A004101 -id:A004101 - cp's OEIS frontend

A327738 Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(i*j^2)).

1, 1, 2, 4, 9, 18, 37, 76, 158, 326, 672, 1386, 2862, 5906, 12187, 25148, 51900, 107103, 221023, 456110, 941256, 1942423, 4008481, 8272094, 17070712, 35227975, 72698206, 150023632, 309596255, 638898274, 1318462339, 2720844607, 5614870612, 11587126980

Offset: 0

Ilya Gutkovskiy, Sep 23 2019

nonn

Invert transform of A046951.

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

Cf. A004101, A046951, A129921, A280451.

a:= proc(n) option remember; `if`(n<1, 1, add(a(n-i)*
      nops(select(issqr, numtheory[divisors](i))), i=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 23 2019

nmax = 33; CoefficientList[Series[1/(1 - Sum[x^(k^2)/(1 - x^(k^2)), {k, 1, Floor[Sqrt[nmax]] + 1}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Length[Select[Divisors[k], IntegerQ[Sqrt[#]] &]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]

G.f.: 1 / (1 - Sum_{k>=1} x^(k^2) / (1 - x^(k^2))).
G.f.: 1 / (1 - Sum_{k>=1} (theta_3(x^k) - 1) / 2), where theta_() is the Jacobi theta function.
a(0) = 1; a(n) = Sum_{k=1..n} A046951(k) * a(n-k).

A343776 G.f.: Product_{k>=1} eta(x^(k^2)).

Original entry on oeis.org

1, -1, -1, 0, -1, 2, 1, 1, -1, -1, 2, 0, -1, 0, -2, -3, -1, 2, 0, 1, 5, -2, 1, -2, 2, -1, 0, 2, 0, -1, -1, 3, -4, 4, 0, -4, -5, -2, -3, 7, 2, 1, -6, -2, 4, -2, 2, 1, 7, -5, 11, 6, 0, -1, 1, -12, -11, 5, -3, -2, -8, 9, 8, 3, 1, 2, -5, -4, 5, -11, -6, 0, 7, 7, 4, -17, 3, -5, 8, 9, -4, -1, -10, 5, -6, 24, -5, 10, 0

Offset: 0

Seiichi Manyama, Apr 29 2021

sign

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

Convolution inverse of A004101.

Cf. A010815, A280451, A288098.

my(N=99, x='x+O('x^N)); Vec(prod(k=1, sqrt(N), eta(x^(k^2))))

A327747 Expansion of Product_{i>=1, j>=1} 1 / (1 + (-x)^(i*j^2)).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 2, 2, 1, 1, 0, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 3, 4, 3, 4, 4, 1, 4, 3, 4, 7, 6, 7, 6, 4, 5, 5, 7, 9, 9, 9, 8, 7, 7, 7, 10, 14, 13, 12, 14, 10, 12, 16, 13, 20, 19, 20, 20, 16, 18, 20, 22, 26, 27, 27, 28, 23, 26, 25, 31, 38, 36, 40

Offset: 0

Ilya Gutkovskiy, Sep 23 2019

nonn

Cf. A004101, A046951, A280451.

nmax = 75; CoefficientList[Series[Product[1/(1 + (-x)^k)^Length[Select[Divisors[k], IntegerQ[Sqrt[#]] &]], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[(-1)^k Sum[(-1)^(k/d) d Length[Select[Divisors[d], IntegerQ[Sqrt[#]] &]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 75}]

G.f.: Product_{k>=1} 1 / (1 + (-x)^k)^A046951(k).

A369763 Decimal expansion of the asymptotic mean of the ratio A000688(k)/A038538(k).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jan 31 2024

The asymptotic mean of the ratio between the number of non-isomorphic abelian groups and the number of non-isomorphic semisimple rings of the same order.
The constant A in Kühleitner's paper (1995).
The ratio is 1 for all biquadratefree numbers (whose asymptotic density is A215267 = 0.923..., see A046100), and smaller than 1 otherwise.

			0.98771484004493763774023068670639349351901075670395...

Manfred Kühleitner, Comparing the number of Abelian groups and of semisimple rings of a given order, Mathematica Slovaca, Vol. 45, No. 5 (1995), pp. 509-518.

Cf. A000041, A000688, A004101, A038538, A046100, A215267.

default(realprecision, 120); my(N=512, x='x+O('x^N), v); v = Vec(1/prod(k=1, sqrtint(N)+1, prod(j=1, 1+N\k^2, 1-x^(j*k^2)))); prodeulerrat((1-1/p)*vecsum(vector(N, i, numbpart(i-1)/(v[i]*p^(i-1))))) \\ after Vaclav Kotesovec at A004101

Equals Product_{p prime} (1 - 1/p)*(1 + Sum_{k>=1} A000041(k)/(A004101(k)*p^k)).

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A327738 Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(i*j^2)).

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A343776 G.f.: Product_{k>=1} eta(x^(k^2)).

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A327747 Expansion of Product_{i>=1, j>=1} 1 / (1 + (-x)^(i*j^2)).

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Mathematica

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A369763 Decimal expansion of the asymptotic mean of the ratio A000688(k)/A038538(k).

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