A103775 -id:A103775 - cp's OEIS frontend

A050326 Number of factorizations of n into distinct squarefree numbers > 1.

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 5, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 5, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 4, 1, 2, 1, 0, 2, 5, 1, 1, 2, 5, 1, 0, 1, 2, 1, 1, 2, 5, 1, 0, 0, 2, 1, 4, 2, 2, 2, 0, 1, 4, 2, 1, 2, 2, 2, 0, 1, 1, 1, 1, 1, 5, 1

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).
a(A212164(n)) = 0; a(A212166(n)) = 1; a(A006881(n)) = 2; a(A190107(n)) = 3; a(A085987(n)) = 4; a(A225228(n)) = 5; a(A179670(n)) = 7; a(A162143(n)) = 8; a(A190108(n)) = 11; a(A212167(n)) > 0; a(A212168(n)) > 1. - Reinhard Zumkeller, May 03 2013
The comment that a(A212164(n)) = 0 is incorrect. For example, 3600 belongs to A212164 but a(3600) = 1. The positions of zeros in this sequence are A293243. - Gus Wiseman, Oct 10 2017

			The a(30) = 5 factorizations are: 2*3*5, 2*15, 3*10, 5*6, 30. The a(180) = 5 factorizations are: 2*3*5*6, 2*3*30, 2*6*15, 3*6*10, 6*30. - _Gus Wiseman_, Oct 10 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001055, A005117, A045778, A046523, A050320, A050327, a(p^k)=0 (p>1), a(A002110)=A000110, a(n!)=A103775(n), A206778, A293243.

import Data.List (subsequences, genericIndex)
a050326 n = genericIndex a050326_list (n-1)
a050326_list = 1 : f 2 where
   f x = (if x /= s then a050326 s
                    else length $ filter (== x) $ map product $
                         subsequences $ tail $ a206778_row x) : f (x + 1)
         where s = a046523 x
-- Reinhard Zumkeller, May 03 2013

N:= 1000: # to get a(1)..a(N)
A:= Vector(N):
A[1]:= 1:
for n from 2 to N do
  if numtheory:-issqrfree(n) then
     S:= [$1..N/n]; T:= n*S; A[T]:= A[T]+A[S]
    fi;
od:
convert(A,list); # Robert Israel, Oct 10 2017

sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[sqfacs[n]],{n,100}] (* Gus Wiseman, Oct 10 2017 *)

Dirichlet g.f.: prod{n is squarefree and > 1}(1+1/n^s).

a(n) = A050327(A101296(n)). - R. J. Mathar, May 26 2017

A103774 Number of ways to write n! as product of squarefree numbers.

Original entry on oeis.org

1, 1, 2, 2, 6, 10, 42, 42, 82, 204, 1196, 1556, 10324, 34668, 104948, 104964, 873540, 1309396, 11855027, 25238220, 91193575, 453628255, 5002616219, 5902762219, 21142729523, 122981607092, 189706055368, 547296181656, 7291700021313, 14330422534833, 202498591157970

Offset: 1

Reinhard Zumkeller, Feb 15 2005

nonn

a(n) = A050320(A000142(n)).
From Gus Wiseman, Aug 20 2020: (Start)
Also the number of set multipartitions (multisets of sets) of the multiset of prime factors of n!. For example, The a(2) = 1 through a(6) = 10 set multipartitions are:
  {1}  {12}    {1}{1}{12}    {1}{1}{123}      {1}{1}{12}{123}
       {1}{2}  {1}{1}{1}{2}  {1}{12}{13}      {1}{12}{12}{13}
                             {1}{1}{1}{23}    {1}{1}{1}{12}{23}
                             {1}{1}{2}{13}    {1}{1}{1}{2}{123}
                             {1}{1}{3}{12}    {1}{1}{2}{12}{13}
                             {1}{1}{1}{2}{3}  {1}{1}{3}{12}{12}
                                              {1}{1}{1}{1}{2}{23}
                                              {1}{1}{1}{2}{2}{13}
                                              {1}{1}{1}{2}{3}{12}
                                              {1}{1}{1}{1}{2}{2}{3}
(End)

			n=5, 5! = 1*2*3*4*5 = 120 = 2 * 2 * 2 * 3 * 5: a(5)=#{2*2*2*3*5,2*2*2*15,2*2*6*5,2*2*30,2*2*3*10,2*6*10}=6.

Index entries for sequences related to factorial numbers.

A103775 is the strict case.
A157612 is the case of superprimorials.
A001055 counts factorizations.
A045778 counts strict factorizations.
A048656 counts squarefree divisors of factorials.
A050320 counts factorizations into squarefree numbers.
A050326 counts strict factorizations into squarefree numbers.
A076716 counts factorizations of factorials.
A089259 counts set multipartitions of integer partitions.
A116540 counts normal set multipartitions.
A157612 counts strict factorizations of factorials.
Cf. A000110, A005117, A008480, A124010, A318360.
Factorial numbers: A000142, A007489, A022559, A027423, A071626, A325272, A325617, A336498.

sub[w_, e_] := Block[{v=w}, v[[e]]--; v]; ric[w_, k_] := ric[w, k] = If[Max[w] == 0, 1, Block[{e, s, p = Flatten@ Position[Sign@w, 1]}, s = Select[ Prepend[#, First@p] & /@ Subsets[Rest@p], Total[1/2^#] <= k &]; Sum[ric[sub[w, e], Total[1/2^e]], {e, s}]]]; a[n_] := ric[ Sort[ Last /@ FactorInteger[n!]], 1]; Array[a, 22] (* Giovanni Resta, Sep 30 2019 *)

a(17)-a(18) from Amiram Eldar, Sep 30 2019

a(19)-a(31) from Giovanni Resta, Sep 30 2019

A294619 a(0) = 0, a(1) = 1, a(2) = 2 and a(n) = 1 for n > 2.

Original entry on oeis.org

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

Offset: 0

Franck Maminirina Ramaharo, Nov 05 2017

Continued fraction expansion of (sqrt(5) + 1)/(2*sqrt(5)).
Inverse binomial transform is {0, 1, 4, 10, 21, 41, 78, 148, ...}, A132925 with one leading zero.
Also the main diagonal in the expansion of (1 + x)^n - 1 + x^2 (A300453).
The partial sum of this sequence is A184985.
a(n) is the number of state diagrams having n components that are obtained from an n-foil [(2,n)-torus knot] shadow. Let a shadow diagram be the regular projection of a mathematical knot into the plane, where the under/over information at every crossing is omitted. A state for the shadow diagram is a diagram obtained by merging either of the opposite areas surrounding each crossing.
a(n) satisfies the identities a(n)^a(n+k) = a(n), 2^a(k) = 2*a(k) and a(k)! = a(k), k > 0.
Also the number of non-isomorphic simple connected undirected graphs with n+1 edges and a longest path of length 2. - Nathaniel Gregg, Nov 02 2021

			For n = 2, the shadow of the Hopf link yields 2 two-component state diagrams (see example in A300453). Thus a(2) = 2.

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.
L. H. Kauffman, Knots and Physics, World Scientific Publishers, 1991.
V. Manturov, Knot Theory, CRC Press, 2004.

I. Altintas, An oriented state model for the Jones polynomial and its applications to alternating links, Appl. Math. Comput. 194 (2007) 168-178.
J. A. Baldwin and A. S. Levine, A combinatorial spanning tree model for knot Floer homology, Advances in Mathematics, Vol. 231 (2012), 1886-1939.
A. Banerjee, Knot theory [Foil knot family].
D. Denton and P. Doyle, Shadow movies not arising from knots, arXiv preprint, arXiv:1106.3545 [math.GT], 2011.
L. H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000007, A000045, A004278, A005096, A005803, A010701, A063524, A103775, A107583, A130130, A132925, A141044, A157928, A158411, A163985, A171386, A179184, A184985, A185012, A300453.

CoefficientList[Series[(x + x^2 - x^3)/(1 - x), {x, 0, 100}], x] (* Wesley Ivan Hurt, Nov 05 2017 *)
f[n_] := If[n > 2, 1, n]; Array[f, 105, 0] (* Robert G. Wilson v, Dec 27 2017 *)
PadRight[{0,1,2},120,{1}] (* Harvey P. Dale, Feb 20 2023 *)

makelist((1 + (-1)^((n + 1)!))/2 + kron_delta(n, 2), n, 0, 100);

PARI
```
a(n) = if(n>2, 1, n);
```

a(n) = ((-1)^2^(n^2 + 3*n + 2) + (-1)^2^(n^2 - n) - (-1)^2^(n^2 - 3*n + 2) + 1)/2.
a(n) = (1 + (-1)^((n + 1)!))/2 + Kronecker(n, 2).
a(n) = min(n, 3) - 2*(max(n - 2, 0) - max(n - 3, 0)).
a(n) = floor(F(n+1)/F(n)) for n > 0, with a(0) = 0, where F(n) = A000045(n) is the n-th Fibonacci number.
a(n) = a(n-1) for n > 3, with a(0) = 0, a(1) = 1, a(2) = 2 and a(3) = 1.
A005803(a(n)) = A005096(a(n)) = A000007(n).
A107583(a(n)) = A103775(n+5).
a(n+1) = 2^A185012(n+1), with a(0) = 0.
a(n) = A163985(n) mod A004278(n+1).
a(n) = A157928(n) + A171386(n+1).
a(n) = A063524(n) + A157928(n) + A185012(n).
a(n) = A010701(n) - A141044(n) - A179184(n).
G.f.: (x + x^2 - x^3)/(1 - x).
E.g.f.: (2*exp(x) - 2 + x^2)/2.

A337073 Number of strict factorizations of the superprimorial A006939(n) into squarefree numbers > 1.

Original entry on oeis.org

1, 1, 1, 2, 14, 422, 59433, 43181280, 178025660042, 4550598470020490, 782250333882971717562, 974196106965358319940100513, 9412280190038329162111356578977100, 751537739224674099813783040471383322758327

Offset: 0

Gus Wiseman, Aug 15 2020

The n-th superprimorial is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1). It has n! divisors.

Also the number of strict set multipartitions (sets of sets) of the multiset of prime factors of the superprimorial A006939(n).

			The a(1) = 1 through a(3) = 10 factorizations:
    2  2*6  2*6*30    2*6*30*210
            2*3*6*10  6*10*30*42
                      2*3*6*30*70
                      2*5*6*30*42
                      2*3*10*30*42
                      2*3*6*10*210
                      2*6*10*15*42
                      2*6*10*21*30
                      2*6*14*15*30
                      3*6*10*14*30
                      2*3*5*6*10*42
                      2*3*5*6*14*30
                      2*3*6*7*10*30
                      2*3*6*10*14*15
The a(1) = 1 through a(3) = 14 set multipartitions:
    {1}  {1}{12}  {1}{12}{123}    {1}{12}{123}{1234}
                  {1}{2}{12}{13}  {12}{13}{123}{124}
                                  {1}{12}{13}{23}{124}
                                  {1}{12}{13}{24}{123}
                                  {1}{12}{14}{23}{123}
                                  {1}{2}{12}{123}{134}
                                  {1}{2}{12}{13}{1234}
                                  {1}{2}{13}{123}{124}
                                  {1}{3}{12}{123}{124}
                                  {2}{12}{13}{14}{123}
                                  {1}{2}{12}{13}{14}{23}
                                  {1}{2}{12}{4}{13}{123}
                                  {1}{2}{3}{12}{13}{124}
                                  {1}{2}{3}{12}{14}{123}

A000142 counts divisors of superprimorials.
A022915 counts permutations of the same multiset.
A103775 is the version for factorials instead of superprimorials.
A337072 is the non-strict version.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A045778 counts strict factorizations.
A050320 counts factorizations into squarefree numbers.
A050326 counts strict factorizations into squarefree numbers.
A050342 counts strict set multipartitions of integer partitions.
A076954 can be used instead of A006939 (cf. A307895, A325337).
A283877 counts non-isomorphic strict set multipartitions.
A317829 counts factorizations of superprimorials.
A337069 counts strict factorizations of superprimorials.
Cf. A000178, A002110, A022559, A103774, A124010, A181818, A318361, A336417, A337070.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
ystfac[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[ystfac[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[ystfac[chern[n]]],{n,0,4}]

\\ See A318361 for count.
a(n) = {if(n==0, 1, count(vector(n, i, i)))} \\ Andrew Howroyd, Sep 01 2020

a(n) = A050326(A006939(n)).

a(n) = A318361(A002110(n)). - Andrew Howroyd, Sep 01 2020

a(7)-a(13) from Andrew Howroyd, Sep 01 2020

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A050326 Number of factorizations of n into distinct squarefree numbers > 1.

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A103774 Number of ways to write n! as product of squarefree numbers.

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A294619 a(0) = 0, a(1) = 1, a(2) = 2 and a(n) = 1 for n > 2.

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A337073 Number of strict factorizations of the superprimorial A006939(n) into squarefree numbers > 1.

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