A000041, A000837, A047966, A055893, A137587, A003606 (Euler transform).
Sequence in context: A002246 A310016 A030014 * A358835 A358833 A322117
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 11 1999
STATUS
approved

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Last modified March 19 09:40 EDT 2024. Contains 370981 sequences. (Running on oeis4.)