A000837 -id:A000837 - cp's OEIS frontend

A320813 Number of non-isomorphic multiset partitions of an aperiodic multiset of weight n such that there are no singletons and all parts are themselves aperiodic multisets.

1, 0, 1, 2, 5, 13, 33, 104, 293, 938, 2892

Offset: 0

Gus Wiseman, Nov 08 2018

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 (1) the row sums are all > 1, (2) the positive entries in each row are relatively prime, and (3) the column-sums are relatively prime.
A multiset is aperiodic if its multiplicities are relatively prime.
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) = 13 multiset partitions:
  {{1,2}}  {{1,2,2}}  {{1,2,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,3,3}}    {{1,2,2,2,2}}
                      {{1,2,3,4}}    {{1,2,2,3,3}}
                      {{1,2},{3,4}}  {{1,2,3,3,3}}
                      {{1,3},{2,3}}  {{1,2,3,4,4}}
                                     {{1,2,3,4,5}}
                                     {{1,2},{1,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,2},{3,4,4}}
                                     {{1,2},{3,4,5}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,3},{1,2,3}}

This is the case of A320804 where the underlying multiset is aperiodic.

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A302545, A303386, A303546, A303707, A303708, A320797-A320813, A321283, A321390.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
aperQ[m_]:=Length[m]==0||GCD@@Length/@Split[Sort[m]]==1;
Table[Length[Union[brute /@ Select[mpm[n],And[Min@@Length/@#>1,aperQ[Join@@#]&&And@@aperQ /@ #]&]]],{n,0,7}] (* Gus Wiseman, Jan 19 2024 *)

Definition corrected by Gus Wiseman, Jan 19 2024

A328220 Number of strict integer partitions of n with no pair of consecutive parts relatively prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 5, 1, 5, 4, 6, 3, 10, 3, 11, 7, 12, 3, 19, 5, 18, 12, 23, 9, 36, 11, 33, 21, 40, 20, 58, 19, 58, 35, 70, 31, 98, 36, 101, 65, 112, 56, 155, 64, 164, 97, 188, 88, 250, 112, 256, 157, 293, 145, 392, 163, 399, 241, 461, 242

Offset: 0

Gus Wiseman, Oct 14 2019

nonn

			The a(2) = 1 through a(20) = 11 partitions (A..K = 10..20):
  2  3  4  5  6   7  8   9   A   B  C    D  E    F   G    H    I    J    K
              42     62  63  64     84      86   96  A6   863  A8   964  C8
                             82     93      A4   A5  C4   962  C6   A63  E6
                                    A2      C2   C3  E2        E4        F5
                                    642     842      862       F3        G4
                                                     A42       G2        I2
                                                               864       A64
                                                               963       A82
                                                               A62       C62
                                                               C42       E42
                                                                         8642

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500

The non-strict case is A328187.
Partitions with all consecutive parts relatively prime are A328172, with strict case A328188.
Strict partitions with relatively prime parts are A078374.
Partitions with no consecutive divisibilities are A328171.
Cf. A000837, A018783, A070211, A289509, A318980, A325545, A328170, A328221, A328335, A328336.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MatchQ[#,{_,x_,y_,_}/;GCD[x,y]==1]&]],{n,0,30}]

A282750 Triangle read by rows: T(n,k) is the number of partitions of n into k parts x_1, x_2, ..., x_k such that gcd(x_1, x_2, ..., x_k) = 1 (where 1 <= k <= n).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 3, 4, 3, 2, 1, 1, 0, 2, 4, 4, 3, 2, 1, 1, 0, 3, 6, 6, 5, 3, 2, 1, 1, 0, 2, 6, 8, 6, 5, 3, 2, 1, 1, 0, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1, 0, 2, 8, 12, 12, 10, 7, 5, 3, 2, 1, 1, 0, 6, 14, 18, 18, 14

Offset: 1

N. J. A. Sloane, Mar 05 2017

Columns 2-10 are A023022-A023030. - Lars Blomberg Mar 08 2017

To base the triangle on (0, 0) a column (1, 0, 0, ...) has to be appended to the left hand side of the triangle. To compute this triangle with Michael De Vlieger's Mathematica program only the ranges of the indices have to be adapted. The SageMath program computes the extended triangle by default. - Peter Luschny, Aug 24 2019

			Triangle begins:
   n/k: 1,  2,  3,  4,  5,  6,  7,  8, ...
   1:   1;
   2:   0,  1;
   3:   0,  1,  1;
   4:   0,  1,  1,  1;
   5:   0,  2,  2,  1,  1;
   6:   0,  1,  2,  2,  1,  1;
   7:   0,  3,  4,  3,  2,  1,  1;
   8:   0,  2,  4,  4,  3,  2,  1,  1;
   9:   0,  3,  6,  6,  5,  3,  2,  1,  1;
  10:   0,  2,  6,  8,  6,  5,  3,  2,  1,  1;
  11:   0,  5, 10, 11, 10,  7,  5,  3,  2,  1,  1;
  12:   0,  2,  8, 12, 12, 10,  7,  5,  3,  2,  1,  1;
  ...
The partitions with their gcd value for n=8, k=2..5:
(1, 7)=1, (2, 6)=2, (3, 5)=1, (4, 4)=4, so T(8,2)=2.
(1, 1, 6)=1, (1, 2, 5)=1, (1, 3, 4)=1, (2, 2, 4)=2, (2, 3, 3)=1, so T(8,2)=4.
(1, 1, 1, 5)=1, (1, 1, 2, 4)=1, (1, 1, 3, 3)=1, (1, 2, 2, 3)=1, (2, 2, 2, 2)=2, so T(8,3)=4.
(1, 1, 1, 1, 4)=1, (1, 1, 1, 2, 3)=1, (1, 1, 2, 2, 2)=1, so T(8,4)=3.
(1, 1, 1, 1, 1, 3)=1, (1, 1, 1, 1, 2, 2)=1, so T(8,5)=2.

Michael De Vlieger, Table of n, a(n) for n = 1..5050 (rows 1 <= n <= 100)

Cf. A023022-A023030, A101391 (analog for compositions), A282749 (triangle of partitions into pairwise relatively prime parts).
Row sums = A000837. See also A051424.
For ordinary partition table see A008284.

Table[Length@ Select[IntegerPartitions[n, {k}], GCD @@ # == 1 &], {n, 13}, {k, n}] // Flatten (* Michael De Vlieger, Mar 08 2017 *)

# uses[DivisorTriangle from A327029, A008284]
DivisorTriangle(moebius, A008284, 13) # Peter Luschny, Aug 24 2019

T(n, k) = Sum_{d|n} Moebius(d) * A008284(n/d, k) for n >= 1, T(0, 0) = 1. - Peter Luschny, Aug 24 2019

Corrected a(30)-a(32) and more terms from Lars Blomberg, Mar 08 2017

A303282 Numbers whose prime indices have no common divisor other than 1 but are not pairwise coprime.

Original entry on oeis.org

18, 36, 42, 45, 50, 54, 72, 75, 78, 84, 90, 98, 99, 100, 105, 108, 114, 126, 130, 135, 144, 150, 153, 156, 162, 168, 174, 175, 180, 182, 195, 196, 198, 200, 207, 210, 216, 222, 225, 228, 230, 231, 234, 242, 245, 250, 252, 258, 260, 266, 270, 275, 279, 285, 288

Offset: 1

Gus Wiseman, Apr 20 2018

nonn

A prime index of n is a number m such that prime(m) divides n. Two or more numbers are coprime if no pair of them has a common divisor other than 1.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of integer partitions whose Heinz numbers belong to this sequence begins (221), (2211), (421), (322), (331), (2221), (22111), (332), (621), (4211), (3221), (441), (522), (3311), (432), (22211).

Cf. A000837, A001222, A018783, A051424, A056239, A078374, A168532, A289508, A289509, A296150, A298748, A300486, A302569, A302696, A302796, A303138, A303139, A303140, A303283.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[400],!CoprimeQ@@primeMS[#]&&GCD@@primeMS[#]===1&]

A303708 Number of aperiodic factorizations of n using elements of A007916 (numbers that are not perfect powers).

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 3, 1, 2, 2, 0, 1, 3, 1, 3, 2, 2, 1, 4, 0, 2, 0, 3, 1, 5, 1, 0, 2, 2, 2, 3, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 0, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 0, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 0, 2, 1, 9, 2, 2, 2, 4, 1, 9, 2

Offset: 1

Gus Wiseman, Apr 29 2018

nonn

An aperiodic factorization of n is a finite multiset of positive integers greater than 1 whose product is n and whose multiplicities are relatively prime.

The positions of zeros in this sequence are the prime powers A000961.

			The a(144) = 8 aperiodic factorizations are (2*2*2*3*6), (2*2*2*18), (2*2*3*12), (2*3*24), (2*6*12), (2*72), (3*48) and (6*24). Missing from this list are (12*12), (2*2*6*6) and (2*2*2*2*3*3).

Cf. A000740, A000837, A001055, A001597, A007716, A007916, A052409, A052410, A100953, A275870, A281116, A301700, A303386, A303431, A303546, A303551, A303707, A303709, A303710.

radQ[n_]:=Or[n===1,GCD@@FactorInteger[n][[All,2]]===1];
facsr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsr[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],radQ]}]];
Table[Length[Select[facsr[n],GCD@@Length/@Split[#]===1&]],{n,100}]

a(n) = Sum_{d in A007916, d|A052409(n)} mu(d) * A303707(n^(1/d)).

A304712 Number of integer partitions of n whose parts are all equal or whose distinct parts are pairwise coprime.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 19, 25, 32, 43, 54, 70, 86, 105, 130, 162, 196, 240, 286, 339, 405, 485, 573, 674, 790, 922, 1072, 1252, 1456, 1685, 1939, 2226, 2557, 2923, 3349, 3822, 4347, 4931, 5593, 6335, 7170, 8092, 9105, 10233, 11495, 12903, 14458, 16169, 18063

Offset: 0

Gus Wiseman, May 17 2018

nonn

Two parts are coprime if they have no common divisor greater than 1.

			The a(6) = 10 partitions whose parts are all equal or whose distinct parts are pairwise coprime are (6), (51), (411), (33), (321), (3111), (222), (2211), (21111), (111111).

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A000837, A007359, A018783, A051424, A056239, A078374, A101268, A289508, A289509, A298748, A300486, A302569, A302696, A302698, A302796, A302797, A304709, A304711.

g:= proc(n, i, s) `if`(n=0, 1, `if`(i<1, 0,
      b(n, i, select(x-> x<=i, s))))
    end:
b:= proc(n, i, s) option remember; g(n, i-1, s)+(f->
     `if`(f intersect s={}, add(g(n-i*j, i-1, s union f)
        , j=1..n/i), 0))(numtheory[factorset](i))
    end:
a:= n-> g(n$2, {}):
seq(a(n), n=0..60);  # Alois P. Heinz, May 17 2018

Table[Select[IntegerPartitions[n],Or[SameQ@@#,CoprimeQ@@Union[#]]&]//Length,{n,20}]
(* Second program: *)
g[n_, i_, s_] := If[n == 0, 1, If[i < 1, 0, b[n, i, Select[s, # <= i &]]]];
b[n_, i_, s_] := b[n, i, s] = g[n, i - 1, s] + Function[f,
     If[f ~Intersection~ s == {}, Sum[g[n - i*j, i - 1, s ~Union~ f],
     {j, 1, n/i}], 0]][FactorInteger[i][[All, 1]]];
a[n_] := g[n, n, {}];
a /@ Range[0, 60] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

A305732 Heinz numbers of reducible integer partitions. Numbers n > 1 that are prime or whose prime indices are relatively prime and such that A181819(n) is already in the sequence.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Gus Wiseman, Jun 22 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A prime index of n is a number m such that prime(m) divides n. A multiset m whose distinct elements are m_1, m_2, ..., m_k with multiplicities y_1, y_2, ..., y_k is reducible if either m is of size 1 or gcd(m_1,...,m_k) = 1 and the multiset {y_1,...,y_k} is also reducible.

			60 has relatively prime prime indices {1,1,2,3} with multiplicities {1,1,2} corresponding to A181819(90) = 12. 12 has relatively prime prime indices {1,1,2} with multiplicities {1,2} corresponding to A181819(12) = 6. 6 has relatively prime prime indices {1,2} with multiplicities {1,1} corresponding to A181819(6) = 4. 4 has relatively prime prime indices {1,1} with multiplicities {2} corresponding to A181819(4) = 3. 3 is prime, so we conclude that 60 belongs to the sequence.

Cf. A000837, A007916, A056239, A181819, A182850, A289508, A289509, A298748, A304465, A304687, A304818, A305563, A305731, A305733.

rdzQ[n_]:=And[n>1,Or[PrimeQ[n],And[rdzQ[Times@@Prime/@FactorInteger[n][[All,2]]],GCD@@PrimePi/@FactorInteger[n][[All,1]]==1]]];
Select[Range[50],rdzQ]

A317085 Number of integer partitions of n whose sequence of multiplicities is a palindrome.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 6, 11, 12, 18, 16, 31, 25, 40, 47, 60, 58, 92, 85, 125, 135, 165, 173, 248, 246, 310, 351, 435, 450, 602, 608, 766, 846, 997, 1098, 1382, 1421, 1713, 1912, 2272, 2413, 2958, 3118, 3732, 4135, 4718, 5127, 6170, 6550, 7638, 8396, 9667, 10433

Offset: 0

Gus Wiseman, Jul 21 2018

nonn

			The a(10) = 18 partitions:
(ten),
(91), (82), (73), (64), (55),
(721), (631), (541), (532),
(5221), (4411), (4321), (3322),
(33211), (32221), (22222),
(1111111111).

Chai Wah Wu, Table of n, a(n) for n = 0..166
Wikipedia, Palindrome

Cf. A000041, A000837, A025065, A124010, A242414, A317086, A317087.

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

from sympy.utilities.iterables import partitions
def A317085(n):
    c = 1
    for d in partitions(n,m=n*2//3):
        l = len(d)
        if l > 0:
            k = sorted(d.keys())
            for i in range(l//2):
                if d[k[i]] != d[k[l-i-1]]:
                    break
            else:
                c += 1
    return c # Chai Wah Wu, Jun 22 2020

A317088 Number of normal integer partitions of n whose multiset of multiplicities is also normal.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 3, 4, 1, 4, 4, 5, 4, 6, 7, 9, 10, 13, 13, 15, 15, 17, 23, 22, 29, 29, 34, 36, 47, 45, 59, 60, 72, 77, 93, 95, 112, 121, 129, 149, 169, 176, 202, 228, 247, 268, 305, 334, 372, 405, 452, 496, 544, 594, 663, 724, 802

Offset: 0

Gus Wiseman, Jul 21 2018

nonn

A multiset is normal if it spans an initial interval of positive integers.

			The a(18) = 7 integer partitions are (543321), (5432211), (4433211), (4432221), (44322111), (4333221), (43322211).

Chai Wah Wu, Table of n, a(n) for n = 0..192

Cf. A000009, A000041, A000837, A055932, A296150, A317081, A317082, A317086.

normalQ[m_]:=Union[m]==Range[Max[m]];
Table[Length[Select[IntegerPartitions[n],And[normalQ[#],normalQ[Length/@Split[#]]]&]],{n,30}]

from sympy.utilities.iterables import partitions
from sympy import integer_nthroot
def A317088(n):
    if n == 0:
        return 1
    c = 0
    for d in partitions(n,k=integer_nthroot(2*n,2)[0]):
        l = len(d)
        if l > 0 and l == max(d):
            v = set(d.values())
            if len(v) == max(v):
                c += 1
    return c # Chai Wah Wu, Jun 23 2020

A319055 Maximum product of an integer partition of n with relatively prime parts.

Original entry on oeis.org

1, 1, 2, 3, 6, 6, 12, 18, 24, 36, 54, 72, 108, 162, 216, 324, 486, 648, 972, 1458, 1944, 2916, 4374, 5832, 8748, 13122, 17496, 26244, 39366, 52488, 78732, 118098, 157464, 236196, 354294, 472392, 708588, 1062882, 1417176, 2125764, 3188646, 4251528, 6377292

Offset: 1

Gus Wiseman, Sep 09 2018

nonn

After a(7), this appears to be the same as A319054.

Cf. A000740, A000792, A000793, A000837, A001414, A007916, A100953, A281116, A289508, A289509, A296302, A319054, A319057.

Table[Max[Times@@@Select[IntegerPartitions[n],GCD@@#==1&]],{n,20}]

cp's OEIS Frontend

A320813 Number of non-isomorphic multiset partitions of an aperiodic multiset of weight n such that there are no singletons and all parts are themselves aperiodic multisets.

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A328220 Number of strict integer partitions of n with no pair of consecutive parts relatively prime.

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Mathematica

A282750 Triangle read by rows: T(n,k) is the number of partitions of n into k parts x_1, x_2, ..., x_k such that gcd(x_1, x_2, ..., x_k) = 1 (where 1 <= k <= n).

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Sage

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A303282 Numbers whose prime indices have no common divisor other than 1 but are not pairwise coprime.

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Mathematica

A303708 Number of aperiodic factorizations of n using elements of A007916 (numbers that are not perfect powers).

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Mathematica

Formula

A304712 Number of integer partitions of n whose parts are all equal or whose distinct parts are pairwise coprime.

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Maple

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A305732 Heinz numbers of reducible integer partitions. Numbers n > 1 that are prime or whose prime indices are relatively prime and such that A181819(n) is already in the sequence.

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A317085 Number of integer partitions of n whose sequence of multiplicities is a palindrome.

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