A000837 -id:A000837 - cp's OEIS frontend

A332004 Number of compositions (ordered partitions) of n into distinct and relatively prime parts.

1, 1, 0, 2, 2, 4, 8, 12, 16, 24, 52, 64, 88, 132, 180, 344, 416, 616, 816, 1176, 1496, 2736, 3232, 4756, 6176, 8756, 11172, 15576, 24120, 30460, 41456, 55740, 74440, 97976, 130192, 168408, 256464, 315972, 429888, 558192, 749920, 958264, 1274928, 1621272, 2120288, 3020256

Offset: 0

Ilya Gutkovskiy, Feb 04 2020

nonn

Moebius transform of A032020.

Ranking these compositions using standard compositions (A066099) gives the intersection of A233564 (strict) with A291166 (relatively prime). - Gus Wiseman, Oct 18 2020

			a(6) = 8 because we have [5, 1], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 5], [1, 3, 2] and [1, 2, 3].
From _Gus Wiseman_, Oct 18 2020: (Start)
The a(1) = 1 through a(8) = 16 compositions (empty column indicated by dot):
  (1)  .  (1,2)  (1,3)  (1,4)  (1,5)    (1,6)    (1,7)
          (2,1)  (3,1)  (2,3)  (5,1)    (2,5)    (3,5)
                        (3,2)  (1,2,3)  (3,4)    (5,3)
                        (4,1)  (1,3,2)  (4,3)    (7,1)
                               (2,1,3)  (5,2)    (1,2,5)
                               (2,3,1)  (6,1)    (1,3,4)
                               (3,1,2)  (1,2,4)  (1,4,3)
                               (3,2,1)  (1,4,2)  (1,5,2)
                                        (2,1,4)  (2,1,5)
                                        (2,4,1)  (2,5,1)
                                        (4,1,2)  (3,1,4)
                                        (4,2,1)  (3,4,1)
                                                 (4,1,3)
                                                 (4,3,1)
                                                 (5,1,2)
                                                 (5,2,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to compositions

Cf. A007360, A032020, A108700, A302698.
A000740 is the non-strict version.
A078374 is the unordered version (non-strict: A000837).
A101271*6 counts these compositions of length 3 (non-strict: A000741).
A337561/A337562 is the pairwise coprime instead of relatively prime version (non-strict: A337462/A101268).
A289509 gives the Heinz numbers of relatively prime partitions.
A333227/A335235 ranks pairwise coprime compositions.
Cf. A001523, A178472, A216652, A289508, A291166, A333228.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&GCD@@#<=1&]],{n,0,15}] (* Gus Wiseman, Oct 18 2020 *)

A337601 Number of unordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 4, 4, 5, 6, 8, 7, 10, 7, 11, 11, 17, 12, 19, 12, 19, 17, 29, 16, 28, 19, 31, 23, 46, 23, 42, 25, 45, 27, 59, 31, 57, 34, 61, 37, 84, 38, 75, 42, 74, 47, 107, 45, 98, 51, 96, 56, 135, 54, 115, 63, 117, 67, 174, 65, 139, 75, 144, 75, 194

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

First differs from A337600 at a(9) = 4, A337600(9) = 5.

			The a(3) = 1 through a(14) = 10 partitions (A = 10, B = 11, C = 12):
  111  211  221  321  322  332  441  433  443  543  544  554
            311  411  331  431  522  532  533  552  553  743
                      511  521  531  541  551  651  661  752
                           611  711  721  722  732  733  761
                                     811  731  741  751  833
                                          911  831  922  851
                                               921  B11  941
                                               A11       A31
                                                         B21
                                                         C11

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

A014612 intersected with A304711 ranks these partitions.
A220377 is the strict case.
A304709 counts these partitions of any length.
A307719 is the strict case except for any number of 1's.
A337600 considers singletons to be coprime.
A337603 is the ordered version.
A000217 counts 3-part compositions.
A000837 counts relatively prime partitions.
A001399/A069905/A211540 count 3-part partitions.
A023023 counts relatively prime 3-part partitions.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A337461 counts pairwise coprime 3-part compositions.
Cf. A001840, A007359, A007360, A087087, A284825, A302696, A304712, A328673, A335238, A337602.

Table[Length[Select[IntegerPartitions[n,{3}],CoprimeQ@@Union[#]&]],{n,0,100}]

For n > 0, a(n) = A337600(n) - A079978(n).

A355393 Number of integer partitions of n such that, for all parts x of multiplicity 1, either x - 1 or x + 1 is also a part.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 6, 7, 10, 14, 17, 23, 32, 39, 51, 67, 83, 105, 134, 165, 206, 256, 312, 385, 475, 573, 697, 849, 1021, 1231, 1483, 1771, 2121, 2534, 3007, 3575, 4245, 5008, 5914, 6979, 8198, 9626, 11292, 13201, 15430, 18010, 20960, 24389, 28346, 32855, 38066

Offset: 0

Gus Wiseman, Aug 26 2022

nonn

These are partitions without a neighborless singleton, where a part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once.

			The a(0) = 1 through a(8) = 10 partitions:
  ()  .  (11)  (21)   (22)    (32)     (33)      (43)       (44)
               (111)  (211)   (221)    (222)     (322)      (332)
                      (1111)  (2111)   (321)     (2221)     (2222)
                              (11111)  (2211)    (3211)     (3221)
                                       (21111)   (22111)    (3311)
                                       (111111)  (211111)   (22211)
                                                 (1111111)  (32111)
                                                            (221111)
                                                            (2111111)
                                                            (11111111)

This is the singleton case of A355394, complement A356236.
The complement is counted by A356235.
These partitions are ranked by the complement of A356237.
The strict case is A356606, complement A356607.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A073491, A325160, A328171, A328172, A328187, A328220, A356233.

Table[Length[Select[IntegerPartitions[n],Function[ptn,!Or@@Table[Count[ptn,x]==1&&!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]

A356235 Number of integer partitions of n with a neighborless singleton.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 5, 8, 12, 16, 25, 33, 45, 62, 84, 109, 148, 192, 251, 325, 421, 536, 690, 870, 1100, 1385, 1739, 2161, 2697, 3334, 4121, 5071, 6228, 7609, 9303, 11308, 13732, 16629, 20101, 24206, 29140, 34957, 41882, 50060, 59745, 71124, 84598, 100365

Offset: 0

Gus Wiseman, Aug 23 2022

nonn

A part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once. Examples of partitions with a neighborless singleton are: (3), (3,1), (3,1,1), (3,3,1). Examples of partitions without a neighborless singleton are: (3,3,1,1), (4,3,1,1), (3,2,1), (2,1), (3,3).

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)  (3)  (4)   (5)    (6)     (7)      (8)
                 (31)  (41)   (42)    (52)     (53)
                       (311)  (51)    (61)     (62)
                              (411)   (331)    (71)
                              (3111)  (421)    (422)
                                      (511)    (431)
                                      (4111)   (521)
                                      (31111)  (611)
                                               (4211)
                                               (5111)
                                               (41111)
                                               (311111)

The complement is counted by A355393.
This is the singleton case of A356236, complement A355394.
These partitions are ranked by A356237.
The strict case is A356607, complement A356606.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A066205, A325160, A328171, A328172, A328187, A328221, A356233.

Table[Length[Select[IntegerPartitions[n],Min@@Length/@Split[Reverse[#],#1>=#2-1&]==1&]],{n,0,30}]

A356236 Number of integer partitions of n with a neighborless part.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 8, 9, 16, 20, 31, 40, 59, 76, 105, 138, 184, 238, 311, 400, 515, 656, 831, 1052, 1322, 1659, 2064, 2572, 3182, 3934, 4837, 5942, 7264, 8872, 10789, 13109, 15865, 19174, 23105, 27796, 33361, 39956, 47766, 56985, 67871, 80675, 95750, 113416

Offset: 0

Gus Wiseman, Aug 24 2022

nonn

A part x of a partition is neighborless if neither x - 1 nor x + 1 are parts.

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (41)     (33)      (52)
                    (31)    (311)    (42)      (61)
                    (1111)  (11111)  (51)      (331)
                                     (222)     (421)
                                     (411)     (511)
                                     (3111)    (4111)
                                     (111111)  (31111)
                                               (1111111)

The complement is counted by A355394, singleton case A355393.
The singleton case is A356235, ranked by A356237.
The strict case is A356607, complement A356606.
These partitions are ranked by the complement of A356736.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A066205, A112798, A319630, A325160, A328171, A328172, A328187, A328221.

Table[Length[Select[IntegerPartitions[n],Function[ptn,Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]

a(n) = A000041(n) - A355394(n).

A371177 Positive integers whose prime indices include all distinct divisors of all prime indices.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 30, 32, 34, 36, 40, 42, 44, 48, 50, 54, 60, 62, 64, 66, 68, 72, 80, 82, 84, 88, 90, 96, 100, 102, 108, 110, 118, 120, 124, 126, 128, 132, 134, 136, 144, 150, 160, 162, 164, 166, 168, 170, 176, 180, 186, 192, 198, 200

Offset: 1

Gus Wiseman, Mar 18 2024

nonn

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.

Also positive integers with as many distinct prime factors (A001221) as distinct divisors of prime indices (A370820).

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}

The LHS is A001221, distinct case of A001222.
The RHS is A370820, for prime factors A303975.
For bigomega on the LHS we have A370802, counted by A371130.
For divisors on the LHS we have A371165, counted by A371172.
Partitions of this type are counted by A371178, strict A371128.
The complement is A371179, counted by A371132.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
A305148 counts partitions without divisors, strict A303362, ranks A316476.
Cf. A000837, A003963, A239312, A285573, A355529, A370813, A371168.

Select[Range[100],PrimeNu[#]==Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

A001221(a(n)) = A370820(a(n)).

A319149 Number of superperiodic integer partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 6, 1, 3, 3, 5, 1, 7, 1, 7, 3, 3, 1, 13, 2, 3, 4, 9, 1, 13, 1, 11, 3, 3, 3, 23, 1, 3, 3, 20, 1, 17, 1, 16, 9, 3, 1, 38, 2, 9, 3, 23, 1, 25, 3, 36, 3, 3, 1, 71, 1, 3, 11, 49, 3, 31, 1, 52, 3, 19

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

An integer partition is superperiodic if either it consists of a single part equal to 1 or its parts have a common divisor > 1 and its multiset of multiplicities is itself superperiodic. For example, (8,8,6,6,4,4,4,4,2,2,2,2) has multiplicities (4,4,2,2) with multiplicities (2,2) with multiplicities (2) with multiplicities (1). The first four of these partitions are periodic and the last is (1), so (8,8,6,6,4,4,4,4,2,2,2,2) is superperiodic.

			The a(24) = 11 superperiodic partitions:
  (24)
  (12,12)
  (8,8,8)
  (9,9,3,3)
  (8,8,4,4)
  (6,6,6,6)
  (10,10,2,2)
  (6,6,6,2,2,2)
  (6,6,4,4,2,2)
  (4,4,4,4,4,4)
  (4,4,4,4,2,2,2,2)
  (3,3,3,3,3,3,3,3)
  (2,2,2,2,2,2,2,2,2,2,2,2)

Cf. A000041, A000837, A018783, A024994, A047966, A098859, A100953, A181819, A182857, A304660, A305563, A317245, A317256, A319151, A319152, A319157.

wotperQ[m_]:=Or[m=={1},And[GCD@@m>1,wotperQ[Sort[Length/@Split[Sort[m]]]]]];
Table[Length[Select[IntegerPartitions[n],wotperQ]],{n,30}]

A325356 Number of integer partitions of n whose augmented differences are weakly increasing.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 6, 5, 5, 6, 8, 6, 10, 9, 8, 10, 13, 10, 15, 14, 13, 15, 21, 15, 19, 21, 20, 25, 25, 20, 31, 30, 30, 32, 35, 28, 40, 44, 36, 42, 50, 43, 54, 53, 49, 57, 67, 58, 68, 66, 66, 78, 84, 71, 86, 92, 82, 99, 109

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

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 aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The Heinz numbers of these partitions are given by A325394.

			The a(1) = 1 through a(8) = 6 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (1111)  (11111)  (222)     (1111111)  (53)
                                     (111111)             (332)
                                                          (2222)
                                                          (11111111)
For example, the augmented differences of (6,6,5,3) are (1,2,3,3), which are weakly increasing, so (6,6,5,3) is counted under a(20).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1500

Cf. A000837, A007294, A049988, A098859, A325350, A325351, A325354, A325357, A325358, A325360, A325394.

Mathematica
```
aug[y_]:=Table[If[i
				
```

A078392 Sum of GCD's of parts in all partitions of n.

Original entry on oeis.org

1, 3, 5, 9, 11, 20, 21, 35, 42, 61, 66, 112, 113, 168, 210, 279, 313, 461, 508, 719, 852, 1088, 1277, 1756, 2006, 2573, 3106, 3937, 4593, 5958, 6872, 8676, 10305, 12655, 15009, 18664, 21673, 26559, 31447, 38217, 44623, 54386, 63303, 76379, 89696, 106879

Offset: 1

Vladeta Jovovic, Dec 24 2002

nonn

Equals row sums of triangle A168534. - Gary W. Adamson, Nov 28 2009

			Partitions of 4 are 1+1+1+1, 1+1+2, 2+2, 1+3, 4, the corresponding GCD's of parts are 1,1,2,1,4 and their sum is a(4) = 9.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000010, A000041, A168534, A181844 (the same for LCM), A319301.

with(numtheory): with(combinat):
a:= n-> add(phi(n/d)*numbpart(d), d=divisors(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Apr 02 2015

a[n_] := Sum[EulerPhi[n/d]*PartitionsP[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jul 01 2015, after Alois P. Heinz *)

a(n) = Sum_{d|n} d * A000837(n/d).
a(n) = Sum_{d|n} phi(n/d)*numbpart(d) = Sum_{d|n} A000010(n/d)*A000041(d). - Vladeta Jovovic, May 08 2003
From Richard L. Ollerton, May 06 2021: (Start)
a(n) = Sum_{k=1..n} A000041(gcd(n,k)).
a(n) = Sum_{k=1..n} A000041(n/gcd(n,k))*A000010(gcd(n,k))/A000010(n/gcd(n,k)). (End)

A318728 Number of cyclic compositions (necklaces of positive integers) summing to n that have only one part or whose adjacent parts (including the last with first) are coprime.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 13, 22, 34, 58, 95, 168, 280, 492, 853, 1508, 2648, 4715, 8350, 14924, 26643, 47794, 85779, 154475, 278323, 502716, 908913, 1646206, 2984547, 5418653, 9847190, 17916001, 32625618, 59470540, 108493150, 198094483, 361965239, 661891580, 1211162271

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(7) = 13 cyclic compositions with adjacent parts coprime:
  7,
  16, 25, 34,
  115,
  1114, 1213, 1132, 1123,
  11113, 11212,
  111112,
  1111111.

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000740, A000837, A008965, A059966, A100953, A167606, A296302, A318726, A318727, A328597.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,neckQ[#]&&And@@CoprimeQ@@@Partition[#,2,1,1]]&]],{n,20}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q,]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i,j)->gcd(i,j)==1))); vector(n, n, (n > 1) + sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 27 2019

a(n) = A328597(n) + 1 for n > 1. - Andrew Howroyd, Oct 27 2019

Terms a(21) and beyond from Andrew Howroyd, Sep 08 2018

Name corrected by Gus Wiseman, Nov 04 2019

cp's OEIS Frontend

A332004 Number of compositions (ordered partitions) of n into distinct and relatively prime parts.

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A337601 Number of unordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is not considered coprime unless it is (1).

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A355393 Number of integer partitions of n such that, for all parts x of multiplicity 1, either x - 1 or x + 1 is also a part.

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A356235 Number of integer partitions of n with a neighborless singleton.

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A356236 Number of integer partitions of n with a neighborless part.

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A371177 Positive integers whose prime indices include all distinct divisors of all prime indices.

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A319149 Number of superperiodic integer partitions of n.

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A325356 Number of integer partitions of n whose augmented differences are weakly increasing.

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A078392 Sum of GCD's of parts in all partitions of n.