A050354 -id:A050354 - cp's OEIS frontend

A050355 Ordered factorizations with one level of parentheses indexed by prime signatures. A050354(A025487).

1, 1, 3, 5, 9, 21, 27, 81, 37, 81, 111, 297, 201, 243, 513, 1053, 945, 729, 2187, 1317, 3645, 365, 2745, 4077, 2187, 8829, 7209, 12393, 2433, 13257, 16605, 6561, 34263, 35397, 41553, 13473, 59697, 10155, 64881, 19683, 44793, 129033, 18993, 71307

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

A051707 Number of factorizations of (n,n) into pairs (j,k).

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 1, 8, 3, 5, 1, 23, 1, 5, 5, 23, 1, 23, 1, 23, 5, 5, 1, 91, 3, 5, 8, 23, 1, 52, 1, 60, 5, 5, 5, 143, 1, 5, 5, 91, 1, 52, 1, 23, 23, 5, 1, 328, 3, 23, 5, 23, 1, 91, 5, 91, 5, 5, 1, 339, 1, 5, 23, 161, 5, 52, 1, 23, 5, 52, 1, 686, 1, 5, 23, 23, 5, 52, 1, 328, 23, 5, 1, 339, 5

Offset: 1

Yasutoshi Kohmoto

Pairs (j,k) must satisfy j>1, k>=1; (a,b)*(x,y)=(a*x,b*y); unit is (1,1).

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).

			(6,6)=(2,1)*(3,6)=(2,6)*(3,1)=(2,2)*(3,3)=(2,3)*(3,2), so a(6)=5.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A050354, A108461, A108455, A348161 (into at most two pairs).
a(A025487) = A108460.
a(p^k) = A108457(k).
a(A002110) = A108459.
Main diagonal of A108455.

Edited by Christian G. Bower, Jun 03 2005

A050356 Number of ordered factorizations of n with 2 levels of parentheses.

Original entry on oeis.org

1, 1, 1, 4, 1, 7, 1, 16, 4, 7, 1, 40, 1, 7, 7, 64, 1, 40, 1, 40, 7, 7, 1, 208, 4, 7, 16, 40, 1, 73, 1, 256, 7, 7, 7, 292, 1, 7, 7, 208, 1, 73, 1, 40, 40, 7, 1, 1024, 4, 40, 7, 40, 1, 208, 7, 208, 7, 7, 1, 544, 1, 7, 40, 1024, 7, 73, 1, 40, 7, 73, 1, 1840, 1, 7, 40, 40, 7, 73, 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).

			For n=6 we have ((6)) = ((3*2)) = ((2*3)) = ((3)*(2)) = ((2)*(3)) = ((3))*((2)) = ((2))*((3)), thus a(6) = 7.

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

Cf. A002033, A050351, A050352, A050353, A050354, A050355, A050357, A050358, A050359.

A050356aux(n) = if(1==n,1/3, 3*sumdiv(n,d, if(dA050356aux(d), 0)));
A050356(n) = if(1==n,n,A050356aux(n)); \\ Antti Karttunen, May 19 2017, after the general recurrence given by Vladeta Jovovic May 25 2005 in A050354.

Dirichlet g.f.: (3-2*zeta(s))/(4-3*zeta(s)).
a(p^k) = 4^(k-1).
a(A002110(n)) = A050352(n).
Sum_{k=1..n} a(k) ~ -n^r / (9*r*Zeta'(r)), where r = 2.52138975790328306967497455387140053675965539610041801606891036... is the root of the equation Zeta(r) = 4/3. - Vaclav Kotesovec, Feb 02 2019

A308076 G.f. A(x) satisfies: A(x) = x + 2A(x^2) + 4A(x^3) + 8A(x^4) + ... + 2^(k-1)A(x^k) + ...

Original entry on oeis.org

1, 2, 4, 12, 16, 48, 64, 168, 272, 576, 1024, 2288, 4096, 8448, 16512, 33456, 65536, 132448, 262144, 526784, 1049088, 2101248, 4194304, 8399232, 16777472, 33570816, 67110976, 134252288, 268435456, 536942336, 1073741824, 2147618976, 4294975488, 8590196736, 17179871232

Offset: 1

Ilya Gutkovskiy, May 11 2019

nonn

Cf. A011782, A050354, A050369, A054599, A074206.

terms = 35; A[] = 0; Do[A[x] = x + Sum[2^(k - 1) A[x^k], {k, 2, terms}] + O[x]^(terms + 1) //Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
a[n_] := If[n == 1, n, Sum[If[d < n, 2^(n/d - 1) a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 35}]

a(1) = 1; a(n) = Sum_{d|n, d

a(n) ~ 2^(n-1). - Vaclav Kotesovec, Oct 16 2019

A328424 a(1) = 1; a(n) = Sum_{d|n, d < n} p(n/d) * a(d), where p = A000041 (partition numbers).

Original entry on oeis.org

1, 2, 3, 9, 7, 23, 15, 50, 39, 70, 56, 187, 101, 195, 218, 420, 297, 625, 490, 949, 882, 1226, 1255, 2533, 2007, 2840, 3217, 4588, 4565, 6966, 6842, 10099, 10479, 13498, 15093, 21507, 21637, 27975, 31791, 41722, 44583, 58022, 63261, 80415, 90799, 110578, 124754

Offset: 1

Ilya Gutkovskiy, Oct 15 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000041, A050354, A050369, A074206, A308076.

a[n_] := If[n == 1, n, Sum[If[d < n, PartitionsP[n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 47}]
terms = 47; A[] = 0; Do[A[x] = x + Sum[PartitionsP[k] A[x^k], {k, 2, terms}] + O[x]^(terms + 1) // Normal,terms + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} p(k) * A(x^k).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). - Vaclav Kotesovec, Oct 16 2019

A329385 Dirichlet g.f.: 1 / (2 - Product_{k>=1} zeta(k*s)).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 8, 3, 3, 1, 11, 1, 3, 3, 22, 1, 11, 1, 11, 3, 3, 1, 36, 3, 3, 8, 11, 1, 13, 1, 59, 3, 3, 3, 45, 1, 3, 3, 36, 1, 13, 1, 11, 11, 3, 1, 116, 3, 11, 3, 11, 1, 36, 3, 36, 3, 3, 1, 57, 1, 3, 11, 160, 3, 13, 1, 11, 3, 13, 1, 164, 1, 3, 11, 11, 3, 13, 1, 116

Offset: 1

Ilya Gutkovskiy, Jun 07 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..100000
Vaclav Kotesovec, Graph - The asymptotic ratio (100000 terms)

Cf. A000688, A001358 (positions of 3's), A008578 (positions of 1's), A050354, A129667.

a[n_] := If[n == 1, n, Sum[If[d < n, FiniteAbelianGroupCount[n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 80}]

G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} A000688(k) * A(x^k).
a(1) = 1; a(n) = Sum_{d|n, d < n} A000688(n/d) * a(d).
Let f(s) = Product_{k>=1} zeta(k*s), then Sum_{k=1..n} a(k) ~ n^r / (-r*f'(r)), where r = A335494 = 1.8868691498777... is the root of the equation f(r) = 2 and f'(r) = -1.8255483309672084429580571100367977185868132697213762608374345719289... - Vaclav Kotesovec, Jun 11 2020

A329803 a(1) = 1; a(n) = Sum_{d|n, d < n} q(n/d) * a(d), where q() = A000009.

Original entry on oeis.org

1, 1, 2, 3, 3, 8, 5, 11, 12, 16, 12, 37, 18, 32, 39, 55, 38, 90, 54, 105, 96, 113, 104, 236, 151, 201, 232, 301, 256, 450, 340, 517, 496, 588, 615, 988, 760, 972, 1054, 1395, 1260, 1766, 1610, 2078, 2240, 2512, 2590, 3653, 3289, 4029, 4249, 5038, 5120, 6526

Offset: 1

Ilya Gutkovskiy, Nov 21 2019

nonn

Cf. A000009, A050354, A050369, A074206, A308076, A328424.

a[n_] := If[n == 1, n, Sum[If[d < n, PartitionsQ[n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 54}]
nmax = 54; A[] = 0; Do[A[x] = x + Sum[PartitionsQ[k] A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} q(k) * A(x^k).

Showing 1-7 of 7 results.

cp's OEIS Frontend

A050355 Ordered factorizations with one level of parentheses indexed by prime signatures. A050354(A025487).

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A051707 Number of factorizations of (n,n) into pairs (j,k).

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A050356 Number of ordered factorizations of n with 2 levels of parentheses.

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A308076 G.f. A(x) satisfies: A(x) = x + 2*A(x^2) + 4*A(x^3) + 8*A(x^4) + ... + 2^(k-1)*A(x^k) + ...

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A328424 a(1) = 1; a(n) = Sum_{d|n, d < n} p(n/d) * a(d), where p = A000041 (partition numbers).

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A329385 Dirichlet g.f.: 1 / (2 - Product_{k>=1} zeta(k*s)).

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Mathematica

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A329803 a(1) = 1; a(n) = Sum_{d|n, d < n} q(n/d) * a(d), where q() = A000009.

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Mathematica

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A308076 G.f. A(x) satisfies: A(x) = x + 2A(x^2) + 4A(x^3) + 8A(x^4) + ... + 2^(k-1)A(x^k) + ...