A325328 -id:A325328 - cp's OEIS frontend

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

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]&]

A325851 Number of (strict) compositions of n whose differences of all degrees are nonzero.

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 12, 21, 35, 58, 102, 171, 284, 485, 819, 1355, 2301, 3884, 6528, 10983, 18380, 30824, 51851

Offset: 0

Gus Wiseman, May 31 2019

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. The differences of all degrees of a sequence are the union of its zeroth through m-th differences, where m is the length of the sequence.

			The a(1) = 1 through a(7) = 21 compositions:
  (1)  (2)  (3)   (4)    (5)    (6)     (7)
            (12)  (13)   (14)   (15)    (16)
            (21)  (31)   (23)   (24)    (25)
                  (121)  (32)   (42)    (34)
                         (41)   (51)    (43)
                         (131)  (132)   (52)
                         (212)  (141)   (61)
                                (213)   (124)
                                (231)   (142)
                                (312)   (151)
                                (1212)  (214)
                                (2121)  (232)
                                        (241)
                                        (313)
                                        (412)
                                        (421)
                                        (1213)
                                        (1312)
                                        (2131)
                                        (3121)
                                        (12121)

The case for only degrees > 1 is A325875.

Cf. A049988, A175342, A238423, A295370, A325328, A325545, A325850, A325852, A325874.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[Union@@Table[Differences[#,i],{i,Length[#]}],0]&]],{n,0,10}]

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}]

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_,_}]&]

cp's OEIS Frontend

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

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A325851 Number of (strict) compositions of n whose differences of all degrees are nonzero.

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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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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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