A046644 -id:A046644 - cp's OEIS frontend

A318444 Numerators of the sequence whose Dirichlet convolution with itself yields A057660(n) = Sum_{k=1..n} n/gcd(n,k).

1, 3, 7, 35, 21, 21, 43, 239, 195, 63, 111, 245, 157, 129, 147, 6851, 273, 585, 343, 735, 301, 333, 507, 1673, 1643, 471, 3011, 1505, 813, 441, 931, 50141, 777, 819, 903, 6825, 1333, 1029, 1099, 5019, 1641, 903, 1807, 3885, 4095, 1521, 2163, 47957, 6555, 4929, 1911, 5495, 2757, 9033, 2331, 10277, 2401, 2439, 3423

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Because A057660 contains only odd values, A046644 gives the sequence of denominators. Because both of those sequences are multiplicative, this is also.

General formula: if k >= 0, m > 0, and the Dirichlet generating function is zeta(s-k)^m * f(s), where f(s) has all possible poles at points less than k+1, then Sum_{j=1..n} a(j) ~ n^(k+1) * log(n)^(m-1) * f(k+1) / ((k+1) * Gamma(m)) * (1 + (m-1)*(m*gamma - 1/(k+1) + f'(k+1)/f(k+1)) / log(n)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function. - Vaclav Kotesovec, May 10 2025

Antti Karttunen, Table of n, a(n) for n = 1..16384
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A057660, A046644 (denominators).

Cf. also A318443.

a57660[n_] := DivisorSigma[2, n^2]/DivisorSigma[1, n^2];
f[1] = 1; f[n_] := f[n] = 1/2 (a57660[n] - Sum[f[d]*f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
Table[f[n] // Numerator, {n, 1, 60}] (* Jean-François Alcover, Sep 13 2018 *)

up_to = 16384;
A057660(n) = sumdivmult(n, d, eulerphi(d)*d); \\ From A057660
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
v318444aux = DirSqrt(vector(up_to, n, A057660(n)));
A318444(n) = numerator(v318444aux[n]);

for(n=1, 100, print1(numerator(direuler(p=2, n, ((1-p*X)/((1-p^2*X)*(1-X)))^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A057660(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A318444(k) / A046644(k) ~ n^3 * Pi^(-3/2) * sqrt(2*zeta(3)/(3*log(n))) * (1 + (1/3 - gamma/2 + 3*zeta'(2)/Pi^2 - zeta'(3)/(2*zeta(3))) / (2*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 10 2025

A318666 a(n) = 2^{the 3-adic valuation of n}.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 16, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2

Offset: 1

Antti Karttunen, Sep 03 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000079, A007949, A046644, A317932.

[2^Valuation(n, 3): n in [1..100]]; // Vincenzo Librandi, Mar 19 2020

Table[2^IntegerExponent[n, 3], {n, 100}] (* Vincenzo Librandi, Mar 19 2020 *)

PARI
```
A318666(n) = 2^valuation(n,3);
```

A318666(n) = { my(f = factor(n), m=1); for(i=1, #f~, if(3 == f[i,1], m *= 2^f[i,2])); (m); };

a(n) = 2^A007949(n).
a(n) = A046644(n)/A317932(n).
Multiplicative with a(3^e) = 2^e, a(p^e) = 1 for any other primes.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Nov 17 2022
Dirichlet g.f.: zeta(s)*(3^s-1)/(3^s-2). - Amiram Eldar, Jan 03 2023
More precise asymptotics: Sum_{k=1..n} a(k) ~ 2*n + zeta(log(2)/log(3)) * n^(log(2)/log(3)) / (2*log(2)). - Vaclav Kotesovec, Jun 25 2024

A346103 Numerators of sequence whose Dirichlet convolution with itself yields A342920.

Original entry on oeis.org

1, 1, 1, 7, 1, 3, 1, 57, 47, 3, 1, 19, 1, 3, 11, 747, 1, 139, 1, 19, 11, 3, 1, 319, 199, 3, 81, 19, 1, 231, 1, -265, 11, 3, 251, 873, 1, 3, 11, 191, 1, 79, 1, 19, 299, 3, 1, -157, 5943, 595, 11, 19, 1, 151, 187, 31, 11, 3, 1, 269, 1, 3, 507, -957, 527, 31, 1, 19, 11, 223, 1, 18787, 1, 3, 8915, 19, 483, 31, 1, 2147, 19355

Offset: 1

Antti Karttunen, Jul 09 2021

Cf. A046644 (gives the denominators).

Cf. A108951, A324886, A342001, A342002, A342920, A346104.

up_to = 2310;
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A342002(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= p^(e>0); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
A342920(n) = A342002(A108951(n));
vA346103aux = DirSqrt(vector(up_to, n, A342920(n)));
A346103(n) = numerator(vA346103aux[n]);

A383658 Denominator of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s)^(3/2).

Original entry on oeis.org

1, 2, 2, 8, 2, 4, 2, 16, 8, 4, 2, 16, 2, 4, 4, 128, 2, 16, 2, 16, 4, 4, 2, 32, 8, 4, 16, 16, 2, 8, 2, 256, 4, 4, 4, 64, 2, 4, 4, 32, 2, 8, 2, 16, 16, 4, 2, 256, 8, 16, 4, 16, 2, 32, 4, 32, 4, 4, 2, 32, 2, 4, 16, 1024, 4, 8, 2, 16, 4, 8, 2, 128, 2, 4, 16, 16, 4

Offset: 1

Vaclav Kotesovec, May 04 2025

Is this a duplicate of A046644 (the first 8192 entries are the same)? - R. J. Mathar, May 06 2025

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

Cf. A383657.
Cf. A046643, A046644, A256688, A256689, A256690, A256691, A256692, A256693.
Cf. A000005, A007425, A007426, A061200, A034695.

coeff=CoefficientList[Series[1/(1-x)^(3/2),{x,0,20}]//Normal,x]; dptTerm[n_]:=Module[{flist=FactorInteger[n]},If[n==1,coeff[[1]],Denominator[Times@@(coeff[[flist[[All,2]]+1]])]]];Array[dptTerm,77] (* Shenghui Yang, May 04 2025 *)

for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-X)^(3/2))[n]), ", "))

Sum_{k=1..n} A383657(k)/A383658(k) ~ 2*n*sqrt(log(n)/Pi) * (1 - (1 - 3*gamma/2)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620.

A318650 Numerators of the sequence whose Dirichlet convolution with itself yields A057521, the powerful part of n.

Original entry on oeis.org

1, 1, 1, 15, 1, 1, 1, 49, 35, 1, 1, 15, 1, 1, 1, 603, 1, 35, 1, 15, 1, 1, 1, 49, 99, 1, 181, 15, 1, 1, 1, 2023, 1, 1, 1, 525, 1, 1, 1, 49, 1, 1, 1, 15, 35, 1, 1, 603, 195, 99, 1, 15, 1, 181, 1, 49, 1, 1, 1, 15, 1, 1, 35, 14875, 1, 1, 1, 15, 1, 1, 1, 1715, 1, 1, 99, 15, 1, 1, 1, 603, 3235, 1, 1, 15, 1, 1, 1, 49, 1, 35, 1, 15, 1, 1, 1, 2023, 1

Offset: 1

Antti Karttunen, Aug 31 2018

Multiplicative because A046644 and A057521 are.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (10^8 terms)

Cf. A057521, A046644 (denominators).

Cf. also A317935, A318511, A318649.

ff[p_, e_] := If[e > 1, p^e, 1]; a[1] = 1; a[n_] := Times @@ ff @@@ FactorInteger[n]; f[1] = 1; f[n_] := f[n] = 1/2 (a[n] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]); Table[Numerator[f[n]], {n, 1, 100}] (* Vaclav Kotesovec, May 11 2025 *)

up_to = 65537;
A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA057521(n)));
A318650(n) = numerator(v318650_aux[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A057521(n) - Sum_{d|n, d>1, d 1.
From Vaclav Kotesovec, May 10 2025, simplified May 11 2025: (Start)
Let f(s) = Product_{p prime} (1 - 1/p^(3*s-2) + 1/p^(3*s-3) + 1/p^s).
Sum_{k=1..n} A318650(k) / A046644(k) ~ n^(3/2) * sqrt(2*f(3/2)/(9*Pi*log(n))) * (1 + (2/3 - gamma - f'(3/2)/(2*f(3/2))) / (2*log(n))), where
f(3/2) = Product_{p prime} (1 + 2/p^(3/2) - 1/p^(5/2)) = A328013 = 3.51955505841710664719752940369857817...
f'(3/2)/f(3/2) = Sum_{p prime} (4*p - 3) * log(p) / (1 - 2*p - p^(5/2)) = -3.90914718020692131140714384422938370058563543737256496...
and gamma is the Euler-Mascheroni constant A001620. (End)

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A318444 Numerators of the sequence whose Dirichlet convolution with itself yields A057660(n) = Sum_{k=1..n} n/gcd(n,k).

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A318666 a(n) = 2^{the 3-adic valuation of n}.

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A346103 Numerators of sequence whose Dirichlet convolution with itself yields A342920.

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A383658 Denominator of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s)^(3/2).

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A318650 Numerators of the sequence whose Dirichlet convolution with itself yields A057521, the powerful part of n.

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