A303709 -id:A303709 - cp's OEIS frontend

A303707 Number of factorizations of n using elements of A007916 (numbers that are not perfect powers).

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

Offset: 1

Gus Wiseman, Apr 29 2018

nonn

First differs from A081707 at a(60) = 9, A081707(60) = 8.

			The a(60) = 9 factorizations are (2*2*3*5), (2*2*15), (2*3*10), (2*5*6), (2*30), (3*20), (5*12), (6*10), (60).

Cf. A000837, A001055, A001597, A007716, A007916, A045778, A052409, A052410, A162247, A281113, A281116, A303708, 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[facsr[n]],{n,100}]

Dirichlet g.f.: Product_{n in A007916} 1/(1 - n^s).

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)).

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]}]

A304250 Perfect powers whose prime factors span an initial interval of prime numbers.

Original entry on oeis.org

4, 8, 16, 32, 36, 64, 128, 144, 216, 256, 324, 512, 576, 900, 1024, 1296, 1728, 2048, 2304, 2916, 3600, 4096, 5184, 5832, 7776, 8100, 8192, 9216, 11664, 13824, 14400, 16384, 20736, 22500, 26244, 27000, 32400, 32768, 36864, 44100, 46656, 57600, 65536, 72900

Offset: 1

Gus Wiseman, May 13 2018

nonn

The multiset of prime indices of a(n) is the a(n)-th row of A112798. This multiset is normal, meaning it spans an initial interval of positive integers, and periodic, meaning its multiplicities have a common divisor greater than 1.

			Sequence of all normal periodic multisets begins
4:    {1,1}
8:    {1,1,1}
16:   {1,1,1,1}
32:   {1,1,1,1,1}
36:   {1,1,2,2}
64:   {1,1,1,1,1,1}
128:  {1,1,1,1,1,1,1}
144:  {1,1,1,1,2,2}
216:  {1,1,1,2,2,2}
256:  {1,1,1,1,1,1,1,1}
324:  {1,1,2,2,2,2}
512:  {1,1,1,1,1,1,1,1,1}
576:  {1,1,1,1,1,1,2,2}
900:  {1,1,2,2,3,3}
1024: {1,1,1,1,1,1,1,1,1,1}

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000837, A000961 A001597, A007916, A018783, A052409, A052410, A055932, A056239, A112798, A178472, A303431, A303709, A304450, A368682.

Select[Range[1000],FactorInteger[#][[-1,1]]==Prime[Length[FactorInteger[#]]]&&GCD@@FactorInteger[#][[All,2]]>1&]

Intersection of A001597 and A055932.

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

A303553 Number of periodic factorizations of n > 1 into positive factors greater than 1; a(1) = 1 by convention.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1

Offset: 1

Gus Wiseman, Apr 26 2018

nonn

A multiset is periodic if its multiplicities have a common divisor greater than 1.

			The a(64)  = 4 periodic factorizations are (2*2*2*2*2*2), (2*2*4*4), (4*4*4), (8*8).
The a(144) = 4 periodic factorizations are (2*2*2*2*3*3), (2*2*6*6), (3*3*4*4), (12*12).
The a(256) = 5 periodic factorizations are (2*2*2*2*2*2*2*2), (2*2*2*2*4*4), (2*2*8*8), (4*4*4*4), (16*16).
The a(576) = 7 periodic factorizations are (2*2*2*2*2*2*3*3), (2*2*2*2*6*6), (2*2*3*3*4*4), (2*2*12*12), (3*3*8*8), (4*4*6*6), (24*24).

Antti Karttunen, Table of n, a(n) for n = 1..20736
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A000740, A000837, A001055, A007916, A018783(n) = a(2^n), A052409, A052410, A275870, A303386, A303431, A303547, A303552, A303709.

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],GCD@@Length/@Split[#]>1&]],{n,2,100}]

gcd_of_multiplicities(lista) = { my(u=length(lista)); if(u<2, u, my(g=0, pe = lista[1], j=1); for(i=2,u,if(lista[i]==pe, j++, g = gcd(j,g); j=1; pe = lista[i])); gcd(g,j)); }; \\ the supplied lista (newfacs) should be monotonic
A303553(n, m=n, facs=List([])) = if(1==n, (gcd_of_multiplicities(facs)!=1), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A303553(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Dec 06 2018

a(n) >= A303709(n). - Antti Karttunen, Dec 06 2018

a(1) = 1 prepended by Antti Karttunen, Dec 06 2018

A304648 Number of different periodic multisets that fit within some normal multiset of weight n.

Original entry on oeis.org

0, 1, 3, 7, 13, 25, 44, 78, 136, 242, 422, 747, 1314, 2326, 4121, 7338, 13052, 23288, 41568, 74329, 133011, 238338, 427278, 766652, 1376258, 2472012, 4441916, 7984990, 14358424, 25826779, 46465956, 83616962, 150497816, 270917035, 487753034, 878244512

Offset: 1

Gus Wiseman, May 15 2018

nonn

A multiset is normal if it spans an initial interval of positive integers. It is periodic if its multiplicities have a common divisor greater than 1.

			The a(5) = 13 periodic multisets:
(11), (22), (33), (44),
(111), (222), (333),
(1111), (1122), (1133), (2222), (2233),
(11111).

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000217, A001597, A018783, A027941, A178472, A210554, A303547, A303709, A303974, A303976.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]>1&]],{n,10}]

seq(n)=Vec(sum(d=2, n, -moebius(d)*x^d/(1 - x - x^d*(2-x)) + O(x*x^n))/(1-x), -n) \\ Andrew Howroyd, Feb 04 2021

From Andrew Howroyd, Feb 04 2021: (Start)
a(n) = A027941(n) - A303976(n).
G.f.: Sum_{d>=2} -mu(d)*x^d/((1 - x - x^d*(2-x))*(1-x)).
(End)

a(12)-a(13) from Robert Price, Sep 15 2018

Terms a(14) and beyond from Andrew Howroyd, Feb 04 2021

cp's OEIS Frontend

A303707 Number of factorizations of n using elements of A007916 (numbers that are not perfect powers).

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

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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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A304250 Perfect powers whose prime factors span an initial interval of prime numbers.

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

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A303553 Number of periodic factorizations of n > 1 into positive factors greater than 1; a(1) = 1 by convention.

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A304648 Number of different periodic multisets that fit within some normal multiset of weight n.

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