A288419 -id:A288419 - cp's OEIS frontend

A288417 a(n) = Sum_{d|n} A000593(n/d).

1, 2, 5, 3, 7, 10, 9, 4, 18, 14, 13, 15, 15, 18, 35, 5, 19, 36, 21, 21, 45, 26, 25, 20, 38, 30, 58, 27, 31, 70, 33, 6, 65, 38, 63, 54, 39, 42, 75, 28, 43, 90, 45, 39, 126, 50, 49, 25, 66, 76, 95, 45, 55, 116, 91, 36, 105, 62, 61, 105, 63, 66, 162, 7, 105, 130, 69

Offset: 1

Seiichi Manyama, Jun 09 2017

Multiplicative because this sequence is the Dirichlet convolution of A000012 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 27 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000012.

Sum_{d|n} d^k*A000593(n/d): this sequence (k=0), A109386 (k=1), A288418 (k=2), A288419 (k=3), A288420 (k=4).

f[p_, e_] := Sum[(i + 1)*p^(e - i), {i, 0, e}]; f[2, e_] := e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2022 *)

a(n)={sumdiv(n, d, sigma(d>>valuation(d,2)))} \\ Andrew Howroyd, Jul 27 2018

L.g.f.: log(Product_{k>=1} (1 + x^k)^(sigma(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jun 19 2018
Multiplicative with a(2^e) = e+1 and a(p^e) = Sum_{i=0..e} (i+1)*p^(e-i) for e >= 0 and prime p > 2. - Werner Schulte, Jan 05 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/144 = 0.676452... . - Amiram Eldar, Oct 22 2022

A288414 Expansion of Product_{k>=1} (1 + x^k)^(sigma_2(k)).

Original entry on oeis.org

1, 1, 5, 15, 41, 107, 286, 700, 1735, 4162, 9803, 22673, 51822, 116376, 258548, 567197, 1230763, 2642958, 5622616, 11850537, 24769248, 51353095, 105662389, 215838649, 437890022, 882562763, 1767741732, 3519599996, 6967592060, 13717874719, 26865949075

Offset: 0

Seiichi Manyama, Jun 08 2017

nonn

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

Cf. A275585, A288389, A288419.

Product_{k>=1} (1 + x^k)^sigma_m(k): A107742 (m=0), A192065 (m=1), this sequence (m=2), A288415 (m=3).

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+q^k)^DivisorSigma(2,k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

with(numtheory): seq(coeff(series(mul((1+x^k)^(sigma[2](k)),k=1..n),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 31 2018

nmax = 40; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[2, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)

m=40; x='x+O('x^m); Vec(prod(k=1, m, (1+x^k)^sigma(k,2))) \\ G. C. Greubel, Oct 30 2018

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A288419(k)*a(n-k) for n > 0.
a(n) ~ exp(2^(5/4) * (7*Zeta(3))^(1/4) * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - 5^(1/4) * Pi * n^(1/4) / (2^(13/4) * 3^(7/4) * (7*Zeta(3))^(1/4))) * (7*Zeta(3))^(1/8) / (2^(15/8) * 15^(1/8) * n^(5/8)). - Vaclav Kotesovec, Mar 23 2018
G.f.: Product_{i>=1, j>=1} (1 + x^(i*j))^(j^2). - Ilya Gutkovskiy, Aug 26 2018

A109386 G.f. is the logarithm of the g.f. of A107742: Sum_{n>=1} (a(n)/n)x^n = log( Sum_{n>=0} A107742(n)x^n ).

Original entry on oeis.org

1, 3, 7, 7, 11, 21, 15, 15, 34, 33, 23, 49, 27, 45, 77, 31, 35, 102, 39, 77, 105, 69, 47, 105, 86, 81, 142, 105, 59, 231, 63, 63, 161, 105, 165, 238, 75, 117, 189, 165, 83, 315, 87, 161, 374, 141, 95, 217, 162, 258, 245, 189, 107, 426, 253, 225, 273, 177, 119, 539, 123, 189, 510, 127, 297

Offset: 1

Paul D. Hanna, Jun 26 2005

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005, A001227, A060640, A107742.

Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), this sequence (k=1), A288418 (k=2), A288419 (k=3), A288420 (k=4).

a[n_] := DivisorSum[n, #*DivisorSum[#, Mod[#, 2]&]&]; Array[a, 65] (* Jean-François Alcover, Dec 23 2015 *)
f[p_, e_] := ((p + e*(p-1) - 2)*p^(e+1) + 1)/(p-1)^2; f[2, e_] := 2^(e+1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 29 2023 *)

PARI
```
a(n)=sumdiv(n,d,d*sumdiv(d,m,m%2))
```

N=66; x='x+O('x^N); /* that many terms */
c=sum(j=1, N, j*x^j);
t=log( 1/prod(j=0, N, eta(x^(2*j+1))) );
gf=serconvol(t, c);
Vec(gf) /* show terms */
/* Joerg Arndt, May 03 2008 */

a(n) = Sum_{d|n} d * Sum_{m|d} (m mod 2).
G.f.: Sum_{n>=1} a(n)/n*x^n = Sum_{j>=1} Sum_{i>=1} log(1+x^(i*j)).
From Vladeta Jovovic, Jul 05 2005:(Start)
Multiplicative with a(2^e) = 2^(e+1)-1 and a(p^e) = (p^(e+2)*(e+1)-p^(e+1)*(e+2)+1)/(p-1)^2 for p>2.
G.f.: Sum_{n>0} n*A000005(n)*x^n/(1+x^n).
G.f.: Sum_{n>0} n*A001227(n)*x^n/(1-x^n).
a(n) = A060640(n) if n is odd, else a(n) = A060640(n) - 2*A060640(n/2).
a(n) = Sum_{d|n} d*A001227(d).
a(n) = Sum_{d|n} d*A000593(n/d).
A107742(n) = (1/n)*Sum_{k=1..n} a(k)*A107742(n-k). (End)

A288418 a(n) = Sum_{d|n} d^2*A000593(n/d).

Original entry on oeis.org

1, 5, 13, 21, 31, 65, 57, 85, 130, 155, 133, 273, 183, 285, 403, 341, 307, 650, 381, 651, 741, 665, 553, 1105, 806, 915, 1210, 1197, 871, 2015, 993, 1365, 1729, 1535, 1767, 2730, 1407, 1905, 2379, 2635, 1723, 3705, 1893, 2793, 4030, 2765, 2257, 4433, 2850, 4030

Offset: 1

Seiichi Manyama, Jun 09 2017

Multiplicative because this sequence is the Dirichlet convolution of A000290 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 27 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000290, A001001, A192065, A183699.

Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), this sequence (k=2), A288419 (k=3), A288420 (k=4).

a[n_] := DivisorSum[n, Function[d, d^2*DivisorSum[n/d, If[OddQ[#], #, 0]&]] ];
Array[a, 50] (* Jean-François Alcover, Jul 03 2017 *)
f[p_, e_] := (p^(e + 1) - 1)*(p^(e + 2) - 1)/((p - 1)*(p^2 - 1)); f[2, e_] := (4^(e + 1) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

a(n) = sumdiv(n, d, d^2*sigma((n/d)>>valuation(n/d, 2))); \\ Michel Marcus, Jul 03 2017; corrected Jun 12 2022

L.g.f.: log(Product_{k>=1} (1 + x^k)^sigma(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jun 19 2018
From Amiram Eldar, Nov 13 2022: (Start)
a(n) = A001001(n) for odd n.
Multiplicative with a(2^e) = (4^(e+1)-1)/3 and a(p^e) = (p^(e+1)-1)*(p^(e+2)-1)/((p-1)*(p^2-1)) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(2)*zeta(3)/4 = A183699 / 4 = 0.494326... . (End)

A288420 a(n) = Sum_{d|n} d^4*A000593(n/d).

Original entry on oeis.org

1, 17, 85, 273, 631, 1445, 2409, 4369, 6898, 10727, 14653, 23205, 28575, 40953, 53635, 69905, 83539, 117266, 130341, 172263, 204765, 249101, 279865, 371365, 394406, 485775, 558778, 657657, 707311, 911795, 923553, 1118481, 1245505, 1420163, 1520079, 1883154

Offset: 1

Seiichi Manyama, Jun 09 2017

Multiplicative because this sequence is the Dirichlet convolution of A000583 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 20 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000583, A013662, A013663, A027848, A288415.

Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), A288418 (k=2), A288419 (k=3), this sequence (k=4).

f[p_, e_] :=  (p^(4*e+7) - (p^3+p^2+p+1)*p^(e+1) + p^2 + p + 1)/(p^7 - (p^3+p^2+p+1)*p + p^2 + p + 1); f[2, e_] := (16^(e+1)-1)/15; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

From Amiram Eldar, Nov 13 2022: (Start)
a(n) = A027848(n) for odd n.
Multiplicative with a(2^e) = (16^(e+1)-1)/15 and a(p^e) = (p^(4*e+7) - (p^3+p^2+p+1)*p^(e+1) + p^2 + p + 1)/(p^7 - (p^3+p^2+p+1)*p + p^2 + p + 1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^5, where c = Pi^4*zeta(5)/480 = (3/16)*zeta(4)*zeta(5) = 0.210429... . (End)

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A288422 Expansion of Product_{k>=1} 1/(1 + x^k)^(sigma_2(k)).

Original entry on oeis.org

1, -1, -4, -6, 0, 24, 51, 89, 47, -152, -578, -1149, -1482, -738, 2384, 8901, 18476, 26774, 24151, -7143, -86804, -226605, -406442, -539872, -441822, 181268, 1671148, 4240334, 7618777, 10551340, 10218856, 1973258, -20190349, -61492391, -121880826

Offset: 0

Seiichi Manyama, Jun 09 2017

sign

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

Cf. A288414, A288419.

Product_{k>=1} 1/(1 + x^k)^sigma_m(k): A288007 (m=0), A288421 (m=1), this sequence (m=2), A288423 (m=3).

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1+q^k)^DivisorSigma(2,k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018

nmax = 40; CoefficientList[Series[Product[1/(1+x^k)^DivisorSigma[2, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)

m=50; x='x+O('x^m); Vec(prod(k=1, m+2, 1/(1+x^k)^sigma(k,2))) \\ G. C. Greubel, Oct 29 2018

Convolution inverse of A288414.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A288419(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma_3(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 29 2018

cp's OEIS Frontend

A288417 a(n) = Sum_{d|n} A000593(n/d).

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A288414 Expansion of Product_{k>=1} (1 + x^k)^(sigma_2(k)).

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A109386 G.f. is the logarithm of the g.f. of A107742: Sum_{n>=1} (a(n)/n)*x^n = log( Sum_{n>=0} A107742(n)*x^n ).

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A288418 a(n) = Sum_{d|n} d^2*A000593(n/d).

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A288420 a(n) = Sum_{d|n} d^4*A000593(n/d).

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A288422 Expansion of Product_{k>=1} 1/(1 + x^k)^(sigma_2(k)).

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Magma

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A109386 G.f. is the logarithm of the g.f. of A107742: Sum_{n>=1} (a(n)/n)x^n = log( Sum_{n>=0} A107742(n)x^n ).