A281113 -id:A281113 - 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).

A318812 Number of total multiset partitions of the multiset of prime indices of n. Number of total factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 6, 1, 3, 1, 3, 1, 1, 1, 11, 1, 1, 2, 3, 1, 4, 1, 20, 1, 1, 1, 15, 1, 1, 1, 11, 1, 4, 1, 3, 3, 1, 1, 51, 1, 3, 1, 3, 1, 11, 1, 11, 1, 1, 1, 21, 1, 1, 3, 90, 1, 4, 1, 3, 1, 4, 1, 80, 1, 1, 3, 3, 1, 4, 1, 51, 6, 1, 1

Offset: 1

Gus Wiseman, Sep 04 2018

nonn

A total multiset partition of m is either m itself or a total multiset partition of a multiset partition of m that is neither minimal nor maximal.

a(n) depends only on the prime signature of n. - Andrew Howroyd, Dec 30 2019

			The a(24) = 11 total multiset partitions:
  {1,1,1,2}
  {{1},{1,1,2}}
  {{2},{1,1,1}}
  {{1,1},{1,2}}
  {{1},{1},{1,2}}
  {{1},{2},{1,1}}
  {{{1}},{{1},{1,2}}}
  {{{1}},{{2},{1,1}}}
  {{{2}},{{1},{1,1}}}
  {{{1,2}},{{1},{1}}}
  {{{1,1}},{{1},{2}}}
The a(24) = 11 total factorizations:
  24,
  (2*12), (3*8), (4*6),
  (2*2*6), (2*3*4),
  ((2)*(2*6)), ((6)*(2*2)), ((2)*(3*4)), ((3)*(2*4)), ((4)*(2*3)).

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000110, A001055, A002846, A005121, A213427, A281113, A281118, A281119, A317145, A318813.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
totfac[n_]:=1+Sum[totfac[Times@@Prime/@f],{f,Select[facs[n],1

MultEulerT(u)={my(v=vector(#u)); v[1]=1; for(k=2, #u, forstep(j=#v\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=binomial(e+u[k]-1, e)*v[i]))); v}
seq(n)={my(v=vector(n, i, isprime(i)), u=vector(n), m=logint(n,2)+1); for(r=1, m, u += v*sum(j=r, m, (-1)^(j-r)*binomial(j-1, r-1)); v=MultEulerT(v)); u[1]=1; u} \\ Andrew Howroyd, Dec 30 2019

a(product of n distinct primes) = A005121(n).

a(prime^n) = A318813(n).

A317145 Number of maximal chains of factorizations of n into factors > 1, ordered by refinement.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 1, 7, 1, 1, 1, 5, 1, 3, 1, 2, 2, 1, 1, 15, 1, 2, 1, 2, 1, 5, 1, 5, 1, 1, 1, 11, 1, 1, 2, 11, 1, 3, 1, 2, 1, 3, 1, 26, 1, 1, 2, 2, 1, 3, 1, 15, 2, 1, 1, 11, 1, 1, 1, 5, 1, 11, 1, 2, 1, 1, 1, 52, 1, 2, 2, 7, 1, 3, 1, 5, 3

Offset: 1

Gus Wiseman, Jul 22 2018

nonn

If x and y are factorizations of the same integer and it is possible to produce x by further factoring the factors of y, flattening, and sorting, then x <= y.

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Oct 08 2018

			The a(36) = 7 maximal chains:
  (2*2*3*3) < (2*2*9) < (2*18) < (36)
  (2*2*3*3) < (2*2*9) < (4*9)  < (36)
  (2*2*3*3) < (2*3*6) < (2*18) < (36)
  (2*2*3*3) < (2*3*6) < (3*12) < (36)
  (2*2*3*3) < (2*3*6) < (6*6)  < (36)
  (2*2*3*3) < (3*3*4) < (3*12) < (36)
  (2*2*3*3) < (3*3*4) < (4*9)  < (36)

Antti Karttunen, Table of n, a(n) for n = 1..11520
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A002846, A007716, A045778, A064988, A162247, A213427, A275024, A281113, A299202, A300385, A317144, A317146, A320105.

A064988(n) = { my(f = factor(n)); for (k=1, #f~, f[k, 1] = prime(f[k, 1]); ); factorback(f); }; \\ From A064988
memoA320105 = Map();
A320105(n) = if(bigomega(n)<=2,1,if(mapisdefined(memoA320105,n), mapget(memoA320105,n), my(f=factor(n), u = #f~, s = 0); for(i=1,u,for(j=i+(1==f[i,2]),u, s += A320105(prime(primepi(f[i,1])*primepi(f[j,1]))*(n/(f[i,1]*f[j,1]))))); mapput(memoA320105,n,s); (s)));
A317145(n) = A320105(A064988(n)); \\ Antti Karttunen, Oct 08 2018

a(prime^n) = A002846(n).

a(n) = A320105(A064988(n)). - Antti Karttunen, Oct 08 2018

Data section extended to 105 terms by Antti Karttunen, Oct 08 2018

A318949 Number of ways to write n as an orderless product of orderless sums.

Original entry on oeis.org

1, 2, 3, 8, 7, 17, 15, 36, 36, 56, 56, 123, 101, 165, 197, 310, 297, 490, 490, 767, 837, 1114, 1255, 1925, 1986, 2638, 3110, 4108, 4565, 6201, 6842, 9043, 10311, 12904, 14988, 19398, 21637, 26995, 31488, 39180, 44583, 55418, 63261, 77627, 89914, 108068, 124754

Offset: 1

Gus Wiseman, Sep 05 2018

nonn

			The a(6) = 17 ways:
  (6)              (2)*(3)
  (3+3)            (2)*(2+1)
  (4+2)            (2)*(1+1+1)
  (5+1)            (1+1)*(3)
  (2+2+2)          (1+1)*(2+1)
  (3+2+1)          (1+1)*(1+1+1)
  (4+1+1)
  (2+2+1+1)
  (3+1+1+1)
  (2+1+1+1+1)
  (1+1+1+1+1+1)

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000041, A001055, A001970, A063834, A065026, A066739, A066815, A281113, A284639, A318948.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
prodsums[n_]:=Union[Sort/@Join@@Table[Tuples[IntegerPartitions/@fac],{fac,facs[n]}]];
Table[Length[prodsums[n]],{n,30}]

MultEulerT(u)={my(v=vector(#u)); v[1]=1; for(k=2, #u, forstep(j=#v\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=binomial(e+u[k]-1, e)*v[i]))); v}
seq(n)={MultEulerT(vector(n, n, numbpart(n)))} \\ Andrew Howroyd, Oct 26 2019

Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^p(k), where p(k) = number of partitions of k (A000041). - Ilya Gutkovskiy, Oct 26 2019

A302505 Numbers whose prime indices are squarefree and have disjoint prime indices.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 15, 16, 17, 20, 22, 24, 26, 29, 30, 31, 32, 33, 34, 40, 41, 43, 44, 47, 48, 51, 52, 55, 58, 59, 60, 62, 64, 66, 67, 68, 73, 79, 80, 82, 83, 85, 86, 88, 93, 94, 96, 101, 102, 104, 109, 110, 113, 116, 118, 120, 123, 124, 127

Offset: 1

Gus Wiseman, Apr 09 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set multisystems.
01: {}
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
06: {{},{1}}
08: {{},{},{}}
10: {{},{2}}
11: {{3}}
12: {{},{},{1}}
13: {{1,2}}
15: {{1},{2}}
16: {{},{},{},{}}
17: {{4}}
20: {{},{},{2}}
22: {{},{3}}
24: {{},{},{},{1}}
26: {{},{1,2}}
29: {{1,3}}
30: {{},{1},{2}}
31: {{5}}
32: {{},{},{},{},{}}

Cf. A000961, A001222, A003963, A005117, A007716, A056239, A275024, A279790, A281113, A294788, A301750, A302242, A302243.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Join@@primeMS/@primeMS[#]&]

A317144 Number of refinement-ordered pairs of factorizations of n into factors > 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 6, 3, 3, 1, 9, 1, 3, 3, 14, 1, 9, 1, 9, 3, 3, 1, 23, 3, 3, 6, 9, 1, 12, 1, 26, 3, 3, 3, 31, 1, 3, 3, 23, 1, 12, 1, 9, 9, 3, 1, 56, 3, 9, 3, 9, 1, 23, 3, 23, 3, 3, 1, 41, 1, 3, 9, 55, 3, 12, 1, 9, 3, 12, 1, 82, 1, 3, 9, 9, 3, 12, 1, 56, 14

Offset: 1

Gus Wiseman, Jul 22 2018

nonn

If x and y are factorizations of the same integer and it is possible to produce x by further factoring the factors of y, flattening, and sorting, then x <= y.

As this is a sequence computed from exponents in factorization of n, distinct values of a(n) in this sequence can be found by computing a(A025487(k)) for k >= 0. - David A. Corneth, Jul 30 2018

			The a(12) = 9 ordered pairs:
  (2*2*3) <= (12)
  (2*2*3) <= (2*6)
  (2*2*3) <= (3*4)
  (2*2*3) <= (2*2*3)
    (2*6) <= (12)
    (2*6) <= (2*6)
    (3*4) <= (12)
    (3*4) <= (3*4)
     (12) <= (12)

Cf. A000005, A001055, A007716, A025487, A045778, A162247, A281113, A317142, A317145, A317146.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
faccaps[fac_]:=Union[Sort/@Apply[Times,mps[fac],{2}]];
Table[Sum[Length[faccaps[fac]],{fac,facs[n]}],{n,100}]

a(n) >= A001055(n) + floor(A000005(n) / 2) - 1. - David A. Corneth, Jul 30 2018

A339564 Number of ways to choose a distinct factor in a factorization of n (pointed factorizations).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 3, 3, 7, 1, 7, 1, 7, 3, 3, 1, 14, 2, 3, 4, 7, 1, 10, 1, 12, 3, 3, 3, 17, 1, 3, 3, 14, 1, 10, 1, 7, 7, 3, 1, 26, 2, 7, 3, 7, 1, 14, 3, 14, 3, 3, 1, 25, 1, 3, 7, 19, 3, 10, 1, 7, 3, 10, 1, 36, 1, 3, 7, 7, 3, 10, 1, 26, 7, 3

Offset: 1

Gus Wiseman, Apr 10 2021

nonn

			The pointed factorizations of n for n = 2, 4, 6, 8, 12, 24, 30:
  ((2))  ((4))    ((6))    ((8))      ((12))     ((24))       ((30))
         ((2)*2)  ((2)*3)  ((2)*4)    ((2)*6)    ((3)*8)      ((5)*6)
                  (2*(3))  (2*(4))    (2*(6))    (3*(8))      (5*(6))
                           ((2)*2*2)  ((3)*4)    ((4)*6)      ((2)*15)
                                      (3*(4))    (4*(6))      (2*(15))
                                      ((2)*2*3)  ((2)*12)     ((3)*10)
                                      (2*2*(3))  (2*(12))     (3*(10))
                                                 ((2)*2*6)    ((2)*3*5)
                                                 (2*2*(6))    (2*(3)*5)
                                                 ((2)*3*4)    (2*3*(5))
                                                 (2*(3)*4)
                                                 (2*3*(4))
                                                 ((2)*2*2*3)
                                                 (2*2*2*(3))

The additive version is A000070 (strict: A015723).
The unpointed version is A001055 (strict: A045778, ordered: A074206, listed: A162247).
Allowing point (1) gives A057567.
Choosing a position instead of value gives A066637.
The ordered additive version is A336875.
A000005 counts divisors.
A001787 count normal multisets with a selected position.
A001792 counts compositions with a selected position.
A006128 counts partitions with a selected position.
A066186 count strongly normal multisets with a selected position.
A254577 counts ordered factorizations with a selected position.
Cf. A007716, A050336, A281113, A281116, A292886, A293627.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Length[Union[fac]],{fac,facs[n]}],{n,50}]

a(n) = A057567(n) - A001055(n).

a(n) = Sum_{d|n, d>1} A001055(n/d).

A302590 Squarefree numbers whose prime indices are prime numbers.

Original entry on oeis.org

1, 3, 5, 11, 15, 17, 31, 33, 41, 51, 55, 59, 67, 83, 85, 93, 109, 123, 127, 155, 157, 165, 177, 179, 187, 191, 201, 205, 211, 241, 249, 255, 277, 283, 295, 327, 331, 335, 341, 353, 367, 381, 401, 415, 431, 451, 461, 465, 471, 509, 527, 537, 545, 547, 561, 563

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

A prime index of n is a number m such that prime(m) divides n.
From David A. Corneth, Feb 05 2021: (Start)
Product_{p in A006450} (p + 1)/p where primepi(p) <= 10^k for k = 3..9 respectively is
3221793975627545730894469494385382768...
3962097386916566795581118542505513350...
4423525010102788492232765893521739629...
4739349879225654126399615785205666552...
4969363158706022367680967716958174889...
5144436325229538304870684054018856517...
5282263225826916578696019016723107071... (End)

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.
001: {}
003: {{1}}
005: {{2}}
011: {{3}}
015: {{1},{2}}
017: {{4}}
031: {{5}}
033: {{1},{3}}
041: {{6}}
051: {{1},{4}}
055: {{2},{3}}
059: {{7}}
067: {{8}}
083: {{9}}
085: {{2},{4}}
093: {{1},{5}}
109: {{10}}
123: {{1},{6}}
127: {{11}}
155: {{2},{5}}
157: {{12}}
165: {{1},{2},{3}}

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

Cf. A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A281113, A302242, A302243, A302568.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[600],SquareFreeQ[#]&&And@@PrimeQ/@primeMS[#]&]

ok(n)={issquarefree(n) && !#select(p->!isprime(primepi(p)), factor(n)[,1])} \\ Andrew Howroyd, Aug 26 2018

Intersection of A005117 and A076610.

Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + 1/p) converges since the sum of the reciprocals of A006450 converges. - Amiram Eldar, Feb 02 2021

A302540 Numbers whose prime indices other than 1 are prime numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 17, 18, 20, 22, 24, 25, 27, 30, 31, 32, 33, 34, 36, 40, 41, 44, 45, 48, 50, 51, 54, 55, 59, 60, 62, 64, 66, 67, 68, 72, 75, 80, 81, 82, 83, 85, 88, 90, 93, 96, 99, 100, 102, 108, 109, 110, 118, 120, 121, 123, 124

Offset: 1

Gus Wiseman, Apr 09 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

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

Cf. A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A281113, A291686, A302242, A302243, A302534, A302539.

Select[Range[400],#===1||And@@(#===1||PrimeQ[#]&)/@PrimePi/@FactorInteger[#][[All,1]]&]

ok(n)={!#select(p->p>2 && !isprime(primepi(p)), factor(n)[,1])} \\ Andrew Howroyd, Aug 26 2018

Sum_{n>=1} 1/a(n) = 2 * Sum_{n>=1} 1/A076610(n) = 2 * Product_{p in A006450} p/(p-1) converges since the sum of the reciprocals of A006450 converges. - Amiram Eldar, Feb 02 2021

A296120 Number of ways to choose a strict factorization of each factor in a strict factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 13, 1, 3, 3, 6, 1, 12, 1, 7, 3, 3, 3, 14, 1, 3, 3, 13, 1, 12, 1, 6, 6, 3, 1, 25, 1, 6, 3, 6, 1, 13, 3, 13, 3, 3, 1, 31, 1, 3, 6, 11, 3, 12, 1, 6, 3, 12, 1, 36, 1, 3, 6, 6, 3, 12, 1, 25, 4, 3

Offset: 1

Gus Wiseman, Dec 05 2017

nonn

			The a(36) = 14 twice-factorizations:
(36), (4*9), (3*12), (2*18), (2*3*6),
(4)*(9), (3)*(12), (3)*(3*4), (3)*(2*6), (2)*(18), (2)*(3*6), (2)*(2*9),
(2)*(3)*(6), (2)*(3)*(2*3).

Cf. A000009, A005117, A045778, A050345, A279785, A281113, A294788, A296118, A296119, A296121.

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

Dirichlet g.f.: Product_{n > 1}(1 + A045778(n)/n^s).

cp's OEIS Frontend

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

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A318812 Number of total multiset partitions of the multiset of prime indices of n. Number of total factorizations of n.

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A317145 Number of maximal chains of factorizations of n into factors > 1, ordered by refinement.

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A318949 Number of ways to write n as an orderless product of orderless sums.

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A302505 Numbers whose prime indices are squarefree and have disjoint prime indices.

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A317144 Number of refinement-ordered pairs of factorizations of n into factors > 1.

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A339564 Number of ways to choose a distinct factor in a factorization of n (pointed factorizations).

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A302590 Squarefree numbers whose prime indices are prime numbers.

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