A244115 -id:A244115 - cp's OEIS frontend

A341343 Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = (zeta(s))^3 / (zeta(3*s))^2.

1, 3, 3, 6, 3, 9, 3, 8, 6, 9, 3, 18, 3, 9, 9, 9, 3, 18, 3, 18, 9, 9, 3, 24, 6, 9, 8, 18, 3, 27, 3, 9, 9, 9, 9, 36, 3, 9, 9, 24, 3, 27, 3, 18, 18, 9, 3, 27, 6, 18, 9, 18, 3, 24, 9, 24, 9, 9, 3, 54, 3, 9, 18, 9, 9, 27, 3, 18, 9, 27, 3, 48, 3, 9, 18, 18, 9, 27, 3, 27, 9, 9, 3, 54, 9

Offset: 1

Werner Schulte, Feb 09 2021

There is a family of multiplicative sequences based on trinomial numbers (A027907) and some fixed integer k >= 0. Let a(k,n), k >= 0, n > 0, be multiplicative with a(k,p^e) = Sum_{i=0..e} trinomial(k,i) for prime p and e >= 0, where trinomial(n,k) = 0 if 2*n < k. These sequences have the Dirichlet g.f.: Sum_{n>=1} a(k,n)/n^s = (zeta(s))^(k+1) / (zeta(3*s))^k. For several members of the family see A000012 (k=0), A073184 (k=1), and this sequence (k=2).

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

Cf. A000012, A027907, A073184.

{T(n,k) = if( n<0, 0, polcoeff( (1 + x + x^2)^n, k))}; \\ A027907
a(n)={my(f=factor(n));prod(k=1,#f[,1],sum(i=0,f[k,2],T(2,i)))};
for(j=1,75,print1(a(j),", ")) \\ Hugo Pfoertner, Feb 13 2021

for(n=1, 100, print1(direuler(p=2, n, (1 - X^3)^2/(1 - X)^3)[n], ", ")) \\ Vaclav Kotesovec, Nov 20 2021

Multiplicative with a(p^e) = Sum_{i=0..e} trinomial(2,i) for prime p and e >= 0, where trinomial(n,k) = 0 if 2*n < k.
Let b(n), n > 0, be the Dirichlet inverse of a(n); b(n) is multiplicative with b(p^(3*e)) = 1 for e >= 0, and b(p^(3*e-2)) = -3*e and b(p^(3*e-1)) = 3*e for e > 0 and prime p.
Sum_{k=1..n} a(k) ~ n * (log(n)^2/2 + (3*gamma - 6*zeta'(3)/zeta(3) - 1)*log(n) + 1 - 3*gamma + 3*gamma^2 + 6*(1 - 3*gamma)*zeta'(3)/zeta(3) + 27*zeta'(3)^2 / zeta(3)^2 - 9*zeta''(3)/zeta(3) - 3*sg1) / zeta(3)^2, where gamma is the Euler-Mascheroni constant A001620, zeta(3) = A002117, zeta'(3) = -A244115, zeta''(3) = A340442 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Nov 20 2021

A369306 The number of cubefree divisors d of n such that n/d is also cubefree.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 2, 3, 4, 2, 6, 2, 4, 4, 1, 2, 6, 2, 6, 4, 4, 2, 4, 3, 4, 2, 6, 2, 8, 2, 0, 4, 4, 4, 9, 2, 4, 4, 4, 2, 8, 2, 6, 6, 4, 2, 2, 3, 6, 4, 6, 2, 4, 4, 4, 4, 4, 2, 12, 2, 4, 6, 0, 4, 8, 2, 6, 4, 8, 2, 6, 2, 4, 6, 6, 4, 8, 2, 2, 1, 4, 2, 12, 4, 4, 4

Offset: 1

Amiram Eldar, Jan 19 2024

The analogous sequence with squarefree divisors (the number of squarefree divisors d of n such that n/d is also squarefree) is abs(A007427(n)).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A004709, A005117, A007427, A069492, A073184.

Cf. A001620, A002117, A244115.

f[p_,e_] := Switch[e, 1, 2, 2, 3, 3, 2, 4, 1, , 0]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> [2, 3, 2, 1, 0][min(x, 5)], factor(n)[,2]));

Multiplicative with a(p) = 2, a(p^2) = 3, a(p^3) = 2, a(p^4) = 1, and a(p^e) = 0 for e >= 5.
a(n) >= 0, with equality if and only if n is a 5-full number (A069492) larger than 1.
a(n) = 1 if and only if n is the 4th power of a squarefree number (A005117).
a(n) <= A000005(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(s)^2/zeta(3*s)^2.
Sum_{k=1..n} a(k) ~ (n/zeta(3)^2) * (log(n) + 2*gamma - 1 - 6*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620).

A375503 Decimal expansion of the absolute value of zeta'(3/2), first derivative of the Riemann zeta function at s=3/2.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 18 2024

			zeta'(3/2) = -3.93223973743110151070638857840601520269274355489257...

Cf. A073002 (at s=2), A244115 (s=3).

Maple
```
Zeta(1,3/2); evalf(%) ;
```

RealDigits[Zeta'[3/2], 10, 120][[1]] (* Amiram Eldar, Aug 19 2024 *)

A349641 Decimal expansion of the Sum_{k>=2} 1/(k^3*log(k)).

Original entry on oeis.org

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

Offset: 0

Jianing Song, Nov 24 2021

			Sum_{k>=2} 1/(k^3*log(k)) = 0.23799610019862130199...

Similar sequences: A013661, A002117, A073002, A244115, A168218.

(* following Jean-François Alcover's Mathematica program for A168218 *) digits = 110; NSum[ 1/(n^3*Log[n]), {n, 2, Infinity}, NSumTerms -> 500000, WorkingPrecision -> digits + 5, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 12}}] // RealDigits[#, 10, digits] & // First

intnum(x=3, [oo, log(3)], zeta(x)-1) \\ following Charles R Greathouse IV's program for A168218

sumpos(k=2, 1/(k^3*log(k))) \\ Michel Marcus, Nov 27 2021

Equals Integral_{s=3..oo} (zeta(s) - 1) ds.

A349770 a(n) = Sum_{d|n} usigma(d) * usigma(n/d).

Original entry on oeis.org

1, 6, 8, 19, 12, 48, 16, 48, 36, 72, 24, 152, 28, 96, 96, 113, 36, 216, 40, 228, 128, 144, 48, 384, 88, 168, 136, 304, 60, 576, 64, 258, 192, 216, 192, 684, 76, 240, 224, 576, 84, 768, 88, 456, 432, 288, 96, 904, 164, 528, 288, 532, 108, 816, 288, 768, 320, 360, 120, 1824

Offset: 1

Ilya Gutkovskiy, Nov 29 2021

Dirichlet convolution of A034448 with itself.

Cf. A034448, A034761, A322328, A328485, A343569.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; a[n_] := Sum[usigma[d] usigma[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 60}]

Dirichlet g.f.: ( zeta(s) * zeta(s-1) / zeta(2*s-1) )^2.
Multiplicative with a(p^e) = e * (p^e + 1) + (p+1) * (p^e - 1)/(p-1). - Amiram Eldar, Nov 29 2021
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / zeta(3)^2 * (Pi^2 * log(n)/72 + gamma * Pi^2/36 - Pi^2/144 + zeta'(2)/6 - Pi^2 * zeta'(3)/(18*zeta(3))), where zeta(3) = A002117, zeta'(2) = -A073002, zeta'(3) = -A244115 and gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 05 2021

A358336 Multiplicative sequence with a(p^e) = ((p-1) * (1 + e(e+1)/2) + e) p^(e-1) for prime p and e > 0.

Original entry on oeis.org

1, 3, 5, 12, 9, 15, 13, 40, 30, 27, 21, 60, 25, 39, 45, 120, 33, 90, 37, 108, 65, 63, 45, 200, 90, 75, 153, 156, 57, 135, 61, 336, 105, 99, 117, 360, 73, 111, 125, 360, 81, 195, 85, 252, 270, 135, 93, 600, 182, 270, 165, 300, 105, 459, 189, 520, 185, 171, 117, 540, 121, 183, 390, 896

Offset: 1

Werner Schulte, Nov 09 2022

Cf. A000010, A005361, A018804, A112526.

Cf. A001620, A002117, A013664, A244115, A157289, A306016.

f[p_, e_] := ((p - 1)*(1 + e*(e + 1)/2) + e)*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2022 *)

a(n) = { my (f=factor(n), p, e, v=1); for (k=1, #f~, p=f[k,1]; e=f[k,2]; v *= ((p-1) * (1 + e*(e+1)/2) + e) * p^(e-1)); return (v) } \\ Rémy Sigrist, Jan 18 2023

a(n) = Sum_{k=1..n} gcd(k, n) * A005361(gcd(k, n)) for n > 0.
Equals Dirichlet convolution of A000010 and n * A005361.
Dirichlet g.f.: (zeta(s-1)^2 * zeta(2*s-2) * zeta(3*s-3)) / (zeta(s) * zeta(6*s-6)).
Equals Dirichlet convolution of A018804 and A112526.
Sum_{k=1..n} a(k) ~ (zeta(3)/(2*zeta(6))) * n^2 * (log(n) + 2*gamma - 1/2 + zeta'(2)/zeta(2) + 3*zeta'(3)/zeta(3) + 6*zeta'(6)/zeta(6)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 13 2024

cp's OEIS Frontend

A341343 Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = (zeta(s))^3 / (zeta(3*s))^2.

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A369306 The number of cubefree divisors d of n such that n/d is also cubefree.

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A375503 Decimal expansion of the absolute value of zeta'(3/2), first derivative of the Riemann zeta function at s=3/2.

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A349641 Decimal expansion of the Sum_{k>=2} 1/(k^3*log(k)).

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A349770 a(n) = Sum_{d|n} usigma(d) * usigma(n/d).

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A358336 Multiplicative sequence with a(p^e) = ((p-1) * (1 + e*(e+1)/2) + e) * p^(e-1) for prime p and e > 0.

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A358336 Multiplicative sequence with a(p^e) = ((p-1) * (1 + e(e+1)/2) + e) p^(e-1) for prime p and e > 0.