A319162 -id:A319162 - cp's OEIS frontend

A047968 a(n) = Sum_{d|n} p(d), where p(d) = A000041 = number of partitions of d.

1, 3, 4, 8, 8, 17, 16, 30, 34, 52, 57, 99, 102, 153, 187, 261, 298, 432, 491, 684, 811, 1061, 1256, 1696, 1966, 2540, 3044, 3876, 4566, 5846, 6843, 8610, 10203, 12610, 14906, 18491, 21638, 26508, 31290, 38044, 44584, 54133, 63262, 76241

Offset: 1

N. J. A. Sloane, Dec 11 1999

nonn

Inverse Moebius transform of A000041.
Row sums of triangle A137587. - Gary W. Adamson, Jan 27 2008
Row sums of triangle A168021. - Omar E. Pol, Nov 20 2009
Row sums of triangle A168017. Row sums of triangle A168018. - Omar E. Pol, Nov 25 2009
Sum of the partition numbers of the divisors of n. - Omar E. Pol, Feb 25 2014
Conjecture: for n > 6, a(n) is strictly increasing. - Franklin T. Adams-Watters, Apr 19 2014
Number of constant multiset partitions of multisets spanning an initial interval of positive integers with multiplicities an integer partition of n. - Gus Wiseman, Sep 16 2018

			For n = 10 the divisors of 10 are 1, 2, 5, 10, hence the partition numbers of the divisors of 10 are 1, 2, 7, 42, so a(10) = 1 + 2 + 7 + 42 = 52. - _Omar E. Pol_, Feb 26 2014
From _Gus Wiseman_, Sep 16 2018: (Start)
The a(6) = 17 constant multiset partitions:
  (111111)  (111)(111)    (11)(11)(11)  (1)(1)(1)(1)(1)(1)
  (111222)  (12)(12)(12)
  (111122)  (112)(112)
  (112233)  (123)(123)
  (111112)
  (111123)
  (111223)
  (111234)
  (112234)
  (112345)
  (123456)
(End)

T. D. Noe, Table of n, a(n) for n=1..1000
N. J. A. Sloane, Transforms

Cf. A000041, A000837, A047966, A055893, A137587, A003606 (Euler transform).

Cf. A002033, A003238, A018783, A034729, A052409, A078392, A100953, A319162.

with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l) do c := c+numbpart(l[i]) od: RETURN(c): end: for j from 1 to 60 do printf(`%d, `, a(j)) od: # Zerinvary Lajos, Apr 14 2007

a[n_] := Sum[ PartitionsP[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 44}] (* Jean-François Alcover, Oct 03 2013 *)

G.f.: Sum_{k>0} (-1+1/Product_{i>0} (1-z^(k*i))). - Vladeta Jovovic, Jun 22 2003
G.f.: sum(n>0,A000041(n)*x^n/(1-x^n)). - Mircea Merca, Feb 24 2014.
a(n) = A168111(n) + A000041(n). - Omar E. Pol, Feb 26 2014
a(n) = Sum_{y is a partition of n} A000005(GCD(y)). - Gus Wiseman, Sep 16 2018

A320810 Number of non-isomorphic multiset partitions of weight n whose part-sizes have a common divisor > 1.

Original entry on oeis.org

0, 2, 3, 12, 7, 84, 15, 410, 354, 3073, 56, 28300, 101, 210036, 126839, 2070047, 297, 25295952, 490, 269662769, 89071291, 3449056162, 1255, 51132696310, 400625539, 713071048480, 145126661415, 11351097702297, 4565, 199926713003444, 6842, 3460838122540969

Offset: 1

Gus Wiseman, Nov 15 2018

nonn

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the column sums are not relatively prime.
Also the number of non-isomorphic multiset partitions of weight n in which the multiset union of the parts is periodic, where a multiset is periodic if its multiplicities have a common divisor > 1.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 7 multiset partitions whose part-sizes have a common divisor:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}
  {{1,2}}  {{1,2,2}}  {{1,1,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}
                      {{1,2,3,3}}    {{1,2,2,3,3}}
                      {{1,2,3,4}}    {{1,2,3,3,3}}
                      {{1,1},{1,1}}  {{1,2,3,4,4}}
                      {{1,1},{2,2}}  {{1,2,3,4,5}}
                      {{1,2},{1,2}}
                      {{1,2},{2,2}}
                      {{1,2},{3,3}}
                      {{1,2},{3,4}}
                      {{1,3},{2,3}}
Non-isomorphic representatives of the a(2) = 1 through a(5) = 7 multiset partitions with periodic multiset union:
  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}        {{1,1,1,1,1}}
  {{1},{1}}  {{1},{1,1}}    {{1,1,2,2}}        {{1},{1,1,1,1}}
             {{1},{1},{1}}  {{1},{1,1,1}}      {{1,1},{1,1,1}}
                            {{1,1},{1,1}}      {{1},{1},{1,1,1}}
                            {{1},{1,2,2}}      {{1},{1,1},{1,1}}
                            {{1,1},{2,2}}      {{1},{1},{1},{1,1}}
                            {{1,2},{1,2}}      {{1},{1},{1},{1},{1}}
                            {{1},{1},{1,1}}
                            {{1},{1},{2,2}}
                            {{1},{2},{1,2}}
                            {{1},{1},{1},{1}}
                            {{1},{1},{2},{2}}

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A007716, A018783, A047966, A120733, A303546, A303547, A305563, A319149, A319162, A319164, A319810.

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n));Vec(OgfSeries(sCartProd(sExp(A), -sum(d=2, n, moebius(d) * (-1 + sExp(O(x*x^n) + sum(i=1, n\d, polcoef(A,i*d)*x^(i*d)))) ))), -n)} \\ Andrew Howroyd, Jan 17 2023

a(n) = A007716(n) - A321283(n). - Andrew Howroyd, Jan 17 2023

Terms a(11) and beyond from Andrew Howroyd, Jan 17 2023

A319164 Number of integer partitions of n that are neither relatively prime nor aperiodic.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 5, 0, 2, 2, 5, 0, 6, 0, 9, 2, 2, 0, 17, 1, 2, 3, 17, 0, 18, 0, 22, 2, 2, 2, 48, 0, 2, 2, 48, 0, 34, 0, 58, 11, 2, 0, 111, 1, 14, 2, 103, 0, 65, 2, 141, 2, 2, 0, 264, 0, 2, 19, 231, 2, 116, 0, 299, 2, 42

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

A partition is aperiodic if its multiplicities are relatively prime.

			The a(24) = 17 integer partitions:
  (12,12),
  (8,8,8),
  (6,6,6,6), (8,8,4,4), (9,9,3,3), (10,10,2,2),
  (4,4,4,4,4,4), (6,6,3,3,3,3), (6,6,4,4,2,2), (6,6,6,2,2,2), (8,8,2,2,2,2),
  (3,3,3,3,3,3,3,3), (4,4,4,4,2,2,2,2), (6,6,2,2,2,2,2,2),
  (4,4,4,2,2,2,2,2,2),
  (4,4,2,2,2,2,2,2,2,2),
  (2,2,2,2,2,2,2,2,2,2,2,2).

Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319162, A319165.

Table[Length[Select[IntegerPartitions[n],And[GCD@@#>1,GCD@@Length/@Split[#]>1]&]],{n,30}]

A319163 Perfect powers whose prime multiplicities appear with relatively prime multiplicities.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 144, 169, 243, 256, 289, 324, 343, 361, 400, 512, 529, 576, 625, 729, 784, 841, 961, 1024, 1331, 1369, 1600, 1681, 1728, 1849, 1936, 2025, 2048, 2187, 2197, 2209, 2304, 2401, 2500, 2704, 2809, 2916, 3125

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

Perfect powers n such that A181819(n) is not a perfect power (i.e. belongs to A007916).

			The sequence of integer partitions whose Heinz numbers are in the sequence begins: (11), (111), (22), (1111), (33), (222), (11111), (44), (111111), (2222), (55), (333), (1111111), (221111), (66), (22222), (11111111), (77), (222211), (444), (88), (331111), (111111111), (99), (22111111), (3333), (222222), (441111).

Cf. A001597, A007916, A056239, A072774, A181819, A289509, A296150, A298748, A319151, A319161, A319162, A319165.

Select[Range[1000],And[GCD@@FactorInteger[#][[All,2]]>1,GCD@@Length/@Split[Sort[FactorInteger[#][[All,2]]]]==1]&]

A319160 Number of integer partitions of n whose multiplicities appear with relatively prime multiplicities.

Original entry on oeis.org

1, 2, 2, 4, 5, 7, 11, 16, 22, 31, 45, 58, 83, 108, 142, 188, 250, 315, 417, 528, 674, 861, 1094, 1363, 1724, 2152, 2670, 3311, 4105, 5021, 6193, 7561, 9216, 11219, 13614, 16419, 19886, 23920, 28733, 34438, 41272, 49184, 58746, 69823, 82948, 98380, 116567

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

From Gus Wiseman, Jul 11 2023: (Start)
A partition is aperiodic (A000837) if its multiplicities are relatively prime. This sequence counts partitions whose multiplicities are aperiodic.
For example:
- The multiplicities of (5,3) are (1,1), with multiplicities (2), with common divisor 2, so it is not counted under a(8).
- The multiplicities of (3,2,2,1) are (2,1,1), with multiplicities (2,1), which are relatively prime, so it is counted under a(8).
- The multiplicities of (3,3,1,1) are (2,2), with multiplicities (2), with common divisor 2, so it is not counted under a(8).
- The multiplicities of (4,4,4,3,3,3,2,1) are (3,3,1,1), with multiplicities (2,2), which have common divisor 2, so it is not counted under a(24).
(End)

			The a(8) = 16 partitions:
  (8),
  (44),
  (332), (422), (611),
  (2222), (3221), (4211), (5111),
  (22211), (32111), (41111),
  (221111), (311111),
  (2111111),
  (11111111).
Missing from this list are: (53), (62), (71), (431), (521), (3311).

These partitions have ranks A319161.
For distinct instead of relatively prime multiplicities we have A325329.
Cf. A000837, A001597, A007916, A047966, A071625, A098859, A100953, A181819, A182850, A182857, A305563, A319149, A319162, A319164.

Table[Length[Select[IntegerPartitions[n], GCD@@Length/@Split[Sort[Length/@Split[#]]]==1&]],{n,30}]

A319810 Number of fully periodic integer partitions of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 5, 4, 6, 2, 11, 2, 8, 7, 11, 2, 17, 2, 18, 9, 15, 2, 32, 5, 22, 12, 34, 2, 54, 2, 49, 16, 51, 10, 94, 2, 77, 23, 112, 2, 152, 2, 148, 47, 165, 2, 258, 7, 247, 52, 286, 2, 400, 17, 402, 78, 439, 2, 657, 2, 594, 131, 711, 24

Offset: 1

Gus Wiseman, Sep 28 2018

nonn

An integer partition is fully periodic iff either it is a singleton or it is a periodic partition (meaning its multiplicities have a common divisor > 1) with fully periodic multiplicities.

			The a(12) = 11 fully periodic integer partitions:
  (12)
  (6,6)
  (4,4,4)
  (5,5,1,1)
  (4,4,2,2)
  (3,3,3,3)
  (3,3,3,1,1,1)
  (3,3,2,2,1,1)
  (2,2,2,2,2,2)
  (2,2,2,2,1,1,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1)
Periodic partitions missing from this list are:
  (4,4,1,1,1,1)
  (3,3,1,1,1,1,1,1)
  (2,2,2,1,1,1,1,1,1)
  (2,2,1,1,1,1,1,1,1,1)
The first non-uniform fully periodic partition is (4,4,3,3,2,2,2,2,1,1,1,1).
The first periodic integer partition that is not fully periodic is (2,2,1,1,1,1).

Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319162, A319163, A319164, A319811.

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

A108572 Number of partitions of n which, as multisets, are nontrivial repetitions of a multiset.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 4, 2, 7, 0, 13, 0, 15, 8, 21, 0, 37, 0, 44, 16, 56, 0, 93, 6, 101, 29, 137, 0, 217, 0, 230, 57, 297, 20, 450, 0, 490, 102, 643, 0, 918, 0, 1004, 202, 1255, 0, 1783, 14, 1992, 298, 2438, 0, 3364, 61, 3734, 491, 4565, 0, 6251, 0, 6842, 818

Offset: 1

Len Smiley, Jul 25 2005

nonn

The singleton and the all-ones partitions are ignored, so that a(n)=0 if n is prime. If a partition is listed as m_1^am_2^bm_3^c..., then it is counted exactly when gcd(a,b,c,...)>1. These are equinumerous (conjugate) with those partitions for which gcd(m_1,m_2,...)>1 (less 1, the singleton), hence the formula.

			a(25) = 6: 1^(15)2^5 = 5{1, 1, 1, 2}, 1^52^(10) = 5{1, 2, 2}, 1^(10)3^5 = 5{3, 1, 1}, 2^53^5 = 5{3, 2}, 1^44^4 = 5{4, 1}, 5^5 = 5{5}.
Note that A000041(25)=P(25)=1958, only 6 of which satisfy the criterion.

Cf. A000837, A018783, A047966, A100953, A303386, A303547, A303553, A319149, A319162, A319164, A319810.

with(combinat):PartMulti:=proc(n::nonnegint) local count,a,i,j,b,m,k,part_vec;
bigcount:=0; if isprime(n) then return(bigcount) else ps:=partition(n); b:=nops(ps);
for m from 2 to b-1 do p:=ps[m]; a:=nops(p); part_vec:=array(1..n);
for k from 1 to n do part_vec[k]:=0 od;
for i from 1 to a do j:=p[i]; part_vec[j]:=part_vec[j]+1 od;
g:=0; for j from 1 to n do g:=igcd(g,part_vec[j]) od;
if g>1 then bigcount:=bigcount+1 fi od; return(bigcount) end if end proc;
seq(PartMulti(q),q=1..49);

Table[Length[Select[IntegerPartitions[n],And[Length[#]1]&]],{n,20}] (* Gus Wiseman, Dec 06 2018 *)

a(n) = A018783(n)-1, n>1. - Vladeta Jovovic, Jul 28 2005

More terms from Gus Wiseman, Dec 06 2018

A319179 Number of integer partitions of n that are relatively prime but not aperiodic. Number of integer partitions of n that are aperiodic but not relatively prime.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 2, 6, 1, 9, 1, 14, 7, 17, 1, 32, 1, 36, 15, 55, 1, 77, 6, 100, 27, 121, 1, 200, 1, 209, 56, 296, 19, 403, 1, 489, 101, 596, 1, 885, 1, 947, 192, 1254, 1, 1673, 14, 1979, 297, 2336, 1, 3300, 60, 3594, 490, 4564, 1, 5988, 1, 6841, 800

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

An integer partition is aperiodic if its multiplicities are relatively prime.

			The a(12) = 9 integer partitions that are relatively prime but not aperiodic:
  (5511),
  (332211), (333111), (441111),
  (22221111), (33111111),
  (222111111),
  (2211111111),
  (111111111111).
The a(12) = 9 integer partitions that are aperiodic but not relatively prime:
  (12),
  (8,4), (9,3), (10,2),
  (6,3,3), (6,4,2), (8,2,2),
  (6,2,2,2),
  (4,2,2,2,2).

Cf. A000837, A018783, A047966, A100953, A305563, A319149, A319160, A319162, A319164.

Table[Length[Select[IntegerPartitions[n],And[GCD@@#==1,GCD@@Length/@Split[#]>1]&]],{n,30}]

A319811 Number of totally aperiodic integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 6, 7, 14, 17, 27, 34, 55, 63, 99, 117, 162, 203, 286, 333, 469, 558, 737, 903, 1196, 1414, 1860, 2232, 2839, 3422, 4359, 5144, 6531, 7762, 9617, 11479, 14182, 16715, 20630, 24333, 29569, 34890, 42335, 49515, 59871, 70042, 83810, 98105, 117152

Offset: 1

Gus Wiseman, Sep 28 2018

nonn

An integer partition is totally aperiodic iff either it is strict or it is aperiodic with totally aperiodic multiplicities.

			The a(6) = 7 aperiodic integer partitions are: (6), (51), (42), (411), (321), (3111), (21111). The first aperiodic integer partition that is not totally aperiodic is (432211).

Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319162, A319163, A319164, A319810.

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

cp's OEIS Frontend

A047968 a(n) = Sum_{d|n} p(d), where p(d) = A000041 = number of partitions of d.

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A320810 Number of non-isomorphic multiset partitions of weight n whose part-sizes have a common divisor > 1.

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A319164 Number of integer partitions of n that are neither relatively prime nor aperiodic.

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A319163 Perfect powers whose prime multiplicities appear with relatively prime multiplicities.

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A319160 Number of integer partitions of n whose multiplicities appear with relatively prime multiplicities.

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A319810 Number of fully periodic integer partitions of n.

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A108572 Number of partitions of n which, as multisets, are nontrivial repetitions of a multiset.

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A319179 Number of integer partitions of n that are relatively prime but not aperiodic. Number of integer partitions of n that are aperiodic but not relatively prime.

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