A049988 -id:A049988 - cp's OEIS frontend

A049992 a(n) is the number of arithmetic progressions of 3 or more positive integers, nondecreasing with sum n.

0, 0, 1, 1, 1, 3, 1, 2, 4, 3, 1, 7, 1, 3, 8, 4, 1, 10, 1, 6, 10, 4, 1, 14, 4, 4, 12, 7, 1, 19, 1, 6, 14, 5, 7, 22, 1, 5, 16, 12, 1, 24, 1, 8, 25, 6, 1, 27, 4, 12, 21, 9, 1, 29, 9, 12, 23, 7, 1, 40, 1, 7, 30, 11, 10, 35, 1, 10, 27, 21, 1, 42, 1, 8, 39, 11, 7, 40, 1, 22, 35, 9, 1, 49, 12, 9, 34

Offset: 1

Clark Kimberling

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12580
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.
Jon Maiga, Computer-generated formulas for A049992, Sequence Machine.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
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.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.

Cf. A007862, A014405, A023645, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A049991, A049992, A049994, A175676, A240026, A240027, A307824, A320466.

A049992(n) = (A014405(n)+A023645(n)); \\ (Uses the programs given in A014405 and A023645) - Antti Karttunen, Feb 20 2023

G.f.: Sum_{k>=3} x^k/(1-x^(k*(k-1)/2))/(1-x^k). [Leroy Quet from A049988] - Petros Hadjicostas, Sep 29 2019

a(n) = A014405(n) + A023645(n) = A049994(n) + A175676(n). [Two of the formulas listed by Sequence Machine, both obviously true] - Antti Karttunen, Feb 20 2023

More terms from Petros Hadjicostas, Sep 29 2019

A068322 Number of arithmetic progressions of positive odd integers, strictly increasing with sum n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 3, 3, 5, 1, 4, 1, 5, 4, 5, 1, 7, 2, 6, 5, 8, 1, 7, 1, 9, 6, 8, 2, 11, 1, 9, 7, 12, 1, 10, 1, 12, 10, 11, 1, 15, 2, 12, 9, 15, 1, 13, 3, 16, 10, 14, 1, 18, 1, 15, 12, 20, 4, 17, 1, 19, 12, 17, 1, 22, 1, 18, 16, 22, 2, 20, 1, 24, 15, 20, 1, 25, 5, 21, 15, 26

Offset: 1

Naohiro Nomoto, Feb 27 2002

			From _Petros Hadjicostas_, Sep 29 2019: (Start)
a(12) = 3 because we have the following arithmetic progressions of odd numbers, strictly increasing with sum n=12: 1+11, 3+9, and 5+7.
a(13) = 1 because we have only the following arithmetic progressions of odd numbers, strictly increasing with sum n=13: 13.
a(14) = 3 because we have the following arithmetic progressions of odd numbers, strictly increasing with sum n=14: 1+13, 3+11, and 5+9.
a(15) = 3 because we have the following arithmetic progressions of odd numbers, strictly increasing with sum n=15: 15, 3+5+7, and 1+5+9.
(End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000
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.
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.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A068323, A068324, A070211, A127938, A175327, A325328, A325407, A325545, A325546, A325547, A325548.

From Petros Hadjicostas, Oct 01 2019: (Start)
a(n) = A068324(n) - A001227(n) + (1/2) * (1 - (-1)^n).
G.f.: x/(1 - x^2) + Sum_{m >= 2} x^(m^2)/((1 - x^(2*m)) * (1 - x^(m*(m-1)))).
(End)

A332668 Number of strict integer partitions of n without three consecutive parts in arithmetic progression.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 11, 15, 20, 19, 26, 31, 34, 41, 50, 53, 67, 78, 84, 99, 120, 130, 154, 177, 193, 226, 262, 291, 332, 375, 419, 479, 543, 608, 676, 765, 859, 961, 1075, 1202, 1336, 1495, 1672, 1854, 2050, 2301, 2536, 2814, 3142, 3448, 3809

Offset: 0

Gus Wiseman, Mar 28 2020

nonn

Also the number of strict integer partitions of n whose first differences are an anti-run, meaning there are no adjacent equal differences.

			The a(1) = 1 through a(10) = 9 partitions (A = 10):
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)
            (21)  (31)  (32)  (42)  (43)   (53)   (54)   (64)
                        (41)  (51)  (52)   (62)   (63)   (73)
                                    (61)   (71)   (72)   (82)
                                    (421)  (431)  (81)   (91)
                                           (521)  (621)  (532)
                                                         (541)
                                                         (631)
                                                         (721)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..450
Wikipedia, Arithmetic progression

Anti-run compositions are counted by A003242.
Normal anti-runs of length n + 1 are counted by A005649.
Strict partitions with equal differences are A049980.
Partitions with equal differences are A049988.
The non-strict version is A238424.
The version for permutations is A295370.
Anti-run compositions are ranked by A333489.
Cf. A006560, A007862, A238423, A307824, A325328, A325852, A325874, A333195.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MatchQ[Differences[#],{_,x_,x_,_}]&]],{n,0,30}]

A049994 a(n) is the number of arithmetic progressions of 4 or more positive integers, nondecreasing with sum n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 1, 3, 1, 3, 1, 3, 3, 4, 1, 4, 1, 6, 3, 4, 1, 6, 4, 4, 3, 7, 1, 9, 1, 6, 3, 5, 7, 10, 1, 5, 3, 12, 1, 10, 1, 8, 10, 6, 1, 11, 4, 12, 4, 9, 1, 11, 9, 12, 4, 7, 1, 20, 1, 7, 9, 11, 10, 13, 1, 10, 4, 21, 1, 18, 1, 8, 14, 11, 7, 14, 1, 22, 8, 9, 1, 21, 12, 9, 5, 15, 1, 29, 8

Offset: 1

Clark Kimberling

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12580
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.
Jon Maiga, Computer-generated formulas for A049994, Sequence Machine.
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.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.

Cf. A014405, A014406, A175676, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A049991, A049992, A049993, A127938, A321014.

A049994(n) = (A049992(n)-if(n%3, 0, n/3)); \\ Antti Karttunen, Feb 20 2023

G.f.: Sum_{k >= 4} x^k/(1-x^(k*(k-1)/2))/(1-x^k). [Leroy Quet from A049988] - Petros Hadjicostas, Sep 29 2019

a(n) = A049992(n) - A175676(n) = A049986(n) + A321014(n). [Two of the formulas listed by Sequence Machine, both obviously true] - Antti Karttunen, Feb 20 2023

More terms from Petros Hadjicostas, Sep 29 2019

A068324 Number of nondecreasing arithmetic progressions of positive odd integers with sum n.

Original entry on oeis.org

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

Offset: 1

Naohiro Nomoto, Feb 27 2002

			From _Petros Hadjicostas_, Sep 29 2019: (Start)
a(6) = 3 because we have the following nondecreasing arithmetic progressions of positive odd integers with sum n=6: 1+5, 3+3, and 1+1+1+1+1+1.
a(7) = 2 because we have the following nondecreasing arithmetic progressions of positive odd integers with sum n=7: 7 and 1+1+1+1+1+1+1.
a(8) = 3 because we have the following nondecreasing arithmetic progressions of positive odd integers with sum n=8: 1+7, 3+5, and 1+1+1+1+1+1+1+1.
(End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000
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.
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.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A068322, A068323, A127938, A175327, A325328, A325407, A325545, A325546, A325547, A325548.

From Petros Hadjicostas, Oct 01 2019: (Start)
a(n) = A068322(n) + A001227(n) - (1/2) * (1 - (-1)^n).
G.f.: x/(1 - x^2) + Sum_{m >= 2} x^m/((1 - x^(2*m)) * (1 - x^(m*(m-1)))).
(End)

Extended and edited by John W. Layman, Mar 15 2002

A325459 Sum of numbers of nontrivial divisors (greater than 1 and less than k) of k for k = 1..n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 3, 5, 6, 8, 8, 12, 12, 14, 16, 19, 19, 23, 23, 27, 29, 31, 31, 37, 38, 40, 42, 46, 46, 52, 52, 56, 58, 60, 62, 69, 69, 71, 73, 79, 79, 85, 85, 89, 93, 95, 95, 103, 104, 108, 110, 114, 114, 120, 122, 128, 130, 132, 132, 142

Offset: 0

Gus Wiseman, May 04 2019

nonn

Also the number of integer partitions of n that are not hooks but whose augmented differences are hooks (original name). The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and otherwise aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

This sequence counts integer partitions with any number of ones and one part > 1 which appears at least twice. The Heinz numbers of these partitions are given by A325359.

			The a(4) = 1 through a(10) = 8 partitions:
  (22)  (221)  (33)    (331)    (44)      (333)      (55)
               (222)   (2221)   (2222)    (441)      (3331)
               (2211)  (22111)  (3311)    (22221)    (4411)
                                (22211)   (33111)    (22222)
                                (221111)  (222111)   (222211)
                                          (2211111)  (331111)
                                                     (2221111)
                                                     (22111111)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A049988, A093641, A325349, A325351, A325355, A325356, A325357, A325358.

Cf. A070824, A006218.

a:= proc(n) option remember; `if`(n<2, 0,
      numtheory[tau](n)-2+a(n-1))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 11 2019

Table[Length[Select[IntegerPartitions[n],MatchQ[#,{x_,y__,1...}/;x>1&&SameQ[x,y]]&]],{n,0,30}]
(* Second program: *)
a[n_] := a[n] = If[n<2, 0, DivisorSigma[0, n] - 2 + a[n-1]];
a /@ Range[0, 100] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

from math import isqrt
def A325459(n): return 0 if n == 0 else (lambda m: 2*(sum(n//k for k in range(1, m+1))-n)+(1-m)*(1+m))(isqrt(n)) # Chai Wah Wu, Oct 07 2021

From M. F. Hasler, Oct 11 2019: (Start)
a(n) = A006218(n) - 2*n + 1, in terms of partial sums of number of divisors.
a(n) = Sum_{k=1..n} A070824(k): partial sums of A070824 = number of nontrivial divisors. (End)

Name changed at the suggestion of Patrick James Smalley-Wall and Luc Rousseau by Gus Wiseman, Oct 11 2019

A325588 Number of necklace compositions of n with equal circular differences up to sign.

Original entry on oeis.org

1, 2, 3, 4, 4, 7, 5, 9, 8, 10, 8, 17, 9, 14, 15, 22, 12, 23, 14, 31, 23, 25, 19, 48, 25, 35, 36, 56, 33, 59, 43, 86, 64, 74, 76, 136, 95, 127, 138, 219, 178, 245, 249, 372, 370, 445, 506, 747, 730, 907, 1069, 1431, 1544, 1927, 2268, 2981, 3332, 4074, 4896, 6320

Offset: 1

Gus Wiseman, May 11 2019

nonn

A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.

The circular differences of a sequence c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2).

			The a(1) = 1 through a(8) = 9 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (111)  (22)    (23)     (24)      (25)       (26)
                    (1111)  (11111)  (33)      (34)       (35)
                                     (222)     (1111111)  (44)
                                     (1212)               (1232)
                                     (111111)             (1313)
                                                          (2222)
                                                          (11111111)

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A000740, A008965, A049988, A175342, A325549, A325556, A325558, A325590.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&SameQ@@Abs[Differences[Append[#,First[#]]]]&]],{n,15}]

step(R,n,s)={matrix(n, n, i, j, if(i>j, if(j>s, R[i-j, j-s]) + if(j+s<=n, R[i-j, j+s])) )}
w(n,s)={sum(k=1, n, my(R=matrix(n,n,i,j,i==j&&abs(i-k)==s), t=0, m=1); while(R, R=step(R,n,s); m++; t+=sumdiv(n, d, R[d,k]*d*eulerphi(n/d))/m ); t/n)}
a(n) = {numdiv(max(1,n)) + sum(s=1, n-1, w(n,s))} \\ Andrew Howroyd, Aug 24 2019

Terms a(26) and beyond from Andrew Howroyd, Aug 24 2019

A333195 Numbers with three consecutive prime indices in arithmetic progression.

Original entry on oeis.org

8, 16, 24, 27, 30, 32, 40, 48, 54, 56, 60, 64, 72, 80, 81, 88, 96, 104, 105, 108, 110, 112, 120, 125, 128, 135, 136, 144, 150, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 210, 216, 220, 224, 232, 238, 240, 243, 248, 250, 256, 264, 270, 272, 273, 280, 288

Offset: 1

Gus Wiseman, Mar 29 2020

nonn

Also numbers whose first differences of prime indices do not form an anti-run, meaning there are adjacent equal differences.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
    8: {1,1,1}          105: {2,3,4}
   16: {1,1,1,1}        108: {1,1,2,2,2}
   24: {1,1,1,2}        110: {1,3,5}
   27: {2,2,2}          112: {1,1,1,1,4}
   30: {1,2,3}          120: {1,1,1,2,3}
   32: {1,1,1,1,1}      125: {3,3,3}
   40: {1,1,1,3}        128: {1,1,1,1,1,1,1}
   48: {1,1,1,1,2}      135: {2,2,2,3}
   54: {1,2,2,2}        136: {1,1,1,7}
   56: {1,1,1,4}        144: {1,1,1,1,2,2}
   60: {1,1,2,3}        150: {1,2,3,3}
   64: {1,1,1,1,1,1}    152: {1,1,1,8}
   72: {1,1,1,2,2}      160: {1,1,1,1,1,3}
   80: {1,1,1,1,3}      162: {1,2,2,2,2}
   81: {2,2,2,2}        168: {1,1,1,2,4}
   88: {1,1,1,5}        176: {1,1,1,1,5}
   96: {1,1,1,1,1,2}    184: {1,1,1,9}
  104: {1,1,1,6}        189: {2,2,2,4}

Wikipedia, Arithmetic progression

Anti-run compositions are counted by A003242.
Normal anti-runs of length n + 1 are counted by A005649.
Strict partitions with equal differences are A049980.
Partitions with equal differences are A049988.
These are the Heinz numbers of the partitions *not* counted by A238424.
Permutations avoiding triples in arithmetic progression are A295370.
Strict partitions avoiding triples in arithmetic progression are A332668.
Anti-run compositions are ranked by A333489.
Cf. A006560, A007862, A238423, A307824, A325328, A325849, A325852.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],MatchQ[Differences[primeMS[#]],{_,x_,x_,_}]&]

A049993 a(n) is the number of arithmetic progressions of 3 or more positive integers, nondecreasing with sum <= n.

Original entry on oeis.org

0, 0, 1, 2, 3, 6, 7, 9, 13, 16, 17, 24, 25, 28, 36, 40, 41, 51, 52, 58, 68, 72, 73, 87, 91, 95, 107, 114, 115, 134, 135, 141, 155, 160, 167, 189, 190, 195, 211, 223, 224, 248, 249, 257, 282, 288, 289, 316, 320, 332, 353, 362, 363, 392, 401, 413, 436, 443, 444, 484, 485, 492, 522, 533, 543, 578

Offset: 1

Clark Kimberling

nonn

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.
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.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.

Cf. A007862, A014405, A014406, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A049991, A049992, A240026, A240027, A307824, A320466, A325325.

From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = Sum_{k = 1..n} A049992(k).
G.f.: (g.f. of A049992)/(1-x). (End)

More terms from Petros Hadjicostas, Sep 29 2019

A325333 Number of integer partitions of n whose multiplicities all appear the same number of times.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 19, 23, 32, 39, 50, 63, 82, 96, 125, 152, 186, 226, 271, 326, 392, 473, 552, 663, 771, 918, 1065, 1261, 1448, 1710, 1953, 2283, 2608, 3062, 3455, 4013, 4552, 5271, 5974, 6884, 7774, 8937, 10065, 11570, 12953, 14838, 16710, 18979

Offset: 0

Gus Wiseman, May 01 2019

nonn

			The a(0) = 1 through a(7) = 14 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)
           (11)  (21)   (22)    (32)     (33)      (43)
                 (111)  (31)    (41)     (42)      (52)
                        (211)   (221)    (51)      (61)
                        (1111)  (311)    (222)     (322)
                                (2111)   (321)     (331)
                                (11111)  (411)     (421)
                                         (2211)    (511)
                                         (3111)    (2221)
                                         (21111)   (4111)
                                         (111111)  (22111)
                                                   (31111)
                                                   (211111)
                                                   (1111111)
For example, the partition (4,3,3,3,2,2,2,1) has multiplicities (1,3,3,1), and since both multiplicities 1 and 3 appear twice, (4,3,3,3,2,2,2,1) is counted under a(20).

Cf. A000041, A007862, A047966, A049988, A098859, A323022, A325329, A325369.

Table[Length[Select[IntegerPartitions[n],SameQ@@Length/@Split[Sort[Length/@Split[#]]]&]],{n,0,30}]

cp's OEIS Frontend

A049992 a(n) is the number of arithmetic progressions of 3 or more positive integers, nondecreasing with sum n.

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A068322 Number of arithmetic progressions of positive odd integers, strictly increasing with sum n.

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A332668 Number of strict integer partitions of n without three consecutive parts in arithmetic progression.

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A049994 a(n) is the number of arithmetic progressions of 4 or more positive integers, nondecreasing with sum n.

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A068324 Number of nondecreasing arithmetic progressions of positive odd integers with sum n.

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A325459 Sum of numbers of nontrivial divisors (greater than 1 and less than k) of k for k = 1..n.

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A325588 Number of necklace compositions of n with equal circular differences up to sign.

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A333195 Numbers with three consecutive prime indices in arithmetic progression.

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