A284639 -id:A284639 - cp's OEIS frontend

A066739 Number of representations of n as a sum of products of positive integers. 1 is not allowed as a factor, unless it is the only factor. Representations which differ only in the order of terms or factors are considered equivalent.

1, 1, 2, 3, 6, 8, 14, 19, 32, 44, 67, 91, 139, 186, 269, 362, 518, 687, 960, 1267, 1747, 2294, 3106, 4052, 5449, 7063, 9365, 12092, 15914, 20422, 26639, 34029, 44091, 56076, 72110, 91306, 116808, 147272, 187224, 235201, 297594, 372390, 468844, 584644, 732942

Offset: 0

Naohiro Nomoto, Jan 16 2002

			For n=5, 5 = 4+1 = 2*2+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1, so a(5) = 8.
For n=8, 8 = 4*2 = 2*2*2 = ... = 4+4 = 2*2+4 = 2*2+2*2 = ...; note that there are 3 ways to factor the terms of 4+4. In general, if a partition contains a number k exactly r times, then the number of ways to factor the k's is the binomial coefficient C(A001055(k)+r-1,r).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Cf. A000041, A001055.
Cf. A066815, A066816, A066806.
Cf. A001970, A050336, A063834, A065026, A066739, A281113, A284639, A318948, A318949.

with(numtheory):
b:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= proc(n) option remember;
      `if`(n=0, 1, add(add(d*b(d, d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60); # Alois P. Heinz, Apr 22 2012

p[ n_, 1 ] := If[ n==1, 1, 0 ]; p[ 1, k_ ] := 1; p[ n_, k_ ] := p[ n, k ]=p[ n, k-1 ]+If[ Mod[ n, k ]==0, p[ n/k, k ], 0 ]; A001055[ n_ ] := p[ n, n ]; a[ n_, 1 ] := 1; a[ 0, k_ ] := 1; a[ n_, k_ ] := If[ k>n, a[ n, n ], a[ n, k ]=a[ n, k-1 ]+Sum[ Binomial[ A001055[ k ]+r-1, r ]a[ n-k*r, k-1 ], {r, 1, Floor[ n/k ]} ] ]; a[ n_ ] := a[ n, n ]; (* p[ n, k ]=number of factorizations of n with factors <= k. a[ n, k ]=number of representations of n as a sum of products of positive integers, with summands <= k *)
b[n_, k_] := b[n, k] = If[n>k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d>k, 0, b[n/d, d]], {d, Divisors[n] ~Complement~ {1, n}}]]; a[0] = 1; a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSum[j, #*b[#, #]&]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 10 2015, after Alois P. Heinz *)
facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Length[Union[Sort/@Join@@Table[Tuples[facs/@ptn],{ptn,IntegerPartitions[n]}]]],{n,50}] (* Gus Wiseman, Sep 05 2018 *)

from sympy.core.cache import cacheit
from sympy import divisors, isprime
@cacheit
def b(n, k): return (0 if n>k else 1) + (0 if isprime(n) else sum([0 if d>k else b(n//d, d) for d in divisors(n)[1:-1]]))
@cacheit
def a(n): return 1 if n==0 else sum(sum(d*b(d, d) for d in divisors(j))*a(n - j)  for j in range(1, n + 1))//n
print([a(n) for n in range(61)]) # Indranil Ghosh, Aug 19 2017, after Maple code

a(n) = Sum_{pi} Product_{m=1..n} binomial(k(m)+A001055(m)-1, k(m)), where pi runs through all partitions k(1) + 2 * k( 2) + ... + n * k(n) = n. a(n)=1/n*Sum_{m=1..n} a(n-m)*b(m), n > 0, a(0)=1, b(m)=Sum_{d|m} d*A001055(d). Euler transform of A001055(n): Product_{m=1..infinity} (1-x^m)^(-A001055(m)). - Vladeta Jovovic, Jan 21 2002

Edited by Dean Hickerson, Jan 19 2002

A301700 Number of aperiodic rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 21, 52, 120, 290, 697, 1713, 4200, 10446, 26053, 65473, 165257, 419357, 1068239, 2732509, 7013242, 18059960, 46641983, 120790324, 313593621, 816046050, 2128101601, 5560829666, 14557746453, 38177226541, 100281484375, 263815322761, 695027102020

Offset: 1

Gus Wiseman, Apr 23 2018

nonn

An unlabeled rooted tree is aperiodic if the multiset of branches of the root is an aperiodic multiset, meaning it has relatively prime multiplicities, and each branch is also aperiodic.

			The a(6) = 10 aperiodic trees are (((((o))))), (((o(o)))), ((o((o)))), ((oo(o))), (o(((o)))), (o(o(o))), ((o)((o))), (oo((o))), (o(o)(o)), (ooo(o)).

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

Cf. A000081, A000740, A000837, A001678, A003238, A004111, A007716, A007916, A100953, A276625, A284639, A290689, A298422, A303386, A303431.

arut[n_]:=arut[n]=If[n===1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[arut/@c]],GCD@@Length/@Split[#]===1&]]/@IntegerPartitions[n-1]];
Table[Length[arut[n]],{n,20}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
MoebiusT(v)={vector(#v, n, sumdiv(n,d,moebius(n/d)*v[d]))}
seq(n)={my(v=[1]); for(n=2, n, v=concat([1], MoebiusT(EulerT(v)))); v} \\ Andrew Howroyd, Sep 01 2018

Terms a(21) and beyond from Andrew Howroyd, Sep 01 2018

A295935 Number of twice-factorizations of n where the latter factorizations are constant, i.e., type (P,P,R).

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 5, 3, 2, 1, 5, 1, 2, 2, 12, 1, 5, 1, 5, 2, 2, 1, 10, 3, 2, 5, 5, 1, 5, 1, 18, 2, 2, 2, 15, 1, 2, 2, 10, 1, 5, 1, 5, 5, 2, 1, 22, 3, 5, 2, 5, 1, 10, 2, 10, 2, 2, 1, 13, 1, 2, 5, 40, 2, 5, 1, 5, 2, 5, 1, 28, 1, 2, 5, 5, 2, 5, 1, 22, 12, 2, 1

Offset: 1

Gus Wiseman, Nov 29 2017

nonn

a(n) is also the number of ways to choose a perfect divisor of each factor in a factorization of n.

			The a(24) = 10 twice-factorizations are:
(2)*(2)*(2)*(3), (2)*(3)*(2*2), (3)*(2*2*2)
(2)*(2)*(6), (2*2)*(6),
(2)*(3)*(4),
(2)*(12),
(3)*(8),
(4)*(6),
(24).

Cf. A000005, A001055, A052409, A052410, A089723, A279784, A281113, A284639, A295923, A295924, A295931.

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

Dirichlet g.f.: 1/Product_{n > 1}(1 - A089723(n)/n^s).

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

A294336 Number of ways to write n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 28 2017

nonn

Möbius-transform of A294337. - Antti Karttunen, Jun 12 2018

			The a(4096) = 7 ways are: 2^12, 4^6, 8^4, 8^(2^2), 16^3, 64^2, 4096.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000041, A001055, A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294337, A294338, A294339, A375598, A375599.

Array[1+Sum[#0[g],{g,Rest[Divisors[GCD@@FactorInteger[#1][[All,2]]]]}]&,200]

A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A294336(n) = if(1==n,n,sumdiv(A052409(n),d,A294336(d))); \\ Antti Karttunen, Jun 12 2018, after Mathematica-code.

a(1) = 1; for n > 1, a(n) = Sum_{d|A052409(n)} a(d). - Antti Karttunen, Jun 12 2018, after Mathematica-code.

a(n) = A294337(A052409(n)) for n >= 2. - Pontus von Brömssen, Aug 20 2024

More terms from Antti Karttunen, Jun 12 2018

A066815 Number of partitions of n into sums of products.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 14, 19, 33, 45, 69, 94, 148, 197, 289, 390, 575, 762, 1086, 1439, 2040, 2687, 3712, 4874, 6749, 8792, 11918, 15526, 20998, 27164, 36277, 46820, 62367, 80146, 105569, 135326, 177979, 227139, 296027, 377142, 490554, 622526, 804158

Offset: 0

Vladeta Jovovic, Jan 20 2002

nonn

Number of ways to choose a factorization of each part of an integer partition of n. - Gus Wiseman, Sep 05 2018

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1, g(n) = A001055(n). - Seiichi Manyama, Nov 14 2018

			From _Gus Wiseman_, Sep 05 2018: (Start)
The a(6) = 14 partitions of 6 into sums of products:
  6, 2*3,
  5+1, 4+2, 2*2+2, 3+3,
  4+1+1, 2*2+1+1, 3+2+1, 2+2+2,
  3+1+1+1, 2+2+1+1,
  2+1+1+1+1,
  1+1+1+1+1+1.
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000041, A001055, A066739, A321460.

Cf. A001970, A050336, A063834, A065026, A281113, A284639, A318948, A318949.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Length[Join@@Table[Tuples[facs/@ptn],{ptn,IntegerPartitions[n]}]],{n,20}] (* Gus Wiseman, Sep 05 2018 *)

G.f.: Product_{k>=1} 1/(1-A001055(k)*x^k).

a(n) = 1/n*Sum_{k=1..n} a(n-k)*b(k), n > 0, a(0)=1, b(k)=Sum_{d|k} d*(A001055(d))^(k/d).

Renamed by T. D. Noe, May 24 2011

A295924 Number of twice-factorizations of n of type (R,P,R).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 4, 3, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 1, 8, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Nov 30 2017

nonn

a(n) is the number of ways to choose an integer partition of a divisor of A052409(n).

			The a(16) = 8 twice-factorizations are (2)*(2)*(2)*(2), (2)*(2)*(2*2), (2)*(2*2*2), (2*2)*(2*2), (2*2*2*2), (4)*(4), (4*4), (16).

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

Cf. A000005, A000041, A001055, A047968, A052409, A052410, A089723, A281113, A284639, A295923, A295931, A295935, A296134.

Table[DivisorSum[GCD@@FactorInteger[n][[All,2]],PartitionsP],{n,100}]

A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A295924(n) = if(1==n,n,sumdiv(A052409(n),d,numbpart(d))); \\ Antti Karttunen, Jul 29 2018

a(1) = 1; for n > 1, a(n) = Sum_{d|A052409(n)} A000041(d). - Antti Karttunen, Jul 29 2018

More terms from Antti Karttunen, Jul 29 2018

A294337 Number of ways to write 2^n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 6, 4, 4, 2, 7, 2, 4, 4, 10, 2, 7, 2, 7, 4, 4, 2, 10, 4, 4, 6, 7, 2, 8, 2, 12, 4, 4, 4, 12, 2, 4, 4, 10, 2, 8, 2, 7, 7, 4, 2, 15, 4, 7, 4, 7, 2, 10, 4, 10, 4, 4, 2, 13, 2, 4, 7, 16, 4, 8, 2, 7, 4, 8, 2, 16, 2, 4, 7, 7, 4, 8, 2, 15, 10, 4, 2, 13, 4, 4, 4, 10, 2, 13, 4, 7, 4, 4, 4, 18, 2, 7, 7, 12, 2, 8, 2, 10, 8

Offset: 1

Gus Wiseman, Oct 28 2017

nonn

			The a(12) = 7 ways are: 2^12, 4^6, 8^4, 8^(2^2), 16^3, 64^2, 4096.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294336, A294338, A294339.

a[n_]:=1+Sum[a[g],{g,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}];
Table[a[2^n],{n,100}]

A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A294336(n) = if(1==n,n,sumdiv(A052409(n),d,A294336(d)));
A294337(n) = sumdiv(n,d,A294336(d));
\\ Or alternatively, after Mathematica-code as:
A294337(n) = A294336(2^n); \\ Antti Karttunen, Jun 12 2018

a(n) = Sum_{d|n} A294336(d) = A294336(A000079(n)). - Antti Karttunen, Jun 12 2018

More terms from Antti Karttunen, Jun 12 2018

A295931 Number of ways to write n in the form n = (x^y)^z where x, y, and z are positive integers.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Nov 29 2017

nonn

By convention a(1) = 1.

Values can be 1, 3, 6, 9, 10, 15, 18, 21, 27, 28, 30, 36, 45, 54, 60, 63, 84, 90, etc. - Robert G. Wilson v, Dec 10 2017

			The a(256) = 10 ways are:
(2^1)^8    (2^2)^4   (2^4)^2  (2^8)^1
(4^1)^4    (4^2)^2   (4^4)^1
(16^1)^2   (16^2)^1
(256^1)^1

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A007425, A052409, A052410, A089723, A093771, A175082, A277562, A281113, A284639, A294786, A294336, A294338, A295920, A295923, A295924, A295935.

f:= proc(n) local m,d,t;
  m:= igcd(seq(t[2],t=ifactors(n)[2]));
  add(numtheory:-tau(d),d=numtheory:-divisors(m))
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Dec 19 2017

Table[Sum[DivisorSigma[0,d],{d,Divisors[GCD@@FactorInteger[n][[All,2]]]}],{n,100}]

a(A175082(k)) = 1, a(A093771(k)) = 3.

a(n) = Sum_{d|A052409(n)} A000005(d).

A318948 Number of ways to choose an integer partition of each factor in a factorization of n.

Original entry on oeis.org

1, 2, 3, 9, 7, 17, 15, 40, 39, 56, 56, 126, 101, 165, 197, 336, 297, 496, 490, 774, 837, 1114, 1255, 1948, 2007, 2638, 3127, 4123, 4565, 6201, 6842, 9131, 10311, 12904, 14988, 19516, 21637, 26995, 31488, 39250, 44583, 55418, 63261, 77683, 89935, 108068, 124754

Offset: 1

Gus Wiseman, Sep 05 2018

nonn

			The a(4) = 9 ways: (1+1)*(1+1), (1+1+1+1), (1+1)*(2), (2)*(1+1), (2+1+1), (2)*(2), (2+2), (3+1), (4).

Cf. A000041, A001055, A001970, A063834, A065026, A066739, A066815, A121229, A281113, A284639, A318949.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Sum[Times@@PartitionsP/@fac,{fac,facs[n]}],{n,10}]

Dirichlet g.f.: Product_{n > 1} 1 / (1 - P(n) / n^s) where P = A000041. [clarified by Ilya Gutkovskiy, Oct 26 2019]

cp's OEIS Frontend

A066739 Number of representations of n as a sum of products of positive integers. 1 is not allowed as a factor, unless it is the only factor. Representations which differ only in the order of terms or factors are considered equivalent.

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A301700 Number of aperiodic rooted trees with n nodes.

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A295935 Number of twice-factorizations of n where the latter factorizations are constant, i.e., type (P,P,R).

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

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A294336 Number of ways to write n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.

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A066815 Number of partitions of n into sums of products.

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A295924 Number of twice-factorizations of n of type (R,P,R).

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