A014405 -id:A014405 - cp's OEIS frontend

A014407 Numbers k such that s(j) < s(k) for all j < k, where s = A014405.

1, 6, 9, 12, 15, 18, 21, 24, 27, 30, 36, 42, 45, 54, 60, 72, 75, 84, 90, 105, 120, 144, 150, 168, 180, 210, 240, 270, 300, 315, 330, 360, 390, 420, 450, 480, 510, 540, 600, 630, 720, 780, 810, 840, 900, 960, 990, 1050, 1080, 1140, 1170, 1260, 1440, 1470, 1530, 1560, 1620, 1680, 1800, 1890

Offset: 1

Clark Kimberling

nonn

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..622 (terms 1..19 from Clark Kimberling, terms 20..52 from Sean A. Irvine)

Cf. A014405.

More terms from Sean A. Irvine, Oct 22 2018

More terms from Fausto A. C. Cariboni, Feb 24 2019

A023645 a(n) = tau(n)-1 if n is odd or tau(n)-2 if n is even.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Vertex-transitive graphs of valency 2 with n nodes.
Number of values of k such that n+2 divided by k leaves a remainder 2. - Amarnath Murthy, Aug 01 2002
Number of divisors of n that are less than n/2. - Peter Munn, Mar 31 2017, or equivalently, number of divisors of n that are greater than 2. - Antti Karttunen, Feb 20 2023
For n > 2, a(n) is the number of planar arrangements of equal-sized regular n-gons such that their centers lie on a circle and neighboring n-gons have an edge in common. - Peter Munn, Apr 23 2017
Number of partitions of n into two distinct parts such that the smaller divides the larger. - Wesley Ivan Hurt, Dec 21 2017

			x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + x^11 + 4*x^12 + ...

CRC Handbook of Combinatorial Designs, 1996, p. 649.

T. D. Noe, Table of n, a(n) for n = 1..10000
Felix Fröhlich et al., Rings of regular polygons, SeqFan thread, March 26 2017.
Gordon Royle, Transitive Graphs

Cf. A000005, A001620, A023637-A023647.
Cf. A014405, A049992, A069930, A095374, A296955, A321014.
Second column of A072528.

with(numtheory); f := n->if n mod 2 = 1 then tau(n)-1 else tau(n)-2; fi;

Table[s = DivisorSigma[0, n]; If[OddQ[n], s - 1, s - 2], {n, 100}] (* T. D. Noe, Nov 18 2013 *)
Array[DivisorSigma[0, #] - 1 - Boole@ EvenQ@ # &, 104] (* Michael De Vlieger, Apr 25 2017 *)

{a(n) = if( n<1, 0, numdiv(n) - 2 + n%2)} /* Michael Somos, Apr 29 2003 */

a(n) = sumdiv(n, d, d < n/2); \\ Michel Marcus, Apr 01 2017

G.f.: Sum_{k>0} x^(3*k) / (1 - x^k). - Michael Somos, Apr 29 2003.
a(2*n) = A069930(n). a(2*n + 1) = A095374(n). - Michael Somos, Aug 30 2012
a(n) = A072528(n+2,2) for n > 2. - Peter Munn, May 14 2017
From Peter Bala, Jan 13 2021: (Start)
a(n) = Sum_{ d|n, d < n/2 } 1. Cf. A296955.
G.f.: Sum_{k >= 3} x^k/(1 - x^k). (End)
a(n) = A049992(n) - A014405(n). - Antti Karttunen, Feb 20 2023
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 5/2), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024

More terms from Vladeta Jovovic, Dec 03 2001

A049980 a(n) is the number of arithmetic progressions of positive integers, strictly increasing with sum n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 4, 7, 6, 6, 9, 7, 8, 13, 9, 9, 15, 10, 12, 18, 13, 12, 20, 15, 15, 23, 17, 15, 28, 16, 18, 28, 20, 22, 33, 19, 22, 33, 26, 21, 39, 22, 26, 43, 27, 24, 43, 27, 33, 44, 31, 27, 50, 34, 34, 49, 34, 30, 60, 31, 36, 57, 38, 40

Offset: 1

Clark Kimberling

nonn

We need to find the number of pairs of positive integers (b, w) so that there is a positive integer m such that m*b + m*(m-1)*w/2 = n. - Petros Hadjicostas, Sep 27 2019

			a(6) = 4 because we have the following strictly increasing arithmetic progressions of positive integers adding up to n = 6: 6, 1+5, 2+4, and 1+2+3. - _Petros Hadjicostas_, Sep 27 2019

Fausto A. C. Cariboni, 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.

Cf. A000217, A014405, A014406, A049981, A049982, A049983, A049986, A049987, A068322, A068323, A068324, A127938, A175342.

Conjecture: a(n) = 1 + Sum_{m|n, m odd > 1} floor(2 * (n - m)/(m* (m - 1))) + Sum_{m|n} floor((n - m * (5 - (-1)^(n/m))/2 + m^2 * (1 - (-1)^(n/m)))/(2*m * (2*m - 1))). - Petros Hadjicostas, Sep 27 2019

G.f.: x/(1-x) + Sum_{k >= 2} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1) = x/(1-x) + Sum_{k >= 2} x^t(k)/((1 - x^k) * (1 - x^t(k-1))), where t(k) = k*(k+1)/2 = A000217(k) is the k-th triangular number [Graeme McRae]. - Petros Hadjicostas, Sep 29 2019

A049982 Number of arithmetic progressions of 2 or more positive integers, strictly increasing with sum n.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 3, 6, 5, 5, 8, 6, 7, 12, 8, 8, 14, 9, 11, 17, 12, 11, 19, 14, 14, 22, 16, 14, 27, 15, 17, 27, 19, 21, 32, 18, 21, 32, 25, 20, 38, 21, 25, 42, 26, 23, 42, 26, 32, 43, 30, 26, 49, 33, 33, 48, 33, 29, 59, 30, 35, 56, 37, 39, 61, 33, 39, 58, 49, 35, 67, 36, 42

Offset: 1

Clark Kimberling

nonn

Andrew Howroyd, 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.
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.

Cf. A000005, A000217, A014405, A014406, A049980, A049981, A049983, A049986, A049987, A049988, A068322, A127938, A175342.

seq(n)={Vec(sum(k=2, (sqrtint(8*n+1)-1)\2, x^binomial(k+1, 2)/(x^binomial(k+1, 2) - x^binomial(k, 2) - x^k + 1) + O(x*x^n)), -n)} \\ Andrew Howroyd, Sep 28 2019

a(n) has generating function x^3/(x^3 - x - x^2 + 1) + x^6/(x^6 - x^3 - x^3 + 1) + x^10/(x^10 - x^6 - x^4 + 1) + ... = Sum_{k >= 2} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1), where t(k) = A000217(k) is the k-th triangular number. Term k of this generating function generates the number of arithmetic progressions of k positive integers, strictly increasing with sum n. - Graeme McRae, Feb 08 2007
From Petros Hadjicostas, Sep 27 2019: (Start)
a(n) = A049980(n) - 1 = A049988(n) - A000005(n).
a(n) = A049981(n) - A049981(n-1) - 1 for n >= 2.
Conjecture: a(n) = Sum_{m|n, m odd > 1} floor(2 * (n - m)/(m* (m - 1))) + Sum_{m|n} floor((n - m * (5 - (-1)^(n/m))/2 + m^2 * (1 - (-1)^(n/m)))/(2*m * (2*m - 1))).
(End)

More terms from Petros Hadjicostas, Sep 28 2019

A049987 a(n) is the number of arithmetic progressions of 4 or more positive integers, strictly increasing with sum <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 3, 4, 4, 5, 5, 7, 8, 10, 10, 11, 13, 15, 16, 19, 19, 23, 23, 25, 26, 29, 33, 37, 37, 40, 41, 47, 47, 52, 52, 56, 62, 66, 66, 70, 72, 80, 82, 87, 87, 93, 99, 105, 107, 112, 112, 123, 123, 128, 133, 139, 146, 154, 154, 160, 162, 177, 177, 186, 186, 192, 202

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.

Cf. A014405, A014406, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A127938.

From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = Sum_{k = 1..n} A049986(k).
G.f.: (g.f. of A049986)/(1-x). (End)

More terms from Petros Hadjicostas, Sep 29 2019

A049983 a(n) is the number of arithmetic progressions of 2 or more positive integers, strictly increasing with sum <= n.

Original entry on oeis.org

0, 0, 1, 2, 4, 7, 10, 13, 19, 24, 29, 37, 43, 50, 62, 70, 78, 92, 101, 112, 129, 141, 152, 171, 185, 199, 221, 237, 251, 278, 293, 310, 337, 356, 377, 409, 427, 448, 480, 505, 525, 563, 584, 609, 651, 677, 700, 742, 768, 800, 843, 873, 899, 948, 981, 1014, 1062, 1095, 1124, 1183, 1213, 1248, 1304, 1341, 1380

Offset: 1

Clark Kimberling

nonn

			a(7) = 10 because we have the following arithmetic progressions of two or more positive integers, strictly increasing with sum <= n = 7: 1+2, 1+3, 1+4, 1+5, 1+6, 2+3, 2+4, 2+5, 3+4, and 1+2+3. - _Petros Hadjicostas_, Sep 27 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.
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.

Cf. A014405, A014406, A049980, A049981, A049982, A049986, A049987, A068322, A127938, A175342.

From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = Sum_{k = 1..n} A049982(k) = -n + Sum_{k = 1..n} A049980(k) = -n + A049981(k).
G.f.: (g.f. of A049982)/(1-x). (End)

More terms from Petros Hadjicostas, Sep 27 2019

A049986 a(n) is the number of arithmetic progressions of 4 or more positive integers, strictly increasing with sum n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 2, 1, 2, 0, 1, 2, 2, 1, 3, 0, 4, 0, 2, 1, 3, 4, 4, 0, 3, 1, 6, 0, 5, 0, 4, 6, 4, 0, 4, 2, 8, 2, 5, 0, 6, 6, 6, 2, 5, 0, 11, 0, 5, 5, 6, 7, 8, 0, 6, 2, 15, 0, 9, 0, 6, 10, 7, 4, 9, 0, 14, 5, 7, 0, 12, 9, 7, 3

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 A049986, 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.

Cf. A014405, A014406, A049980, A049981, A049982, A049983, A049987 (partial sums), A049988, A049989, A049990, A049991, A049994, A127938, A321014.

\\ Needs also code from A014405 and A023645:
A049994(n) = (A014405(n) + A023645(n) - if(n%3, 0, n/3));
A321014(n) = (numdiv(n) - 3 + !!(n%2) + !!(n%3)); \\ From A321014.
A049986(n) = (A049994(n)-A321014(n)); \\ Antti Karttunen, Feb 20 2023

G.f.: Sum_{k >= 4} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1) = Sum_{k >= 4} x^t(k)/((1 - x^k)*(1 - x^t(k-1))), where t(k) = k*(k+1)/2 = A000217(k) is the k-th triangular number [Graeme McRae]. - Petros Hadjicostas, Sep 29 2019

a(n) = A049994(n) - A321014(n). [Listed by Sequence Machine and obviously true] - Antti Karttunen, Feb 20 2023

A049981 a(n) is the number of arithmetic progressions of positive integers, strictly increasing with sum <= n.

Original entry on oeis.org

1, 2, 4, 6, 9, 13, 17, 21, 28, 34, 40, 49, 56, 64, 77, 86, 95, 110, 120, 132, 150, 163, 175, 195, 210, 225, 248, 265, 280, 308, 324, 342, 370, 390, 412, 445, 464, 486, 519, 545, 566, 605, 627, 653, 696, 723, 747, 790, 817, 850, 894, 925, 952, 1002, 1036, 1070, 1119, 1153, 1183, 1243, 1274, 1310

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.

Cf. A014405, A014406, A049980, A049982, A049983, A049986, A049987, A068322, A068323, A068324, A127938, A175342.

From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = Sum_{k = 1..n} A049980(k) = n + Sum_{k = 1..n} A049982(k).
G.f.: (g.f. of A049980)/(1-x). (End)

More terms from Petros Hadjicostas, Sep 29 2019

A014406 Number of strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 3, 4, 4, 7, 7, 8, 13, 14, 14, 20, 20, 22, 29, 31, 31, 39, 41, 43, 52, 55, 55, 68, 68, 70, 81, 84, 88, 103, 103, 106, 119, 125, 125, 143, 143, 147, 167, 171, 171, 190, 192, 200, 218, 223, 223, 246, 252, 258, 278, 283, 283, 313, 313, 318, 343, 349, 356, 385, 385

Offset: 1

Clark Kimberling

nonn

			From _Petros Hadjicostas_, Sep 29 2019: (Start)
a(8) = 1 because we have only the following strictly increasing arithmetic progression of positive integers with at least 3 terms and sum <= 8: 1+2+3.
a(9) = 3 because we have the following strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= 9: 1+2+3, 1+3+5, and 2+3+4.
a(10) = 4 because we have the following strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= 10: 1+2+3, 1+3+5, 2+3+4, and 1+2+3+4.
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..1000
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. A007862, A014405, A047966, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A049991, A129654, A240026, A240027, A307824, A320466, A325325, A325328.

a(n) = Sum_{k=1..n} A014405(k). - Sean A. Irvine, Oct 22 2018

G.f.: (g.f. of A014405)/(1-x). - Petros Hadjicostas, Sep 29 2019

a(59)-a(67) corrected by Fausto A. C. Cariboni, Oct 02 2018

A049990 a(n) is the number of arithmetic progressions of 2 or more positive integers, nondecreasing with sum n.

Original entry on oeis.org

0, 1, 2, 3, 3, 6, 4, 6, 8, 8, 6, 13, 7, 10, 15, 12, 9, 19, 10, 16, 20, 15, 12, 26, 16, 17, 25, 21, 15, 34, 16, 22, 30, 22, 24, 40, 19, 24, 35, 32, 21, 45, 22, 30, 47, 29, 24, 51, 28, 37, 46, 35, 27, 56, 36, 40, 51, 36, 30, 70, 31, 38, 61, 43

Offset: 1

Clark Kimberling

nonn

			a(6) counts these 6 partitions of 6: [5,1], [4,2], [3,3], [3,2,1], [2,2,2], [1,1,1,1,1,1].

Clark Kimberling, Table of n, a(n) for n = 1..1000
Sadek Bourbaki and Nevrine 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.
Wikipedia, Arithmetic progression.

Cf. A014405, A014406, A049980, A049981, A049982, A049983, A049987, A049988, A049989, A049990, A049991, A111333, A127938.

(* Program 1 *)
Map[Length[Map[#[[2]] &, Select[Map[{Apply[SameQ, Differences[#]], #} &,
IntegerPartitions[#]], #[[1]] &]]] &, Range[40]] - 1
(* Peter J. C. Moses, Dec 24 2016 *)
(* Program 2 *)
enumerateArithmeticPartitions[n_] := Module[{allDivs, oddDivs},
{allDivs, oddDivs} = {#, Select[#, OddQ]} &[Divisors[n]]; Map[Reverse, Union[Flatten[Table[If[OddQ[cDiff], (Flatten[
Map[{If[(2 n - #) cDiff <= # (# - 2), {Table[(cDiff + # - 2 cDiff n/#)/2 +
cDiff term, {term, 0, 2 n/# - 1}]}, {}], If[# (# - 1) cDiff <= 2 (n - #),
{Table[(cDiff + 2 n/# - # cDiff)/2 + cDiff term, {term, 0, # - 1}]},
{}]} &, oddDivs], 2]), (Flatten[Map[If[(n - #) cDiff <= 2 # (# - 1),
{Table[(cDiff + 2 # - n cDiff/#)/2 + cDiff term, {term, 0, n/# - 1}]}, {}] &,
allDivs], 1])], {cDiff, 0, n - 2}], 1]]]];
Join[{0}, Map[Length[enumerateArithmeticPartitions[#]] - 1 &, Range[2, 300]]]
n = 12; enumerateArithmeticPartitions[12] (* shows the desired partition of n *)
(* Peter J. C. Moses, Dec 24 2016 *)

a(A000040(n)) = A111333(n). - Clark Kimberling, Dec 26 2016
From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = A049988(n) - 1. [Note that A049988 has offset 0.]
G.f.: Sum_{k>=2} x^k/(1-x^(k*(k-1)/2))/(1-x^k). [Leroy Quet from A049988]
(End)

cp's OEIS Frontend

A014407 Numbers k such that s(j) < s(k) for all j < k, where s = A014405.

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A023645 a(n) = tau(n)-1 if n is odd or tau(n)-2 if n is even.

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A049980 a(n) is the number of arithmetic progressions of positive integers, strictly increasing with sum n.

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A049982 Number of arithmetic progressions of 2 or more positive integers, strictly increasing with sum n.

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A049987 a(n) is the number of arithmetic progressions of 4 or more positive integers, strictly increasing with sum <= n.

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A049983 a(n) is the number of arithmetic progressions of 2 or more positive integers, strictly increasing with sum <= n.

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A049986 a(n) is the number of arithmetic progressions of 4 or more positive integers, strictly increasing with sum n.

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PARI

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A049981 a(n) is the number of arithmetic progressions of positive integers, strictly increasing with sum <= n.

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A014406 Number of strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= n.

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