A285574 -id:A285574 - cp's OEIS frontend

A238005 Number of partitions of n into distinct parts such that (greatest part) - (least part) = (number of parts).

0, 0, 0, 1, 0, 1, 1, 2, 0, 2, 2, 2, 2, 2, 1, 4, 3, 2, 3, 3, 2, 4, 4, 4, 3, 4, 2, 5, 5, 3, 5, 6, 3, 5, 3, 5, 6, 6, 4, 6, 6, 4, 6, 6, 3, 7, 7, 7, 6, 6, 5, 7, 7, 5, 6, 8, 6, 8, 8, 6, 8, 8, 4, 9, 6, 7, 9, 9, 7, 7, 9, 8, 9, 9, 5, 9, 7, 8, 10, 10

Offset: 1

Clark Kimberling, Feb 17 2014

Note that partitions into distinct parts are also called strict partitions.
a(n) is the number of strict partitions of n into nearly consecutive parts, that is, the number of ways to write n as a sum of terms i, i+1, i+2, ..., i+k (i>=1, k>=2) where one of the interior parts i+1, i+2, ..., i+k-1 is missing. Examples of nearly consecutive partitions (corresponding to the initial nonzero values of a(n)) are 13, 24, 124, 134, 35, 235, 46, ... . - Don Reble, Sep 07 2021
Let T(n) = n*(n+1)/2 = A000217(n) denote the n-th triangular number.
Theorem A. a(n) = b(n) - c(n), where b(n) is the inverse triangular number sequence A003056, that is, b(n) is the maximal i such that T_i <= n, and c(n) is the number of partitions of n into consecutive parts = number of odd divisors of n = A001227(n).
This theorem was conjectured by Omar E. Pol in February 2018, and proved independently by William J. Keith and Roland Bacher on Sep 05 2021. The elegant proof given in the link below is due to Don Reble.

			a(8) = 2 counts these partitions:  53, 431.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
N. J. A. Sloane, Proof of Theorem A in A238005

Cf. A000009, A000217, A001227, A003056, A238006, A238007.
a(n) is also the number of zeros in the n-th row of the triangles A196020, A211343, A231345, A236106, A237048 (simpler), A239662, A261699, A271344, A272026, A280850, A285574, A285891, A285914, A286013, A296508 (and possibly others). Omar E. Pol, Feb 17 2018
Row sums of A347579. - Omar E. Pol, Sep 07 2021

z = 70; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]];
Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A001227 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A003056 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A238005 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A238006 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A238007 *)
{0}~Join~Array[Floor[(Sqrt[1 + 8 #] - 1)/2] - DivisorSum[#, 1 &, OddQ] &, 102] (* Michael De Vlieger, Feb 18 2018 *)

a(n) = if (n, (sqrtint(8*n+1)-1)\2 - sumdiv(n, d, d%2), 0); \\ Michel Marcus, Mar 01 2018

G.f. = (x/(1-x)) * Sum_{k >= 1} x^(k*(k+1)/2) * (1 - x^(k-1)) / (1 - x^k). This follows from Theorem A and the g.f.s for A003056 and A001227. - William J. Keith, Sep 05 2021
a(n) = A238007(n) - A238006(n). - Omar E. Pol, Sep 11 2021
A001227(n) + a(n) + A238006(n) = A000009(n). - R. J. Mathar, Sep 23 2021

Edited by N. J. A. Sloane, Sep 11 2021, mostly to add Theorem A.

A352425 Irregular triangle read by rows in which row n lists the partitions of n into an odd number of consecutive parts.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 3, 2, 1, 7, 8, 9, 4, 3, 2, 10, 11, 12, 5, 4, 3, 13, 14, 15, 6, 5, 4, 5, 4, 3, 2, 1, 16, 17, 18, 7, 6, 5, 19, 20, 6, 5, 4, 3, 2, 21, 8, 7, 6, 22, 23, 24, 9, 8, 7, 25, 7, 6, 5, 4, 3, 26, 27, 10, 9, 8, 28, 7, 6, 5, 4, 3, 2, 1, 29, 30, 11, 10, 9, 8, 7, 6, 5, 4

Offset: 1

Omar E. Pol, Mar 15 2022

Conjecture: the total number of parts in all partitions of n into an odd number of consecutive parts equals the sum of odd divisors of n that are <= A003056(n). In other words: row n has A341309(n) terms.

The first partition with 2*m - 1 parts appears in the row A000384(m), m >= 1.

			Triangle begins:
   [1];
   [2];
   [3],
   [4];
   [5];
   [6], [3, 2, 1];
   [7];
   [8];
   [9], [4, 3, 2];
  [10];
  [11];
  [12], [5, 4, 3];
  [13];
  [14];
  [15], [6, 5, 4], [5, 4, 3, 2, 1];
  [16];
  [17];
  [18], [7, 6, 5];
  [19];
  [20], [6, 5, 4, 3, 2];
  [21], [8, 7, 6];
  [22];
  [23];
  [24], [9, 8, 7];
  [25], [7, 6, 5, 4, 3];
  [26];
  [27], [10, 9, 8];
  [28], [7, 6, 5, 4, 3, 2, 1];
  ...
In the diagram below the m-th staircase walk starts at row A000384(m).
The number of horizontal line segments in the n-th row equals A082647(n), the number of partitions of n into an odd number of consecutive parts, so we can find such partitions as follows: consider the vertical blocks of numbers that start exactly in the n-th row of the diagram, for example: for n = 15 consider the vertical blocks of numbers that start exactly in the 15th row. They are [15], [6, 5, 4]. [5, 4, 3, 2, 1], equaling the 15th row of the above triangle.
                                                           _
                                                         _|1|
                                                       _|2  |
                                                     _|3    |
                                                   _|4      |
                                                 _|5       _|
                                               _|6        |3|
                                             _|7          |2|
                                           _|8           _|1|
                                         _|9            |4  |
                                       _|10             |3  |
                                     _|11              _|2  |
                                   _|12               |5    |
                                 _|13                 |4    |
                               _|14                  _|3   _|
                             _|15                   |6    |5|
                           _|16                     |5    |4|
                         _|17                      _|4    |3|
                       _|18                       |7      |2|
                     _|19                         |6     _|1|
                   _|20                          _|5    |6  |
                 _|21                           |8      |5  |
               _|22                             |7      |4  |
             _|23                              _|6      |3  |
           _|24                               |9       _|2  |
         _|25                                 |8      |7    |
       _|26                                  _|7      |6    |
     _|27                                   |10       |5   _|
    |28                                     |9        |4  |7|
...
The diagram is infinite.
For more information about the diagram see A286000.

Subsequence of A299765.
Row sums give A352257.
Column 1 gives A000027.
Records give A000027.
Row n contains A082647(n) of the mentioned partitions.
Cf. A000384, A003056, A067742, A204217, A237048, A237591, A237593, A240542, A245092, A285574, A285901, A286000, A286001, A320051, A320137, A320142, A341309, A351824.

A347737 Zero together with the partial sums of A238005.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 5, 7, 9, 11, 13, 15, 16, 20, 23, 25, 28, 31, 33, 37, 41, 45, 48, 52, 54, 59, 64, 67, 72, 78, 81, 86, 89, 94, 100, 106, 110, 116, 122, 126, 132, 138, 141, 148, 155, 162, 168, 174, 179, 186, 193, 198, 204, 212, 218, 226, 234, 240, 248, 256, 260

Offset: 0

Omar E. Pol, Sep 11 2021

nonn

a(n) is also the total number of ones in the first n rows of A347579, n >= 1.

a(n) is also the total number of zeros in the first n rows of the triangles A196020, A211343, A231345, A236106, A237048 (simpler), A239662, A261699, A271344, A272026, A280850, A285574, A285891, A285914, A286013, A296508 (and possibly others), n >= 1.

Cf. A001227, A003056, A006463, A060831, A237593, A238005, A347579.

Accumulate@Table[Length@Select[Select[IntegerPartitions@n,DuplicateFreeQ],Differences@MinMax@#=={Length@#}&],{n,60}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)

from math import isqrt
def A347737(n): return (r:=isqrt((n+1<<3)+1)-1>>1)*(6*n+4-r*(r+3))//6-((t:=isqrt(m:=n>>1))+(s:=isqrt(n)))*(t-s)-(sum(n//k for k in range(1,s+1))-sum(m//k for k in range(1,t+1))<<1) # Chai Wah Wu, Jun 07 2025

a(n) = A006463(n+1) - A060831(n).

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A238005 Number of partitions of n into distinct parts such that (greatest part) - (least part) = (number of parts).

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A352425 Irregular triangle read by rows in which row n lists the partitions of n into an odd number of consecutive parts.

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A347737 Zero together with the partial sums of A238005.

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