A066739 -id:A066739 - cp's OEIS frontend

A066806 Expansion of Product_{k>=1} (1+x^k)^A001055(k).

1, 1, 1, 2, 3, 4, 6, 8, 11, 16, 21, 27, 38, 49, 63, 84, 109, 138, 180, 228, 289, 369, 463, 578, 732, 911, 1128, 1407, 1741, 2140, 2646, 3243, 3968, 4862, 5925, 7198, 8770, 10620, 12833, 15524, 18718, 22502, 27075, 32467, 38873, 46537, 55565, 66220, 78946

Offset: 0

Vladeta Jovovic, Jan 19 2002

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A000041, A001055, A066739, A057567, A321460.

with(numtheory):
g:= proc(n, k) option remember; `if`(n>k, 0, 1)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d)),
         d=divisors(n) minus {1, n}))
    end:
b:= proc(n) b(n):= add((-1)^(n/d+1)*d*g(d$2), d=divisors(n)) end:
a:= proc(n) a(n):= `if`(n=0, 1, add(a(n-k)*b(k), k=1..n)/n) end:
seq(a(n), n=0..60);  # Alois P. Heinz, May 16 2014

g[n_, k_] := g[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d > k, 0, g[n/d, d]], {d, Divisors[n] ~Complement~ {1, n}}]];
b[n_] := b[n] = Sum[(-1)^(n/d + 1)*d*g[d, d], {d, Divisors[n]}];
a[n_] := a[n] = If[n == 0, 1, Sum[a[n - k]*b[k], {k, 1, n}]/n];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 23 2017, after Alois P. Heinz *)

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

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

A319855 Minimum number that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 4, 3, 5, 2, 6, 4, 5, 1, 7, 4, 8, 3, 6, 5, 9, 2, 6, 6, 6, 4, 10, 5, 11, 1, 7, 7, 7, 4, 12, 8, 8, 3, 13, 6, 14, 5, 7, 9, 15, 2, 8, 6, 9, 6, 16, 6, 8, 4, 10, 10, 17, 5, 18, 11, 8, 1, 9, 7, 19, 7, 11, 7, 20, 4, 21, 12, 8, 8, 9, 8, 22, 3, 8

Offset: 1

Gus Wiseman, Sep 29 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			a(30) = 5 because the minimum number that can be obtained starting with (3,2,1) is 3+2*1 = 5.

Cf. A000792, A001970, A048249, A056239, A066739, A066815, A070960, A201163, A319850, A318948, A318949, A319841, A319856.

ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
nexos[ptn_]:=If[Length[ptn]==0,{0},Union@@Select[ReplaceListRepeated[{Sort[ptn]},{{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x+y]],{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x*y]]}],Length[#]==1&]];
Table[Min[nexos[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}]

a(1) = 0, a(n) = max(A056239(n) - A007814(n), 1). - Charlie Neder, Oct 03 2018

A319856 Maximum number that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 6, 5, 6, 4, 7, 6, 8, 6, 8, 6, 9, 6, 9, 7, 8, 8, 10, 9, 11, 6, 10, 8, 12, 9, 12, 9, 12, 9, 13, 12, 14, 10, 12, 10, 15, 9, 16, 12, 14, 12, 16, 12, 15, 12, 16, 11, 17, 12, 18, 12, 16, 9, 18, 15, 19, 14, 18, 16, 20, 12, 21, 13

Offset: 1

Gus Wiseman, Sep 29 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			a(30) = 9 because the maximum number that can be obtained starting with (3,2,1) is 3*(2+1) = 9.

Cf. A000792, A001970, A048249, A056239, A066739, A066815, A070960, A201163, A319850, A318948, A318949, A319841, A319855.

ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
nexos[ptn_]:=If[Length[ptn]==0,{0},Union@@Select[ReplaceListRepeated[{Sort[ptn]},{{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x+y]],{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x*y]]}],Length[#]==1&]];
Table[Max[nexos[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}]

A321460 Expansion of Product_{k>0} (1 - x^k)^A001055(k).

Original entry on oeis.org

1, -1, -1, 0, -1, 2, 0, 2, -1, 0, 3, -1, -2, -1, 1, -6, -1, 0, 0, 0, 7, -1, 1, -2, 4, 1, -2, 11, 1, -2, -10, 11, -12, 16, -15, -6, -6, -12, -1, 8, -4, -10, 9, -19, 21, -15, 23, 4, 28, -8, 42, -6, 9, 19, 3, -21, -18, -14, -15, 3, -72, 70, -21, -49, -9, 18, -12, 26, -68, -12

Offset: 0

Seiichi Manyama, Nov 10 2018

sign

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

Convolution inverse of A066739.

Cf. A001055, A066806, A162247, A321376, A321377.

A319841 Number of distinct positive integers that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 4, 2, 2, 1, 5, 2, 2, 2, 4, 1, 5, 1, 6, 2, 2, 2, 6, 1, 2, 2, 7, 1, 6, 1, 4, 4, 2, 1, 8, 2, 5, 2, 4, 1, 6, 2, 8, 2, 2, 1, 7, 1, 2, 4, 9, 2, 6, 1, 4, 2, 6, 1, 8, 1, 2, 6, 4, 2, 6, 1, 9, 4, 2, 1, 10, 2, 2, 2

Offset: 1

Gus Wiseman, Sep 29 2018

nonn

			60 is the Heinz number of (3,2,1,1) and
   5 = (3+2)*1*1
   6 = 3*2*1*1
   7 = 3+2+1+1
   8 = (3+1)*2*1
   9 = 3*(2+1)*1
  10 = (3+2)*(1+1)
  12 = (3+1)*(2+1)
so we have a(60) = 7. It is not possible to obtain 11 by adding or multiplying together the parts of (3,2,1,1).

Cf. A001055, A001970, A048249, A056239, A063834, A066739, A066815, A281113, A318948, A318949, A319855, A319856.

ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
Table[Length[Select[ReplaceListRepeated[{If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]},{{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x+y]],{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x*y]]}],Length[#]==1&]],{n,100}]

a(2^n) = A048249(n).

A370817 Greatest number of multisets that can be obtained by choosing a prime factor of each factor in an integer factorization of n into unordered factors > 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 07 2024

nonn

First differs from A096825 at a(210) = 4, A096825(210) = 6.
First differs from A343943 at a(210) = 4, A343943(210) = 6.
First differs from A345926 at a(90) = 4, A345926(90) = 3.

			For the factorizations of 60 we have the following choices (using prime indices {1,2,3} instead of prime factors {2,3,5}):
  (2*2*3*5): {{1,1,2,3}}
   (2*2*15): {{1,1,2},{1,1,3}}
   (2*3*10): {{1,1,2},{1,2,3}}
    (2*5*6): {{1,1,3},{1,2,3}}
    (3*4*5): {{1,2,3}}
     (2*30): {{1,1},{1,2},{1,3}}
     (3*20): {{1,2},{2,3}}
     (4*15): {{1,2},{1,3}}
     (5*12): {{1,3},{2,3}}
     (6*10): {{1,1},{1,2},{1,3},{2,3}}
       (60): {{1},{2},{3}}
So a(60) = 4.

Max Alekseyev, Table of n, a(n) for n = 1..10000

For all divisors (not just prime factors) we have A370816.
The version for partitions is A370809, for all divisors A370808.
A000005 counts divisors.
A001055 counts factorizations, strict A045778.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 chooses prime factors of prime indices, variations A355744, A355745.
A368413 counts non-choosable factorizations, complement A368414.
A370813 counts non-divisor-choosable factorizations, complement A370814.
Cf. A000792, A048249, A066739, A319055, A319057, A355529, A355535, A367771, A368100, A370585.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Max[Length[Union[Sort/@Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#]]]&/@facs[n]],{n,100}]

A066816 Expansion of Product_{k>=1} (1 + A001055(k)*x^k).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 10, 15, 20, 25, 36, 46, 58, 78, 95, 120, 160, 198, 249, 318, 392, 485, 608, 745, 914, 1140, 1390, 1692, 2092, 2528, 3032, 3709, 4468, 5364, 6494, 7770, 9279, 11161, 13347, 15824, 18920, 22465, 26539, 31607, 37345, 43994, 52016

Offset: 0

Vladeta Jovovic, Jan 20 2002

nonn

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

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

Cf. A001055, A066739, A066806.

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

A321566 Expansion of Product_{1 <= i_1 <= i_2 <= i_3 <= i_4} 1/(1 - x^(i_1i_2i_3*i_4)).

Original entry on oeis.org

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, 44090, 56075, 72108, 91303, 116802, 147264, 187210, 235182, 297562, 372346, 468777, 584553, 732803, 910744

Offset: 0

Seiichi Manyama, Nov 13 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Product_{1 <= i_1 <= i_2 <= ... <= i_b} 1/(1 - x^(i_1 * i_2 * ... * i_b)): A000041 (b=1), A182269 (b=2), A321360 (b=3), this sequence (b=4).

Cf. A066739, A218320, A321567.

Euler transform of A218320.

G.f.: Product_{k>0} 1/(1 - x^k)^A218320(k).

A282249 Number of representations of n as a sum of products of pairs of positive integers: n = Sum_{k=1..m} i_k*j_k with m >= 0, i_k < j_k, j_k > j_{k+1} and all factors distinct.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 3, 4, 4, 6, 5, 6, 8, 8, 9, 11, 10, 14, 15, 14, 14, 21, 18, 21, 25, 25, 30, 34, 33, 42, 45, 41, 55, 62, 58, 66, 79, 76, 94, 95, 97, 115, 131, 120, 148, 153, 159, 175, 203, 189, 226, 232, 243, 268, 299, 271, 340, 349, 363, 389

Offset: 0

Alois P. Heinz, Feb 09 2017

nonn

Or number of partitions of n where part i has multiplicity < i and all multiplicities are distinct and different from all parts.

			a(0) = 1: the empty sum.
a(6) = 2: 1*6 = 2*3.
a(8) = 2: 1*8 = 2*4.
a(10) = 3: 1*10 = 2*5 = 1*4+2*3.
a(11) = 3: 1*11 = 1*5+2*3 = 2*4+1*3.
a(12) = 4: 1*12 = 2*6 = 1*6+2*3 = 3*4.
a(13) = 4: 1*13 = 1*7+2*3 = 2*5+1*3 = 1*5+2*4.
a(14) = 6: 1*14 = 1*8+2*3 = 2*7 = 1*6+2*4 = 2*5+1*4 = 3*4+1*2.
a(15) = 5: 1*15 = 1*9+2*3 = 1*7+2*4 = 2*6+1*3 = 3*5.
a(25) = 14: 1*25 = 1*19+2*3 = 1*17+2*4 = 1*15+2*5 = 1*13+2*6 = 1*13+3*4 = 2*11+1*3 = 1*11+2*7 = 2*10+1*5 = 1*10+3*5 = 2*9+1*7 = 1*9+2*8 = 3*7+1*4 = 1*7+3*6.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000009, A066739, A098859, A182269, A182270, A211856, A211857, A212214, A212215, A212216, A212217, A212218, A212219, A276429, A282379.

h:= proc(n) option remember;
      (((2*n+3)*n-2)*n-`if`(n::odd, 3, 0))/12
    end:
g:= (n, i, s)-> `if`(n=0, 1, `if`(n>h(i), 0,
                b(n, i, select(x-> x<=i, s)))):
b:= proc(n, i, s) option remember; g(n, i-1, s)+
     `if`(i in s, 0, add(`if`(j in s, 0, g(n-i*j,
      min(n-i*j, i-1), s union {j})), j=1..min(i-1, n/i)))
    end:
a:= n-> g(n$2, {}):
seq(a(n), n=0..100);

h[n_] := h[n] = (((2*n + 3)*n - 2)*n - If[OddQ[n], 3, 0])/12;
g[n_, i_, s_] := If[n==0, 1, If[n>h[i], 0, b[n, i, Select[s, # <= i&]]]];
b[n_, i_, s_] := b[n, i, s] = g[n, i - 1, s] + If[MemberQ[s, i], 0, Sum[If[MemberQ[s, j], 0, g[n - i*j, Min[n - i*j, i - 1], s ~Union~ {j}]], {j, 1, Min[i - 1, n/i]}]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 01 2018, after Alois P. Heinz *)

A282379 Number of representations of n as a sum of products of pairs of positive integers: n = Sum_{k=1..m} i_k*j_k with m >= 0, i_k <= j_k, j_k > j_{k+1} and all factors distinct with the exception that i_k = j_k is allowed.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 5, 9, 9, 8, 11, 15, 13, 17, 17, 19, 24, 29, 23, 33, 37, 39, 40, 53, 48, 62, 63, 71, 77, 94, 81, 110, 116, 122, 123, 156, 152, 185, 180, 200, 213, 259, 236, 287, 298, 325, 333, 404, 386, 450, 457, 506, 531, 615, 579, 679, 721

Offset: 0

Alois P. Heinz, Feb 13 2017

nonn

			a(4) = 2: 1*4 = 2*2.
a(5) = 2: 1*5 = 2*2+1*1.
a(6) = 2: 1*6 = 2*3.
a(7) = 3: 1*7 = 2*3+1*1 = 1*3+2*2.
a(8) = 3: 1*8 = 2*4 = 1*4+2*2.
a(9) = 4: 1*9 = 1*5+2*2 = 2*4+1*1 = 3*3.
a(10) = 5: 1*10 = 1*6+2*2 = 2*5 = 1*4+2*3 = 3*3+1*1.
a(11) = 6: 1*11 = 1*7+2*2 = 2*5+1*1 = 1*5+2*3 = 2*4+1*3 = 3*3+1*2.
a(12) = 5: 1*12 = 1*8+2*2 = 2*6 = 1*6+2*3 = 3*4.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000009, A000330, A066739, A098859, A182269, A182270, A211856, A211857, A212214, A212215, A212216, A212217, A212218, A212219, A276429, A282249.

h:= proc(n) option remember;
      n*(n+1)*(2*n+1)/6
    end:
g:= (n, i, s)-> `if`(n=0, 1, `if`(n>h(i), 0,
                b(n, i, select(x-> x<=i, s)))):
b:= proc(n, i, s) option remember; g(n, i-1, s)+
     `if`(i in s, 0, add(`if`(j in s, 0, g(n-i*j,
      min(n-i*j, i-1), s union {j})), j=1..min(i, n/i)))
    end:
a:= n-> g(n$2, {}):
seq(a(n), n=0..100);

h[n_] := h[n] = n(n+1)(2n+1)/6;
g[n_, i_, s_ ] := If[n == 0, 1, If[n > h[i], 0,
     b[n, i, Select[s, # <= i&]]]];
b[n_, i_, s_] := b[n, i, s] = g[n, i - 1, s] +
     If[MemberQ[s, i], 0, Sum[If[MemberQ[s, j], 0, g[n - i*j,
     Min[n - i*j, i - 1], s ~Union~ {j}]], {j, 1, Min[i, n/i]}]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 01 2021, after Alois P. Heinz *)

cp's OEIS Frontend

A066806 Expansion of Product_{k>=1} (1+x^k)^A001055(k).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A319855 Minimum number that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A319856 Maximum number that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A321460 Expansion of Product_{k>0} (1 - x^k)^A001055(k).

Views

Author

Keywords

Links

Crossrefs

A319841 Number of distinct positive integers that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A370817 Greatest number of multisets that can be obtained by choosing a prime factor of each factor in an integer factorization of n into unordered factors > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A066816 Expansion of Product_{k>=1} (1 + A001055(k)*x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A321566 Expansion of Product_{1 <= i_1 <= i_2 <= i_3 <= i_4} 1/(1 - x^(i_1*i_2*i_3*i_4)).

Views

Author

Keywords

Links

Crossrefs

Formula

A282249 Number of representations of n as a sum of products of pairs of positive integers: n = Sum_{k=1..m} i_k*j_k with m >= 0, i_k < j_k, j_k > j_{k+1} and all factors distinct.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

A321566 Expansion of Product_{1 <= i_1 <= i_2 <= i_3 <= i_4} 1/(1 - x^(i_1i_2i_3*i_4)).