A049988 Number of nondecreasing arithmetic progressions of positive integers with sum n.

1, 1, 2, 3, 4, 4, 7, 5, 7, 9, 9, 7, 14, 8, 11, 16, 13, 10, 20, 11, 17, 21, 16, 13, 27, 17, 18, 26, 22, 16, 35, 17, 23, 31, 23, 25, 41, 20, 25, 36, 33, 22, 46, 23, 31, 48, 30, 25, 52, 29, 38, 47, 36, 28, 57, 37, 41, 52, 37, 31, 71, 32, 39, 62, 44, 43, 69, 35, 45, 62, 57, 37, 79, 38

Offset

0,3

Comments

From Gus Wiseman, May 03 2019: (Start)

a(n) is the number of integer partitions of n with equal differences. The Heinz numbers of these partitions are given by A325328. For example, the a(1) = 1 through a(9) = 9 partitions are:

1 2 3 4 5 6 7 8 9

11 21 22 32 33 43 44 54

111 31 41 42 52 53 63

1111 11111 51 61 62 72

222 1111111 71 81

321 2222 333

111111 11111111 432

531

111111111

(End)

From Petros Hadjicostas, Sep 29 2019: (Start)

We show how Leroy Quet's g.f. Sum_{n >= 0} a(n)*x^n = 1/(1-x) + Sum_{k >= 2} x^k/(1-x^(k*(k-1)/2))/(1-x^k) in the Formula section below can be derived from Graeme McRae's g.f. for A049982 (see one of the links below).

Let b(n) = A049982(n) for n >= 1. Then Graeme McRae proved that Sum_{n >= 1} b(n)*x^n = Sum_{k >= 2} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1) = Sum_{k >= 2} x^t(k)/((1 - x^k) * (1 - x^t(k-1))), where t(k) = A000217(k) = k*(k+1)/2.

Since a(n) - b(n) = A000005(n) for n >= 1, to finish the proof, we only need to show that K(x) := 1 + Sum_{n >= 1} a(n)*x^n - Sum_{n >= 1} b(n)*x^n is the g.f. of A000005 (= number of divisors). But it is easy to show that K(x) = 1 + Sum_{k >= 1} x^k/(1 - x^k) = 1 + Sum_{n >= 1} A000005(n)*x^n (Lambert series for the number of divisors function). (End)

Links

Lars Blomberg, Table of n, a(n) for n = 0..10000 (Corrected by Gus Wiseman, May 03 2019)

Lars Blomberg, C# program for calculating b-file (needs to be updated for a(0) = 1 - Gus Wiseman, May 07 2019).

Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.

Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.

F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.

F. Javier de Vega, A Complete Solution of the Partitions of a Number into Arithmetic Progressions, arXiv:2004.09505 [math.NT], 2020.

F. Javier de Vega, On the parabolic partitions of a number, J. Alg., Num. Theor., and Appl. (2023) Vol. 61, No. 2, 135-169.

Graeme McRae, Counting arithmetic sequences whose sum is n.

Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]

Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.

Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.

Wikipedia, Arithmetic progression.

Wikipedia, Lambert series.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Formula

G.f.: 1/(1-x) + Sum_{k>=2} x^k/(1-x^(k*(k-1)/2))/(1-x^k). - Leroy Quet, Apr 08 2010. [Edited by Gus Wiseman, May 03 2019]

a(n) = A049982(n) + A000005(n) = A049980(n) + A000005(n) - 1 for n >= 1. - Petros Hadjicostas, Sep 28 2019

Mathematica

a[n_]:=If[n==0, 1, Block[{i, c=Floor[(n-1)/2]+DivisorSigma[0, n]}, Do[i=1; While[i*k<n, If[Mod[2*(n-i*k), k*(k-1)]==0, c++]; i++], {k, 3, (Sqrt[1+8*n]-1)/2}]; c]]; a/@Range[0, 73] (* Giovanni Resta, Feb 16 2013. Edited by Gus Wiseman, May 07 2019 *)
Table[Length[Select[IntegerPartitions[n], SameQ@@Differences[#]&]], {n, 0, 30}] (* Gus Wiseman, May 03 2019 *)

Prog

(PARI) seq(n)={Vec(1/(1-x) + sum(k=2, n, x^k/(1 - x^(k*(k-1)/2))/(1-x^k) + O(x*x^n)))} \\ Andrew Howroyd, Sep 28 2019

Crossrefs

Cf. A000005, A000217, A007862, A047966, A049982, A049983, A049986, A049987, A129654, A240026, A240027, A307824, A320466, A325325, A325328.

Sequence in context: A056880 A053273 A347700 * A079247 A325588 A244903

Adjacent sequences: A049985 A049986 A049987 * A049989 A049990 A049991

Keyword

nonn

Author

Clark Kimberling

Extensions

Edited by Max Alekseyev, May 03 2010

a(0) = 1 prepended by Gus Wiseman, May 03 2019

Status

approved