A259936 -id:A259936 - cp's OEIS frontend

A050320 Number of ways n is a product of squarefree numbers > 1.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 6, 1, 2, 2, 1, 2, 5, 1, 2, 2, 5, 1, 3, 1, 2, 2, 2, 2, 5, 1, 2, 1, 2, 1, 6, 2, 2, 2, 2, 1, 6, 2, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 5, 1

Offset: 1

Christian G. Bower, Sep 15 1999

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).
Broughan shows (Theorem 8) that the average value of a(n) is k exp(2*sqrt(log n)/sqrt(zeta(2)))/log(n)^(3/4) where k is about 0.18504.  - Charles R Greathouse IV, May 21 2013
From Gus Wiseman, Aug 20 2020: (Start)
Also the number of set multipartitions (multisets of sets) of the multiset of prime indices of n. For example, the a(n) set multipartitions for n = 2, 6, 36, 60, 360 are:
  {1}  {12}    {12}{12}      {1}{123}      {1}{12}{123}
       {1}{2}  {1}{2}{12}    {12}{13}      {12}{12}{13}
               {1}{1}{2}{2}  {1}{1}{23}    {1}{1}{12}{23}
                             {1}{2}{13}    {1}{1}{2}{123}
                             {1}{3}{12}    {1}{2}{12}{13}
                             {1}{1}{2}{3}  {1}{3}{12}{12}
                                           {1}{1}{1}{2}{23}
                                           {1}{1}{2}{2}{13}
                                           {1}{1}{2}{3}{12}
                                           {1}{1}{1}{2}{2}{3}
(End)

			For n = 36 we have three choices as 36 = 2*2*3*3 = 6*6 = 2*3*6 (but no factorizations with factors 4, 9, 12, 18 or 36 are allowed), thus a(36) = 3. - _Antti Karttunen_, Oct 21 2017

Giovanni Resta, Table of n, a(n) for n = 1..10000
Kevin Broughan, Quadrafree factorisatio numerorum, Rocky Mountain J. Math. 44 (3) (2014) 791-807.
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A005117, A050325. a(p^k)=1. a(A002110)=A000110.
a(n!)=A103774(n).
Cf. A206778.
Differs from A259936 for the first time at n=36.
A050326 is the strict case.
A045778 counts strict factorizations.
A089259 counts set multipartitions of integer partitions.
A116540 counts normal set multipartitions.
Cf. A008480, A124010, A317829, A318360.

a050320 n = h n $ tail $ a206778_row n where
   h 1 _          = 1
   h _ []         = 0
   h m fs'@(f:fs) =
     if f > m then 0 else if r > 0 then h m fs else h m' fs' + h m fs
     where (m', r) = divMod m f
-- Reinhard Zumkeller, Dec 16 2013

sub[w_, e_] := Block[{v = w}, v[[e]]--; v]; 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}]]]; sig[w_] := sig[w] = ric[w, 1];  a[n_] := sig@ Sort[Last /@ FactorInteger[n]]; Array[a, 103] (* Giovanni Resta, May 21 2013 *)
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sqfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]]
Table[Length[sqfacs[n]],{n,100}] (* Gus Wiseman, Aug 20 2020 *)

Dirichlet g.f.: Product_{n is squarefree and > 1} (1/(1-1/n^s)).
a(n) = A050325(A101296(n)). - R. J. Mathar, May 26 2017
a(n!) = A103774(n); a(A006939(n)) = A337072(n). - Gus Wiseman, Aug 20 2020

A281116 Number of factorizations of n>=2 into factors greater than 1 with no common divisor other than 1 (a(1)=0 by convention).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 4, 0, 0, 1, 1, 1, 5, 0, 1, 1, 3, 0, 4, 0, 2, 2, 1, 0, 5, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 8, 0, 1, 2, 0, 1, 4, 0, 2, 1, 4, 0, 9, 0, 1, 2, 2, 1, 4, 0, 5, 0, 1, 0, 8, 1, 1, 1, 3, 0, 8, 1, 2, 1, 1, 1, 7, 0, 2, 2, 5

Offset: 1

Gus Wiseman, Jan 15 2017

nonn

Let (e1, e2, ..., ek) be a prime-signature of n (that is, n = p^e1 * q^e2 * ... * r^ek for some primes, p, q, ..., r). Then a(n) is the number of ways of partitioning multiset {e1 x 1, e2 x 2, ..., ek x k} into multisets such that none of the numbers 1 .. k is present in all member multisets of that set partition. - Antti Karttunen, Sep 08 2018

			a(6)=1:  (2*3)
a(12)=2; (2*2*3)       (3*4)
a(24)=3: (2*2*2*3)     (2*3*4)     (3*8)
a(30)=4: (2*3*5)       (2*15)      (3*10)    (5*6)
a(36)=5: (2*2*3*3)     (2*2*9)     (2*3*6)   (3*3*4)   (4*9)
a(96)=7: (2*2*2*2*2*3) (2*2*2*3*4) (2*2*3*8) (2*3*4*4) (2*3*16) (3*4*8) (3*32).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A007916, A089233, A162247, A259936, A281113, A317751.

First column of A317748.

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

A281116(n, m=n, facs=List([])) = if(1==n, (1==gcd(Vec(facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A281116(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Sep 08 2018

Term a(1) = 0 prepended by Antti Karttunen, Sep 08 2018

A303362 Number of strict integer partitions of n with pairwise indivisible parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 5, 4, 6, 7, 7, 9, 11, 12, 13, 15, 17, 20, 23, 25, 27, 32, 35, 40, 45, 50, 55, 58, 67, 78, 84, 95, 101, 113, 124, 137, 153, 169, 180, 198, 219, 242, 268, 291, 319, 342, 374, 412, 450, 492, 535, 573, 632, 685, 746, 813, 868, 944

Offset: 1

Gus Wiseman, Apr 22 2018

nonn

			The a(14) = 7 strict integer partitions are (14), (11,3), (10,4), (9,5), (8,6), (7,5,2), (7,4,3).

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..450, (terms up to a(250) from Andrew Howroyd)

Cf. A000009, A000837, A003238, A006126, A051424, A259936, A275307, A281116, A285572, A285573, A290103, A293606, A293993, A303364.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]==={}&]],{n,60}]

lista(nn)={local(Cache=Map());
  my(excl=vector(nn, n, sumdiv(n, d, 2^(n-d))));
  my(a(n, m=n, b=0)=
     if(n==0, 1,
        while(m>n || bittest(b,0), m--; b>>=1);
        my(hk=[n, m, b], z);
        if(!mapisdefined(Cache, hk, &z),
          z = if(m, self()(n, m-1, b>>1) + self()(n-m, m, bitor(b, excl[m])), 0);
          mapput(Cache, hk, z)); z));
   for(n=1, nn, print1(a(n), ", "))
} \\ Andrew Howroyd, Nov 02 2019

A304714 Number of connected strict integer partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 5, 2, 5, 5, 6, 5, 10, 6, 12, 12, 13, 14, 21, 17, 23, 26, 30, 31, 46, 38, 51, 55, 61, 70, 87, 85, 102, 116, 128, 138, 171, 169, 204, 225, 245, 272, 319, 334, 383, 429, 464, 515, 593, 629, 715, 790, 861, 950, 1082

Offset: 1

Gus Wiseman, May 17 2018

nonn

Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.

			The a(19) = 6 strict integer partitions are (19), (9,6,4), (10,5,4), (10,6,3), (12,4,3), (8,6,3,2). Taking the normalized prime factors of each part (see A112798, A302242), we have the following connected multiset multisystems.
       (19): {{8}}
    (9,6,4): {{2,2},{1,2},{1,1}}
   (10,5,4): {{1,3},{3},{1,1}}
   (10,6,3): {{1,3},{1,2},{2}}
   (12,4,3): {{1,1,2},{1,1},{2}}
  (8,6,3,2): {{1,1,1},{1,2},{2},{1}}

The Heinz numbers of these partitions are given by A328513.

Cf. A000009, A003963, A048143, A054921, A076078, A259936, A281116, A285572, A285573, A286518, A286520, A302242.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c==={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[zsm[#]]===1&]],{n,60}]

A316439 Irregular triangle where T(n,k) is the number of factorizations of n into k factors > 1, with k ranging from 1 to Omega(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 3, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 3

Offset: 1

Gus Wiseman, Jul 03 2018

			The factorizations of 24 are (2*2*2*3), (2*2*6), (2*3*4), (2*12), (3*8), (4*6), (24) so the 24th row is {1, 3, 2, 1}.
Triangle begins:
  {}
  1
  1
  1  1
  1
  1  1
  1
  1  1  1
  1  1
  1  1
  1
  1  2  1
  1
  1  1
  1  1
  1  2  1  1
  1
  1  2  1
  1
  1  2  1
  1  1
  1  1
  1
  1  3  2  1
  1  1
  1  1
  1  1  1
  1  2  1
  1
  1  3  1

Alois P. Heinz, Rows n = 1..20000, flattened

Cf. A001222 (row lengths), A001055 (row sums), A001970, A007716, A045778, A162247, A259936, A281116, A303386.

g:= proc(n, k) option remember; `if`(n>k, 0, x)+
      `if`(isprime(n), 0, expand(x*add(`if`(d>k, 0,
      g(n/d, d)), d=numtheory[divisors](n) minus {1, n})))
    end:
T:= n-> `if`(n=1, [][], (p-> seq(coeff(p, x, i)
        , i=1..degree(p)))(g(n$2))):
seq(T(n), n=1..50);  # Alois P. Heinz, Aug 11 2019

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],Length[#]==k&]],{n,100},{k,PrimeOmega[n]}]

A286520 Number of finite connected sets of pairwise indivisible positive integers greater than one with least common multiple n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 5, 1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 5, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 17, 1, 1, 2, 1, 1, 5, 1, 2, 1, 5, 1, 9, 1, 1, 2, 2, 1, 5, 1, 4, 1, 1, 1, 17, 1, 1, 1

Offset: 2

Gus Wiseman, Jul 24 2017

nonn

Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that are not relatively prime. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph.

			The a(30)=5 sets are: {30}, {6,10}, {6,15}, {10,15}, {6,10,15}.

Cf. A048143, A054921, A076078, A259936, A281116, A285572, A285573, A286518.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c==={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Rest[Divisors[n]]],And[!MemberQ[Tuples[#,2],{x_,y_}/;And[x

A286518 Number of finite connected sets of positive integers greater than one with least common multiple n.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 4, 2, 4, 1, 20, 1, 4, 4, 8, 1, 20, 1, 20, 4, 4, 1, 88, 2, 4, 4, 20, 1, 96, 1, 16, 4, 4, 4, 196, 1, 4, 4, 88, 1, 96, 1, 20, 20, 4, 1, 368, 2, 20, 4, 20, 1, 88, 4, 88, 4, 4, 1, 1824, 1, 4, 20, 32, 4, 96, 1, 20, 4, 96, 1, 1688, 1, 4, 20, 20, 4, 96, 1, 368, 8, 4, 1, 1824, 4, 4, 4, 88, 1, 1824, 4, 20

Offset: 1

Gus Wiseman, Jul 24 2017

nonn

Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that are not relatively prime. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph.

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Feb 17 2024

			The a(6)=4 sets are: {6}, {2,6}, {3,6}, {2,3,6}.

Antti Karttunen, Table of n, a(n) for n = 1..719
Index entries for sequences computed from exponents in factorization of n

Cf. A048143, A054921, A069626, A076078, A259936, A281116, A285572, A285573, A286520, A305193, A318670.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c==={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Rest[Divisors[n]]],zsm[#]==={n}&]],{n,2,20}]

isconnected(facs) = { my(siz=length(facs)); if(1==siz,1,my(m=matrix(siz,siz,i,j,(gcd(facs[i],facs[j])!=1))^siz); for(n=1,siz,if(0==vecmin(m[n,]),return(0))); (1)); };
A286518aux(n, parts, from=1, ss=List([])) = { my(k = #parts, s=0, newss); if(lcm(Vec(ss))==n && isconnected(ss), s++); for(i=from, k, newss = List(ss); listput(newss, parts[i]); s += A286518aux(n, parts, i+1, newss)); (s) };
A286518(n) = if(1==n, n, A286518aux(n, divisors(n))); \\ Antti Karttunen, Feb 17 2024

From Antti Karttunen, Feb 17 2024: (Start)
a(n) <= A069626(n).
It seems that a(n) >= A318670(n), for all n > 1.
(End)

Term a(1)=1 prepended and more terms added by Antti Karttunen, Feb 17 2024

A292886 Number of knapsack factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 6, 2, 2, 2, 8, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 11, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 11, 4, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Sep 26 2017

nonn

A knapsack factorization is a finite multiset of positive integers greater than one such that every distinct submultiset has a different product.

The sequence giving the number of factorizations of n is described as "the multiplicative partition function" (see A001055), so knapsack factorizations are a multiplicative generalization of knapsack partitions. - Gus Wiseman, Oct 24 2017

			The a(36) = 8 factorizations are 2*2*3*3, 2*2*9, 2*18, 3*3*4, 3*12, 4*9, 6*6, 36. The factorization 2*3*6 is not knapsack.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A001055, A045778, a(p^n) = A108917(n), A162247, A259936, A275972, A281116.

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

A038348 Expansion of (1/(1-x^2))*Product_{m>=0} 1/(1-x^(2m+1)).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 11, 14, 19, 24, 31, 39, 49, 61, 76, 93, 114, 139, 168, 203, 244, 292, 348, 414, 490, 579, 682, 801, 938, 1097, 1278, 1487, 1726, 1999, 2311, 2667, 3071, 3531, 4053, 4644, 5313, 6070, 6923, 7886, 8971, 10190, 11561

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of n+2 with exactly one even part. - Vladeta Jovovic, Sep 10 2003
Also, number of partitions of n with at most one even part. - Vladeta Jovovic, Sep 10 2003
Also total number of parts, counted without multiplicity, in all partitions of n into odd parts, offset 1. - Vladeta Jovovic, Mar 27 2005
a(n) = Sum_{k>=1} k*A116674(n+1,k). - Emeric Deutsch, Feb 22 2006
Equals row sums of triangle A173305. - Gary W. Adamson, Feb 15 2010
Equals partial sums of A025147 (observed by Jonathan Vos Post, proved by several correspondents).
Conjecture: The n-th derivative of Gamma(x+1) at x = 0 has a(n+1) terms. For example, d^4/dx^4_(x = 0) Gamma(x+1) = 8*eulergamma*zeta(3) + eulergamma^4 + eulergamma^2*Pi^2 + 3*Pi^4/20 which has a(5) = 4 terms. - David Ulgenes, Dec 05 2023

			From _Gus Wiseman_, Sep 23 2019: (Start)
Also the number of integer partitions of n that are strict except possibly for any number of 1's. For example, the a(1) = 1 through a(7) = 11 partitions are:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (31)    (32)     (42)      (43)
             (111)  (211)   (41)     (51)      (52)
                    (1111)  (311)    (321)     (61)
                            (2111)   (411)     (421)
                            (11111)  (3111)    (511)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (31111)
                                               (211111)
                                               (1111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Cristina Ballantine and Mircea Merca, On identities of Watson type, Ars Mathematica Contemporanea (2019) Vol. 17, 277-290.
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
P. Flajolet and B. Salvy, Euler sums and contour integral representations, Experimental Mathematics, Vol. 7 Issue 1 (1998).
J. Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.), 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 2, with k=1. - _N. J. A. Sloane_, Aug 31 2014
Rebekah Ann Gilbert, A Fine Rediscovery, 2014.
Amrik Singh Nimbran and Paul Levrie, Series of the form Sum {a_n*binomial(2n, n)}, Math. Student (2023) Vol. 92, Nos. 3-4, 155-173. See pp. 10, 16.

Cf. A067588, A090867, A116674, A173305.

Cf. A000009, A007360, A051424, A259936, A266591, A302569, A306200.

f:=1/(1-x^2)/product(1-x^(2*j-1),j=1..32): fser:=series(f,x=0,62): seq(coeff(fser,x,n),n=0..58); # Emeric Deutsch, Feb 22 2006

mmax = 47; CoefficientList[ Series[ (1/(1-x^2))*Product[1/(1-x^(2m+1)), {m, 0, mmax}], {x, 0, mmax}], x] (* Jean-François Alcover, Jun 21 2011 *)

# uses[EulerTransform from A166861]
def g(n): return n % 2 if n > 2 else 1
a = EulerTransform(g)
print([a(n) for n in range(48)]) # Peter Luschny, Dec 04 2020

a(n) = A036469(n) - a(n-1) = Sum_{k=0..n} (-1)^k*A036469(n-k). - Vladeta Jovovic, Sep 10 2003
a(n) = A000009(n) + a(n-2). - Vladeta Jovovic, Feb 10 2004
G.f.: 1/((1-x^2)*Product_{j>=1} (1 - x^(2*j-1))). - Emeric Deutsch, Feb 22 2006
From Vaclav Kotesovec, Aug 16 2015: (Start)
a(n) ~ (1/2) * A036469(n).
a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). (End)
Euler transform of the sequence [1, 1, period(1, 0)] (A266591). - Georg Fischer, Dec 04 2020

A114592 Sum_{n>=1} a(n)/n^s = Product_{k>=2} (1 - 1/k^s).

Original entry on oeis.org

1, -1, -1, -1, -1, 0, -1, 0, -1, 0, -1, 1, -1, 0, 0, 0, -1, 1, -1, 1, 0, 0, -1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, 0, 0, 1, -1, 0, 0, 1, -1, 1, -1, 1, 1, 0, -1, 1, -1, 1, 0, 1, -1, 1, 0, 1, 0, 0, -1, 1, -1, 0, 1, 0, 0, 1, -1, 1, 0, 1, -1, 1, -1, 0, 1, 1, 0, 1

Offset: 1

Leroy Quet, Dec 11 2005

sign

For n >= 2, Sum_{k|n} A001055(n/k) * a(k) = 0. A114591(n) = Sum_{k|n} a(k).

First entry greater than 1 in absolute value is a(360) = -2. - Gus Wiseman, Sep 15 2018

			24 can be factored into distinct integers (each >= 2) as 24; as 4*6, 3*8 and 2*12; and as 2*3*4. (A045778(24) = 5).
So a(24) = (-1)^1 + 3*(-1)^2 + (-1)^3 = 1, where the 1 exponent is due to the 1 factor of the 24 = 24 factorization and the 2 exponent is due to the 3 cases of 2 factors each of the 24 = 4*6 = 3*8 = 2*12 factorizations and the 3 exponent is due to the 24 = 2*3*4 factorization.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001055, A045778, A114591.

Cf. A001222, A162247, A190938, A259936, A281116, A303386, A316441, A319237, A319238.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[(-1)^Length[f],{f,strfacs[n]}],{n,100}] (* Gus Wiseman, Sep 15 2018 *)

A114592aux(n, k) = if(1==n, 1, sumdiv(n, d, if(d > 1 && d <= k && d < n, (-1)*A114592aux(n/d, d-1))) - (n<=k)); \\ After code in A045778.
A114592(n) = A114592aux(n,n); \\ Antti Karttunen, Jul 23 2017

a(1) = 1; for n>= 2, a(n) = sum, over ways to factor n into any number of distinct integers >= 2, of (-1)^(number of integers in a factorization). (See example.)

More terms from Antti Karttunen, Jul 23 2017

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