A049980 -id:A049980 - cp's OEIS frontend

A342515 Number of strict partitions of n with constant (equal) first-quotients.

1, 1, 1, 2, 2, 3, 3, 5, 4, 5, 5, 6, 6, 8, 8, 9, 8, 9, 9, 11, 10, 13, 11, 12, 12, 13, 14, 14, 15, 15, 16, 18, 16, 17, 17, 19, 18, 20, 20, 22, 21, 21, 23, 23, 22, 24, 23, 24, 24, 27, 25, 26, 27, 27, 27, 28, 29, 31, 29, 30, 31, 32, 33, 35, 32, 35, 33, 35, 34, 35

Offset: 0

Gus Wiseman, Mar 19 2021

nonn

Also the number of reversed strict partitions of n with constant (equal) first-quotients.

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the quotients of (6,3,1) are (1/2,1/3).

			The a(1) = 1 through a(15) = 9 partitions (A..F = 10..15):
  1   2   3    4    5    6    7     8    9    A    B    C    D     E     F
          21   31   32   42   43    53   54   64   65   75   76    86    87
                    41   51   52    62   63   73   74   84   85    95    96
                              61    71   72   82   83   93   94    A4    A5
                              421        81   91   92   A2   A3    B3    B4
                                                   A1   B1   B2    C2    C3
                                                             C1    D1    D2
                                                             931   842   E1
                                                                         8421

Wikipedia, Arithmetic progression
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for differences instead of quotients is A049980.
The non-strict ordered version is A342495.
The non-strict version is A342496.
The distinct instead of equal version is A342520.
A000005 counts constant partitions.
A000041 counts partitions (strict: A000009).
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A154402 counts partitions with adjacent parts x = 2y.
A167865 counts strict chains of divisors > 1 summing to n.
A175342 counts compositions with equal differences.
Cf. A003242, A005117, A049988, A057567, A067824, A253249, A307824, A318991, A318992, A325328, A342086.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&SameQ@@Divide@@@Partition[#,2,1]&]],{n,0,30}]

A049991 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, 3, 6, 9, 15, 19, 25, 33, 41, 47, 60, 67, 77, 92, 104, 113, 132, 142, 158, 178, 193, 205, 231, 247, 264, 289, 310, 325, 359, 375, 397, 427, 449, 473, 513, 532, 556, 591, 623, 644, 689, 711, 741, 788, 817, 841, 892, 920, 957, 1003, 1038, 1065, 1121, 1157, 1197, 1248, 1284, 1314, 1384, 1415

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

From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = Sum_{k = 1..n} A049990(k).
G.f.: (g.f. of A049990)/(1-x). (End)

More terms from Petros Hadjicostas, Sep 29 2019

A342522 Heinz numbers of integer partitions with constant (equal) first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97

Offset: 1

Gus Wiseman, Mar 23 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The prime indices of 2093 are {4,6,9}, with first quotients (3/2,3/2), so 2093 is in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
   12: {1,1,2}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   30: {1,2,3}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   50: {1,3,3}
   52: {1,1,6}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Wikipedia, Arithmetic progression
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

For multiplicities (prime signature) instead of quotients we have A072774.
The version counting strict divisor chains is A169594.
For differences instead of quotients we have A325328 (count: A049988).
These partitions are counted by A342496 (strict: A342515, ordered: A342495).
The distinct instead of equal version is A342521.
A000005 count constant partitions.
A000041 counts partitions (strict: A000009).
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A318991/A318992 rank reversed partitions with/without integer quotients.
A342086 counts strict chains of divisors with strictly increasing quotients.
Cf. A003242, A047966, A049980, A056239, A067824, A112798, A118914, A124010, A130091, A181819, A325351, A325352.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],SameQ@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]

A049992 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, 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

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

cp's OEIS Frontend

A342515 Number of strict partitions of n with constant (equal) first-quotients.

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A049991 a(n) is the number of arithmetic progressions of 2 or more positive integers, nondecreasing with sum <= n.

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A342522 Heinz numbers of integer partitions with constant (equal) first quotients.

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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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A333195 Numbers with three consecutive prime indices in arithmetic progression.

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