A320424 -id:A320424 - cp's OEIS frontend

A320423 Number of set partitions of {1,...,n} where each block's elements are pairwise coprime.

1, 1, 1, 2, 2, 8, 4, 28, 18, 120, 60, 888, 252, 5220, 1860, 22224, 9552, 311088, 59616, 2473056, 565920, 13627008, 4051872, 235039392, 33805440, 1932037632, 465239808, 20604487680, 4294865664, 386228795904, 35413136640

Offset: 0

Gus Wiseman, Jan 08 2019

Two or more numbers are pairwise coprime if no pair of them has a common divisor > 1. A single number is not considered to be pairwise coprime unless it is equal to 1.

			The a(5) = 8 set partitions:
  {{1},{2,3},{4,5}}
  {{1},{2,5},{3,4}}
   {{1,2},{3,4,5}}
   {{1,4},{2,3,5}}
   {{1,2,3},{4,5}}
   {{1,2,5},{3,4}}
   {{1,3,4},{2,5}}
   {{1,4,5},{2,3}}

Cf. A000110, A051424, A084422, A085945, A186974, A187106, A302569, A302696, A303139, A303140, A320424, A320426, A320430, A320768.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],CoprimeQ@@#&],Range[n]]],{n,10}]

a(17)-a(18) from Alois P. Heinz, Jan 17 2019

a(19)-a(30) from Christian Sievers, Nov 28 2024

A320430 Number of set partitions of [n] where the elements of each non-singleton block are pairwise coprime.

Original entry on oeis.org

1, 1, 2, 5, 10, 37, 60, 295, 658, 2621, 5368, 38535, 66506, 551529, 1234264, 5004697, 13721836, 143935131, 256835337, 2971237021, 6485081140, 35162930303, 95872321543, 1315397878401, 2399236456202, 25866803180347, 72374386475590, 563368417647305, 1479943119911866

Offset: 0

Gus Wiseman, Jan 08 2019

nonn

Two or more numbers are pairwise coprime if no pair of them has a common divisor > 1.

			The a(4) = 10 set partitions: 1|2|3|4, 14|2|3, 13|2|4, 12|3|4, 1|23|4, 1|2|34, 134|2, 123|4, 14|23, 12|34.

Alois P. Heinz, Table of n, a(n) for n = 0..38

Cf. A000110, A000258, A008277, A051424, A085945, A186974, A187106, A302569, A303139, A320424, A320426, A320423, A333517.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],Length[#]==1||CoprimeQ@@#&],Range[n]]],{n,10}]

a(14)-a(15) from Alois P. Heinz, Jan 08 2019
a(16) from Alois P. Heinz, Mar 26 2020
a(17)-a(24) from Giovanni Resta, Mar 27 2020
a(25)-a(28) from Alois P. Heinz, Aug 03 2023

A320435 Regular triangle read by rows where T(n,k) is the number of relatively prime k-subsets of {1,...,n}, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 9, 10, 5, 1, 1, 11, 19, 15, 6, 1, 1, 17, 34, 35, 21, 7, 1, 1, 21, 52, 69, 56, 28, 8, 1, 1, 27, 79, 125, 126, 84, 36, 9, 1, 1, 31, 109, 205, 251, 210, 120, 45, 10, 1, 1, 41, 154, 325, 461, 462, 330, 165, 55, 11, 1, 1, 45, 196

Offset: 1

Gus Wiseman, Jan 08 2019

Two or more numbers are relatively prime if they have no common divisor > 1. A single number is not considered to be relatively prime unless it is equal to 1.

			Triangle begins:
    1
    1    1
    1    3    1
    1    5    4    1
    1    9   10    5    1
    1   11   19   15    6    1
    1   17   34   35   21    7    1
    1   21   52   69   56   28    8    1
    1   27   79  125  126   84   36    9    1
    1   31  109  205  251  210  120   45   10    1
    1   41  154  325  461  462  330  165   55   11    1
    1   45  196  479  786  923  792  495  220   66   12    1
    1   57  262  699 1281 1715 1716 1287  715  286   78   13    1
The T(6,2) = 11 sets are: {1,2}, {1,3}, {1,4}, {1,5}, {1,6}, {2,3}, {2,5}, {3,4}, {3,5}, {4,5}, {5,6}. Missing from this list are: {2,4}, {2,6}, {3,6}, {4,6}.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A085945.
Second column is A015614.
Cf. A000837, A186974, A187106, A289508,  A289509, A300486, A303139, A320424, A320436.

Table[Length[Select[Subsets[Range[n],{k}],GCD@@#==1&]],{n,10},{k,n}]

T(n,k) = sum(d=1, n\k, moebius(d)*binomial(n\d, k)) \\ Andrew Howroyd, Jan 19 2023

T(n,k) = Sum_{d=1..floor(n/k)} mu(d)*binomial(floor(n/d), k). - Andrew Howroyd, Jan 19 2023

A320439 Number of factorizations of n into factors > 1 where each factor's prime indices are relatively prime. Number of factorizations of n using elements of A289509.

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 1, 5, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 2, 0, 7, 1, 1, 1, 3, 0, 1, 0, 4, 0, 1, 0, 2, 1, 1, 0, 7, 0, 1, 1, 2, 0, 1, 1, 4, 0, 1, 0, 5, 0, 1, 0, 11, 0, 2, 0, 2, 1, 2, 0, 6, 0, 1, 1, 2, 1, 1, 0, 7, 0, 1, 0, 3, 1, 1, 0

Offset: 1

Gus Wiseman, Jan 08 2019

nonn

Also the number of multiset partitions of the multiset of prime indices of n using multisets each of which is relatively prime.
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.
Two or more numbers are relatively prime if they have no common divisor > 1. A single number is not considered to be relatively prime unless it is equal to 1.

			The a(72) = 6 factorizations are (2*2*18), (2*6*6), (2*36), (4*18), (6*12), (72).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Positions of 0's are A318978.

Cf. A000837, A001055, A007359, A085945, A289508, A289509, A302569, A302696, A320424.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facsrp[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[facsrp[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],GCD@@primeMS[#]==1&]}]];
Table[Length[facsrp[n]],{n,100}]

A320439(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d<=m)&&(1==gcd(apply(x->primepi(x), factor(d)[, 1]))), s += A320439(n/d, d))); (s)); \\ Antti Karttunen, Dec 06 2021

A320438 Irregular triangle read by rows where T(n,k) is the number of set partitions of {1,...,n} with all block-sums equal to d, where d is the k-th divisor of n*(n+1)/2 that is >= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 3, 7, 1, 1, 9, 1, 1, 1, 1, 43, 35, 1, 1, 102, 62, 1, 1, 1, 1, 68, 595, 1, 1, 17, 187, 871, 1480, 361, 1, 1, 2650, 657, 1, 1, 9294, 1, 1, 23728, 1, 1, 27763, 4110, 1, 1, 1850, 25035, 108516, 157991, 7636, 1, 1, 11421, 411474, 1

Offset: 1

Gus Wiseman, Jan 08 2019

			Triangle begins:
    1
    1
    1    1
    1    1
    1    1
    1    1
    1    4    1
    1    3    7    1
    1    9    1
    1    1
    1   43   35    1
    1  102   62    1
    1    1
    1   68  595    1
    1   17  187  871 1480  361    1
    1 2650  657    1
Row 8 counts the following set partitions:
  {{18}{27}{36}{45}}  {{1236}{48}{57}}  {{12348}{567}}  {{12345678}}
                      {{138}{246}{57}}  {{12357}{468}}
                      {{156}{237}{48}}  {{12456}{378}}
                                        {{1278}{3456}}
                                        {{1368}{2457}}
                                        {{1458}{2367}}
                                        {{1467}{2358}}

Row lengths are A164978. Row sums are A035470.

Cf. A000110, A000258, A008277, A112956, A164977, A275714, A279375, A300335, A320423, A320424, A321455, A321469.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],Total[#]==d&],Range[n]]],{n,12},{d,Select[Divisors[n*(n+1)/2],#>=n&]}]

More terms from Jinyuan Wang, Feb 27 2025

Name edited by Peter Munn, Mar 06 2025

cp's OEIS Frontend

A320423 Number of set partitions of {1,...,n} where each block's elements are pairwise coprime.

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A320430 Number of set partitions of [n] where the elements of each non-singleton block are pairwise coprime.

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A320435 Regular triangle read by rows where T(n,k) is the number of relatively prime k-subsets of {1,...,n}, 1 <= k <= n.

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A320439 Number of factorizations of n into factors > 1 where each factor's prime indices are relatively prime. Number of factorizations of n using elements of A289509.

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A320438 Irregular triangle read by rows where T(n,k) is the number of set partitions of {1,...,n} with all block-sums equal to d, where d is the k-th divisor of n*(n+1)/2 that is >= n.

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