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A047968 a(n) = Sum_{d|n} p(d), where p(d) = A000041 = number of partitions of d. 40
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

T. D. Noe, Table of n, a(n) for n=1..1000

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-Fran├žois Alcover, Oct 03 2013 *)

CROSSREFS

Cf. A000041, A000837, A047966, A055893, A137587, A003606 (Euler transform).

Cf. A002033, A003238, A018783, A034729, A052409, A078392, A100953, A319162.

Sequence in context: A002246 A310016 A030014 * A322117 A181778 A245026

Adjacent sequences:  A047965 A047966 A047967 * A047969 A047970 A047971

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Dec 11 1999

STATUS

approved

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Last modified October 22 00:52 EDT 2019. Contains 328315 sequences. (Running on oeis4.)