A303707 -id:A303707 - cp's OEIS frontend

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

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]}]

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

A376268 Sorted positions of first appearances in the first differences (A053289) of perfect-powers (A001597).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 16, 17, 18, 19, 21, 23, 24, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81

Offset: 1

Gus Wiseman, Sep 28 2024

nonn

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, ...
with positions of first appearances (A376268):
  1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 16, 17, 18, 19, 21, 23, 24, 27, 28, 29, ...

These are the sorted positions of first appearances in A053289 (union A023055).
The complement is A376519.
A053707 lists first differences of consecutive prime-powers.
A333254 lists run-lengths of differences between consecutive primes.
Other families of numbers and their first differences:
For prime numbers (A000040) we have A001223.
For composite numbers (A002808) we have A073783.
For nonprime numbers (A018252) we have A065310.
For perfect powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
For squarefree numbers (A005117) we have A076259.
For nonsquarefree numbers (A013929) we have A078147.
For prime-powers inclusive (A000961) we have A057820.
For prime-powers exclusive (A246655) we have A057820(>1).
For non-prime-powers inclusive (A024619) we have A375735.
For non-prime-powers exclusive (A361102) we have A375708.
Cf. A025475, A045542, A052410, A069623, A093555, A174965, A216765, A303707, A305630, A305631, A375736.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
q=Differences[Select[Range[1000],perpowQ]];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A376519 Positions of terms not appearing for the first time in the first differences (A053289) of perfect-powers (A001597).

Original entry on oeis.org

8, 14, 15, 20, 22, 25, 26, 31, 40, 46, 52, 59, 68, 75, 88, 96, 102, 110, 111, 112, 114, 128, 136, 144, 145, 162, 180, 188, 198, 216, 226, 235, 246, 264, 265, 275, 285, 295, 305, 316, 317, 325, 328, 338, 350, 360, 367, 373, 385, 406, 416, 417, 419, 431, 443

Offset: 1

Gus Wiseman, Sep 28 2024

nonn

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, ...
with positions of latter appearances (A376519):
  8, 14, 15, 20, 22, 25, 26, 31, 40, 46, 52, 59, 68, 75, 88, 96, 102, 110, 111, ...

These are the sorted positions of latter appearances in A053289 (union A023055).
The complement is A376268.
A053707 lists first differences of consecutive prime-powers.
A333254 lists run-lengths of differences between consecutive primes.
Other families of numbers and their first differences:
For prime numbers (A000040) we have A001223.
For composite numbers (A002808) we have A073783.
For nonprime numbers (A018252) we have A065310.
For perfect powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
For squarefree numbers (A005117) we have A076259.
For nonsquarefree numbers (A013929) we have A078147.
For prime-powers inclusive (A000961) we have A057820.
For prime-powers exclusive (A246655) we have A057820(>1).
For non-prime-powers inclusive (A024619) we have A375735.
For non-prime-powers exclusive (A361102) we have A375708.
Cf. A025475, A045542, A046933, A052410, A069623, A174965, A216765, A303707, A305630, A305631, A375736.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
q=Differences[Select[Range[1000],perpowQ]];
Select[Range[Length[q]],MemberQ[Take[q,#-1],q[[#]]]&]

A320803 Number of non-isomorphic multiset partitions of weight n in which all parts are aperiodic multisets.

Original entry on oeis.org

1, 1, 3, 7, 21, 56, 174, 517, 1664, 5383, 18199, 62745, 223390, 813425, 3040181, 11620969, 45446484, 181537904, 740369798, 3079779662, 13059203150, 56406416004, 248027678362, 1109626606188, 5048119061134, 23342088591797, 109648937760252, 523036690273237

Offset: 0

Gus Wiseman, Nov 06 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(1) = 1 through a(4) = 21 multiset partitions with aperiodic parts:
  {{1}}  {{1,2}}    {{1,2,2}}      {{1,2,2,2}}
         {{1},{1}}  {{1,2,3}}      {{1,2,3,3}}
         {{1},{2}}  {{1},{2,3}}    {{1,2,3,4}}
                    {{2},{1,2}}    {{1},{1,2,2}}
                    {{1},{1},{1}}  {{1,2},{1,2}}
                    {{1},{2},{2}}  {{1},{2,3,3}}
                    {{1},{2},{3}}  {{1},{2,3,4}}
                                   {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

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

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303546, A303707, A303708, A303709, A303710, A320800-A320810.

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))}
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)))), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2023

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

A320809 Number of non-isomorphic multiset partitions of weight n in which each part and each part of the dual, as well as the multiset union of the parts, is an aperiodic multiset.

Original entry on oeis.org

1, 1, 2, 5, 13, 40, 99, 344, 985, 3302, 10583

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 column sums 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}}.
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(1) = 1 through a(4) = 13 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{2}}  {{1},{2,3}}    {{1},{2,3,4}}
                    {{2},{1,2}}    {{1,2},{3,4}}
                    {{1},{2},{2}}  {{1,3},{2,3}}
                    {{1},{2},{3}}  {{2},{1,2,2}}
                                   {{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.

A323090 Number of strict factorizations of n using elements of A007916 (numbers that are not perfect powers).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 04 2019

nonn

			The a(72) = 4 factorizations are (2*3*12), (3*24), (6*12), (72). Missing from this list and not strict are (2*2*2*3*3), (2*2*3*6), (2*6*6), (2*2*18), while missing from the list and using perfect powers are (2*36), (2*4*9), (3*4*6), (4*18), (8*9).

Positions of 0's are A246547.
Positions of 1's are A000040.
Positions of 2's are A084227.
Positions of 3's are A085986.
Positions of 4's are A143610.
Cf. A001055, A001597, A007916, A025147, A045778, A052410, A303707, A323054, A323087, A323088, A323089.

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

A304328 a(n) = n/(largest perfect power divisor of n).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 1, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 3, 1, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 1, 37, 38, 39, 5, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 2

Offset: 1

Gus Wiseman, May 10 2018

nonn

Not all terms are squarefree numbers; for example, a(500) = 4.

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

Cf. A000961, A001055, A001222, A001597, A001694, A005117, A007716, A007916, A052485, A052486, A059404, A066636, A091050, A160400, A203025, A294068, A303707, A304327, A304339.

Table[n/Last[Select[Divisors[n],#===1||GCD@@FactorInteger[#][[All,2]]>1&]],{n,100}]

a(n)={my(m=1); fordiv(n, d, if(ispower(d), m=max(m,d))); n/m} \\ Andrew Howroyd, Aug 26 2018

a(n) * A203025(n) = n.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A376268 Sorted positions of first appearances in the first differences (A053289) of perfect-powers (A001597).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A376519 Positions of terms not appearing for the first time in the first differences (A053289) of perfect-powers (A001597).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A320803 Number of non-isomorphic multiset partitions of weight n in which all parts are aperiodic multisets.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A320809 Number of non-isomorphic multiset partitions of weight n in which each part and each part of the dual, as well as the multiset union of the parts, is an aperiodic multiset.

Views

Author

Keywords

Comments

Examples

Crossrefs

A323090 Number of strict factorizations of n using elements of A007916 (numbers that are not perfect powers).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A304328 a(n) = n/(largest perfect power divisor of n).

Views

Author

Keywords