A303386 -id:A303386 - cp's OEIS frontend

A317748 Irregular triangle where T(n,k) is the number of factorizations of n into factors > 1 with GCD d = A027750(n, k).

0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 2, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 3, 1, 0, 1, 0, 1, 2, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 3, 3, 0, 0, 0, 0, 0, 1, 0, 1, 1

Offset: 1

Gus Wiseman, Aug 06 2018

			Triangle begins:
   1:  0
   2:  0  1
   3:  0  1
   4:  0  1  1
   5:  0  1
   6:  1  0  0  1
   7:  0  1
   8:  0  2  0  1
   9:  0  1  1
  10:  1  0  0  1
  11:  0  1
  12:  2  1  0  0  0  1
  13:  0  1
  14:  1  0  0  1
  15:  1  0  0  1
  16:  0  3  1  0  1
  17:  0  1
  18:  2  0  1  0  0  1
  19:  0  1
  20:  2  1  0  0  0  1

Row lengths are A000005. Row sums are A001055. First column is A281116. Number of nonzero terms in each row is A317751.
Cf. A007562, A007716, A014963, A027750, A045778, A050370, A162247, A289509, A303386, A303546.
Cf. A317752, A317755, A317757.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
goc[n_,m_]:=Length[Select[facs[n],And[And@@(Divisible[#,m]&/@#),GCD@@(#/m)==1]&]];
Table[goc[n,d],{n,30},{d,Divisors[n]}]

Name edited by Peter Munn, Mar 05 2025

A319237 Positions of nonzero terms in A114592, the list of coefficients in the expansion of Product_{n > 1} (1 - 1/n^s).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 17, 18, 19, 20, 23, 24, 25, 28, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 83, 84, 88, 89, 90, 92, 97, 98, 99, 100, 101, 102

Offset: 1

Gus Wiseman, Sep 15 2018

nonn

Complement of A319238.

Cf. A001055, A045778, A114592, A162247, A190938, A281116, A281118, A303386, A316441, A319239.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Join@@Position[Table[Sum[(-1)^Length[f],{f,Select[facs[n],UnsameQ@@#&]}],{n,100}],_Integer?(Abs[#]>0&)]

A319269 Number of uniform factorizations of n into factors > 1, where a factorization is uniform if all factors appear with the same multiplicity.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 5, 1, 4, 2, 2, 2, 7, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 8, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 4, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Sep 16 2018

nonn

			The a(144) = 17 factorizations:
  (144),
  (2*72), (3*48), (4*36), (6*24), (8*18), (9*16), (12*12),
  (2*3*24), (2*4*18), (2*6*12), (2*8*9), (3*4*12), (3*6*8),
  (2*2*6*6), (2*3*4*6), (3*3*4*4).

Cf. A001055, A045778, A047966, A052409, A072774, A296132, A303386, A317710, A317712, A317717, A317718.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],SameQ@@Length/@Split[#]&]],{n,100}]

a(n) = Sum_{d|A052409(n)} A045778(n^(1/d)).

A320804 Number of non-isomorphic multiset partitions of weight n with no singletons in which all parts are aperiodic multisets.

Original entry on oeis.org

1, 0, 1, 2, 6, 13, 41, 104, 326, 958, 3096, 9958, 33869, 116806, 417741, 1526499, 5732931, 22015642, 86543717, 347495480, 1424832602, 5959123908, 25407212843, 110344848622, 487879651220, 2194697288628, 10039367091586, 46675057440634, 220447539120814

Offset: 0

Gus Wiseman, Nov 06 2018

nonn

Also the number of nonnegative integer matrices with (1) sum of elements equal to n, (2) no zero columns, (3) no rows summing to 0 or 1, and (4) no rows whose nonzero entries have a common divisor > 1, up to row and column permutations.
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 with aperiodic parts and no singletons:
  {{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},{1,2}}  {{1,2,3,3,3}}
                      {{1,2},{3,4}}  {{1,2,3,4,4}}
                      {{1,3},{2,3}}  {{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}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

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

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
S(q, t, k)={Vec(sum(j=1, #q, if(t%q[j]==0, q[j]*x^t))  + O(x*x^k), -k)}
a(n)={if(n==0, 1, my(mbt=vector(n, d, moebius(d)), s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(dirmul(mbt, sum(t=1, n, K(q, t, n)/t)) - sum(t=1, n, S(q, t, n)/t) )), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2023

Terms a(11) and beyond from Andrew Howroyd, Jan 16 2023

A303551 Number of aperiodic multisets of compositions of total weight n.

Original entry on oeis.org

1, 2, 6, 15, 41, 95, 243, 567, 1366, 3189, 7532, 17428, 40590, 93465, 215331, 493150, 1127978, 2569049, 5841442, 13240351, 29953601, 67596500, 152258270, 342235866, 767895382, 1719813753, 3845442485, 8584197657, 19133459138, 42583565928, 94641591888

Offset: 1

Gus Wiseman, Apr 26 2018

nonn

A multiset is aperiodic if its multiplicities are relatively prime.

			The a(4) = 15 aperiodic multisets of compositions are:
{4}, {31}, {22}, {211}, {13}, {121}, {112}, {1111},
{1,3}, {1,21}, {1,12}, {1,111}, {2,11},
{1,1,2}, {1,1,11}.
Missing from this list are {1,1,1,1}, {2,2}, and {11,11}.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000740, A000837, A007716, A007916, A034691, A100953, A255906, A269134, A301700, A303386, A303431, A303546, A303552.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*2^(d-1), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> add(mobius(d)*b(n/d), d=divisors(n)):
seq(a(n), n=1..35);  # Alois P. Heinz, Apr 26 2018

nn=20;
ser=Product[1/(1-x^n)^2^(n-1),{n,nn}]
Table[Sum[MoebiusMu[d]*SeriesCoefficient[ser,{x,0,n/d}],{d,Divisors[n]}],{n,1,nn}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(u=EulerT(vector(n, n, 2^(n-1)))); vector(n, n, sumdiv(n, d, moebius(d)*u[n/d]))} \\ Andrew Howroyd, Sep 15 2018

a(n) = Sum_{d|n} mu(d) * A034691(n/d).

A303710 Number of factorizations of numbers that are not perfect powers using only numbers that are not perfect powers.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 2, 3, 1, 5, 1, 2, 2, 2, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 2, 1, 9, 2, 2, 2, 4, 1, 9, 2, 3, 2, 2, 2, 6, 1, 3, 3

Offset: 1

Gus Wiseman, Apr 29 2018

nonn

Note that a factorization of a number that is not a perfect power (A007916) is always itself aperiodic, meaning the multiplicities of its factors are relatively prime.

			The a(19) = 4 factorizations of 24 are (2*2*2*3), (2*2*6), (2*12), (24).
The a(23) = 5 factorizations of 30 are (2*3*5), (2*15), (3*10), (5*6), (30).

Cf. A001055, A001597, A007716, A007916, A052409, A052410, A281116, A303365, A303386, A303707, A303708, A303709.

radQ[n_] := And[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[Divisors[n], radQ]}]]; Table[Length[facsr[n]], {n, Select[Range[100], radQ]}]

A319240 Positions of zeros in A316441, the list of coefficients in the expansion of Product_{n > 1} 1/(1 + 1/n^s).

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 25, 26, 28, 33, 34, 35, 38, 39, 44, 45, 46, 48, 49, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 108, 111, 112, 115, 116, 117, 118, 119, 121, 122

Offset: 1

Gus Wiseman, Sep 15 2018

From Tian Vlasic, Dec 31 2021: (Start)
Numbers that have an equal number of even and odd-length unordered factorizations.
There are infinitely many terms since p^2 is a term for prime p.
Out of all numbers of the form p^k with p prime (listed in A000961), only the numbers of the form p^2 (A001248) are terms.
Out of all numbers of the form p*q^k, p and q prime, only the numbers of the form p*q (A006881), p*q^2 (A054753), p*q^4 (A178739) and p*q^6 (A189987) are terms.
Similar methods can be applied to all prime signatures. (End)

			12 = 2*6 = 3*4 = 2*2*3 has an equal number of even-length factorizations and odd-length factorizations (2). - _Tian Vlasic_, Dec 09 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Complement of A319239.

Cf. A001055, A025487, A045778, A114592, A162247, A190938, A281116, A281118, A303386, A316441, A319238.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Join@@Position[Table[Sum[(-1)^Length[f],{f,facs[n]}],{n,100}],0]

A320806 Number of non-isomorphic multiset partitions of weight n in which each of the parts and each part of the dual, as well as both the multiset union of the parts and the multiset union of the dual parts, is an aperiodic multiset.

Original entry on oeis.org

1, 1, 1, 4, 10, 39, 81, 343, 903, 3223, 9989

Offset: 0

Gus Wiseman, Nov 07 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 positive entries in each row and column are relatively prime and (2) the row sums and column sums are relatively prime.
The last condition (aperiodicity of the multiset union of the dual) is equivalent to the parts having relatively prime sizes.
A multiset is aperiodic if its multiplicities are relatively prime.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
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(1) = 1 through a(4) = 10 multiset partitions:
  {{1}}  {{1},{2}}  {{1},{2,3}}    {{1},{2,3,4}}
                    {{2},{1,2}}    {{2},{1,2,2}}
                    {{1},{2},{2}}  {{3},{1,2,3}}
                    {{1},{2},{3}}  {{1},{1},{2,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303546, A303707, A303708, A316983, A320800-A320810, A321283.

A320811 Number of non-isomorphic multiset partitions with no singletons of aperiodic multisets of size n.

Original entry on oeis.org

1, 0, 1, 2, 7, 21, 57, 200, 575, 1898, 5893

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 and (2) 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) = 21 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},{2,2}}  {{1,2,3,3,3}}
                      {{1,2},{3,3}}  {{1,2,3,4,4}}
                      {{1,2},{3,4}}  {{1,2,3,4,5}}
                      {{1,3},{2,3}}  {{1,1},{1,2,2}}
                                     {{1,1},{2,2,2}}
                                     {{1,1},{2,3,3}}
                                     {{1,1},{2,3,4}}
                                     {{1,2},{1,2,2}}
                                     {{1,2},{2,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,2},{3,3,3}}
                                     {{1,2},{3,4,4}}
                                     {{1,2},{3,4,5}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,2},{1,2,2}}
                                     {{2,3},{1,2,3}}
                                     {{3,3},{1,2,3}}

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

A320812 Number of non-isomorphic aperiodic multiset partitions of weight n with no singletons.

Original entry on oeis.org

1, 0, 2, 3, 10, 23, 79, 204, 670, 1974, 6521, 21003, 71944, 248055, 888565, 3240552, 12152093, 46527471, 182337383, 729405164, 2979114723, 12407307929, 52670334237, 227725915268, 1002285201807, 4487915293675, 20434064047098, 94559526594316, 444527729321513

Offset: 0

Gus Wiseman, Nov 08 2018

nonn

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) = 2 through a(5) = 23 multiset partitions:
  {{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},{2,2}}  {{1,2,3,4,4}}
                      {{1,2},{2,2}}  {{1,2,3,4,5}}
                      {{1,2},{3,3}}  {{1,1},{1,1,1}}
                      {{1,2},{3,4}}  {{1,1},{1,2,2}}
                      {{1,3},{2,3}}  {{1,1},{2,2,2}}
                                     {{1,1},{2,3,3}}
                                     {{1,1},{2,3,4}}
                                     {{1,2},{1,2,2}}
                                     {{1,2},{2,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,2},{3,3,3}}
                                     {{1,2},{3,4,4}}
                                     {{1,2},{3,4,5}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,2},{1,2,2}}
                                     {{2,3},{1,2,3}}
                                     {{3,3},{1,2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

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

a(n) = Sum_{d|n} mu(d)*A302545(n/d) for n > 0. - Andrew Howroyd, Jan 16 2023

Terms a(11) and beyond from Andrew Howroyd, Jan 16 2023

cp's OEIS Frontend

A317748 Irregular triangle where T(n,k) is the number of factorizations of n into factors > 1 with GCD d = A027750(n, k).

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A319237 Positions of nonzero terms in A114592, the list of coefficients in the expansion of Product_{n > 1} (1 - 1/n^s).

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Mathematica

A319269 Number of uniform factorizations of n into factors > 1, where a factorization is uniform if all factors appear with the same multiplicity.

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Mathematica

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A320804 Number of non-isomorphic multiset partitions of weight n with no singletons in which all parts are aperiodic multisets.

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PARI

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A303551 Number of aperiodic multisets of compositions of total weight n.

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Maple

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A303710 Number of factorizations of numbers that are not perfect powers using only numbers that are not perfect powers.

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A319240 Positions of zeros in A316441, the list of coefficients in the expansion of Product_{n > 1} 1/(1 + 1/n^s).

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Mathematica

A320806 Number of non-isomorphic multiset partitions of weight n in which each of the parts and each part of the dual, as well as both the multiset union of the parts and the multiset union of the dual parts, is an aperiodic multiset.

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A320811 Number of non-isomorphic multiset partitions with no singletons of aperiodic multisets of size n.

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