A137587 -id:A137587 - cp's OEIS frontend

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

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

Offset: 1

N. J. A. Sloane, Dec 11 1999

nonn

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

			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)

T. D. Noe, Table of n, a(n) for n=1..1000
N. J. A. Sloane, Transforms

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

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

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

a[n_] := Sum[ PartitionsP[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 44}] (* Jean-François Alcover, Oct 03 2013 *)

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

A083710 Number of integer partitions of n with a part dividing all the other parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 11, 12, 20, 25, 37, 43, 70, 78, 114, 143, 196, 232, 330, 386, 530, 641, 836, 1003, 1340, 1581, 2037, 2461, 3127, 3719, 4746, 5605, 7038, 8394, 10376, 12327, 15272, 17978, 22024, 26095, 31730, 37339, 45333, 53175, 64100, 75340, 90138

Offset: 0

N. J. A. Sloane, Jun 16 2003

Since the summand (part) which divides all the other summands is necessarily the smallest, an equivalent definition is: "Number of partitions of n such that smallest part divides every part." - Joerg Arndt, Jun 08 2009
The first few partitions that fail the criterion are 5=3+2, 7=5+2=4+3=3+2+2. So a(5) = A000041(5) - 1 = 6, a(7) = A000041(7) - 3 = 12. - Vladeta Jovovic, Jun 17 2003
Starting with offset 1 = inverse Mobius transform (A051731) of the partition numbers, A000041. - Gary W. Adamson, Jun 08 2009

			From _Gus Wiseman_, Apr 18 2021: (Start)
The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (41)     (33)      (61)
             (111)  (31)    (221)    (42)      (331)
                    (211)   (311)    (51)      (421)
                    (1111)  (2111)   (222)     (511)
                            (11111)  (321)     (2221)
                                     (411)     (3211)
                                     (2211)    (4111)
                                     (3111)    (22111)
                                     (21111)   (31111)
                                     (111111)  (211111)
                                               (1111111)
(End)

L. M. Chawla, M. O. Levan and J. E. Maxfield, On a restricted partition function and its tables, J. Natur. Sci. and Math., 12 (1972), 95-101.

Cf. A018783, A137587.
Cf. A000041, A051731. - Gary W. Adamson, Jun 08 2009
The case with no 1's is A083711.
The strict case is A097986.
The version for "divisible by" instead of "dividing" is A130689.
The case where there is also a part divisible by all the others is A130714.
The complement of these partitions is counted by A338470.
The Heinz numbers of these partitions are dense, complement of A342193.
The case where there is also no part divisible by all the others is A343345.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A001792, A098965, A264401, A339563, A343340, A343341, A343378.

with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l)-0 do c := c+numbpart(l[i]-1) od: RETURN(c): end: for j from 0 to 60 do printf(`%d, `, a(j)) od: # Zerinvary Lajos, Apr 14 2007

Table[Length[Select[IntegerPartitions[n],And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}] (* Gus Wiseman, Apr 18 2021 *)

Equals left border of triangle A137587 starting (1, 2, 3, 5, 6, 11, ...). - Gary W. Adamson, Jan 27 2008
G.f.: 1 + Sum_{n>=1} x^n/eta(x^n). The g.f. for partitions into parts that are a multiple of n is x^n/eta(x^n), now sum over n. - Joerg Arndt, Jun 08 2009
Gary W. Adamson's comment is equivalent to the formula a(n) = Sum_{d|n} p(d-1) where p(i) = number of partitions of i (A000041(i)). Hence A083710 has g.f. Sum_{d>=1} p(d-1)*x^d/(1-x^d), - N. J. A. Sloane, Jun 08 2009

More terms from Vladeta Jovovic, Jun 17 2003

Name shortened by Gus Wiseman, Apr 18 2021

cp's OEIS Frontend

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

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