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

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

0,3

Comments

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

References

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.

Links

Table of n, a(n) for n=0..46.

Formula

Equals left border of triangle A137587 starting (1, 2, 3, 5, 6, 11, ...). - Gary W. Adamson, Jan 27 2008

Comment from Joerg Arndt, Jun 08 2009: Sequence has 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.

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

Example

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)

Maple

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

Mathematica

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

Crossrefs

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.

Sequence in context: A332275 A318689 A365311 * A127524 A117086 A344551

Adjacent sequences: A083707 A083708 A083709 * A083711 A083712 A083713

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 16 2003

Extensions

More terms from Vladeta Jovovic, Jun 17 2003

Name shortened by Gus Wiseman, Apr 18 2021

Status

approved