A317925 -id:A317925 - cp's OEIS frontend

A317937 Numerators of sequence whose Dirichlet convolution with itself yields sequence A001221 (omega n) + A063524 (1, 0, 0, 0, ...).

1, 1, 1, 3, 1, 3, 1, 5, 3, 3, 1, 7, 1, 3, 3, 35, 1, 7, 1, 7, 3, 3, 1, 11, 3, 3, 5, 7, 1, 3, 1, 63, 3, 3, 3, 9, 1, 3, 3, 11, 1, 3, 1, 7, 7, 3, 1, 75, 3, 7, 3, 7, 1, 11, 3, 11, 3, 3, 1, 1, 1, 3, 7, 231, 3, 3, 1, 7, 3, 3, 1, 19, 1, 3, 7, 7, 3, 3, 1, 75, 35, 3, 1, 1, 3, 3, 3, 11, 1, 1, 3, 7, 3, 3, 3, 133, 1, 7, 7, 9, 1, 3, 1, 11, 3

Offset: 1

Antti Karttunen, Aug 12 2018

The first negative term is a(210) = -7.

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

Cf. A001221, A063524, A046644 (denominators).

Cf. also A317831, A317925, A317933, A317845, A317846, A317936, A317938, A317939.

A317937aux(n) = if(1==n,n,(omega(n)-sumdiv(n,d,if((d>1)&&(dA317937aux(d)*A317937aux(n/d),0)))/2);
A317937(n) = numerator(A317937aux(n));

\\ DirSqrt(v) finds u such that v = v[1]*dirmul(u, u).
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&&dAndrew Howroyd, Aug 13 2018

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A001221(n) - Sum_{d|n, d>1, d 1.

A317846 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A051953 (cototient of n) + A063524 (1, 0, 0, 0, ...).

Original entry on oeis.org

1, 1, 1, 7, 1, 7, 1, 25, 11, 11, 1, 43, 1, 15, 13, 363, 1, 71, 1, 67, 17, 23, 1, 139, 19, 27, 61, 91, 1, 57, 1, 1335, 25, 35, 21, 365, 1, 39, 29, 215, 1, 81, 1, 139, 131, 47, 1, 1875, 27, 199, 37, 163, 1, 367, 29, 291, 41, 59, 1, 235, 1, 63, 171, 9923, 33, 129, 1, 211, 49, 137, 1, 1055, 1, 75, 235, 235, 33, 153, 1, 2883, 1363, 83, 1, 335, 41

Offset: 1

Antti Karttunen, Aug 12 2018

The first negative term is a(420) = -1269.

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

Cf. A051953, A063524, A046644 (denominators).

Cf. also A317845, A317925.

A317846aux(n) = if(1==n,n,((n-eulerphi(n))-sumdiv(n,d,if((d>1)&&(dA317846aux(d)*A317846aux(n/d),0)))/2);
A317846(n) = numerator(A317846aux(n));

\\ Memoized implementation:
memo317846 = Map();
A317846aux(n) = if(1==n,n,if(mapisdefined(memo317846,n),mapget(memo317846,n),my(v = ((n-eulerphi(n))-sumdiv(n,d,if((d>1)&&(dA317846aux(d)*A317846aux(n/d),0)))/2); mapput(memo317846,n,v); (v)));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A051953(n) - Sum_{d|n, d>1, d 1.

A317926 Denominators of rational valued sequence whose Dirichlet convolution with itself yields Euler's phi (A000010).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 11 2018

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A317925 (numerators).

Cf. also A046644, A317832.

f[1] = 1; f[n_] := f[n] = (EulerPhi[n] - DivisorSum[n, f[#]*f[n/#] &, 1 < # < n &])/2; Denominator @ Array[f, 100] (* Amiram Eldar, Dec 12 2022 *)

A317925perA317926(n) = if(1==n,n,(eulerphi(n)-sumdiv(n,d,if((d>1)&&(dA317925perA317926(d)*A317925perA317926(n/d),0)))/2);
A317926(n) = denominator(A317925perA317926(n));

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

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A000010(n) - Sum_{d|n, d>1, d 1.

A317933 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A034444 (number of unitary divisors of n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 12 2018

Multiplicative because A034444 is.
The first 2^20 terms are positive. Is the sequence nonnegative?
Records seem to be A001790, occurring at A000302 (apart from 4).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A001790, A034444, A317934 (denominators).

Cf. also A046643, A317831, A317925, A317937.

A034444(n) = (2^omega(n));
A317933perA317934(n) = if(1==n,n,(A034444(n)-sumdiv(n,d,if((d>1)&&(dA317933perA317934(d)*A317933perA317934(n/d),0)))/2);
A317933(n) = numerator(A317933perA317934(n));

up_to = 65537;
\\ Faster:
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.
v317933aux = DirSqrt(vector(up_to, n, A034444(n)));
A317933(n) = numerator(v317933aux[n]);

for(n=1, 100, print1(numerator(direuler(p=2, n, ((1+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) * (A034444(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A317933(k) / A317934(k) ~ sqrt(6)*n/Pi. - Vaclav Kotesovec, May 10 2025

A317938 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A001222 (bigomega n) + A063524 (1, 0, 0, 0, ...).

Original entry on oeis.org

1, 1, 1, 7, 1, 3, 1, 17, 7, 3, 1, 11, 1, 3, 3, 139, 1, 11, 1, 11, 3, 3, 1, 15, 7, 3, 17, 11, 1, 3, 1, 263, 3, 3, 3, 17, 1, 3, 3, 15, 1, 3, 1, 11, 11, 3, 1, 83, 7, 11, 3, 11, 1, 15, 3, 15, 3, 3, 1, -3, 1, 3, 11, 995, 3, 3, 1, 11, 3, 3, 1, 11, 1, 3, 11, 11, 3, 3, 1, 83, 139, 3, 1, -3, 3, 3, 3, 15, 1, -3, 3, 11, 3, 3, 3, 189, 1, 11, 11, 17, 1, 3, 1, 15, 3

Offset: 1

Antti Karttunen, Aug 12 2018

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

Cf. A001222, A063524, A046644 (denominators).

Cf. also A317831, A317925, A317933, A317845, A317846, A317937.

A317938aux(n) = if(1==n,n,(bigomega(n)-sumdiv(n,d,if((d>1)&&(dA317938aux(d)*A317938aux(n/d),0)))/2);
A317938(n) = numerator(A317938aux(n));

\\ Memoized implementation:
memo317938 = Map();
A317938aux(n) = if(1==n,n,if(mapisdefined(memo317938,n),mapget(memo317938,n),my(v = (bigomega(n)-sumdiv(n,d,if((d>1)&&(dA317938aux(d)*A317938aux(n/d),0)))/2); mapput(memo317938,n,v); (v)));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A001222(n) - Sum_{d|n, d>1, d 1.

A318317 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A173557.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 3, 5, 1, 1, 5, 3, 6, 3, 2, 35, 8, 1, 9, 3, 3, 5, 11, 5, 0, 3, 1, 9, 14, 1, 15, 63, 5, 4, 6, 3, 18, 9, 6, 5, 20, 3, 21, 15, 1, 11, 23, 35, -3, 0, 8, 9, 26, 1, 10, 15, 9, 7, 29, 3, 30, 15, 3, 231, 12, 5, 33, 3, 11, 3, 35, 5, 36, 9, 0, 27, 15, 3, 39, 35, 3, 10, 41, 9, 16, 21, 14, 25, 44, 1, 18, 33, 15, 23, 18, 63, 48, -3, 5, 0, 50, 4, 51, 15, 6

Offset: 1

Antti Karttunen, Aug 24 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A173557, A318318 (denominators).

Cf. also A317925, A317935.

f[1] = 1; f[n_] := f[n] = 1/2 (Module[{fac = FactorInteger[n]}, If[n == 1, 1, Product[fac[[i, 1]] - 1, {i, Length[fac]}]]] - Sum[f[d]*f[n/d], {d, Divisors[n][[2 ;; -2]]}]); Table[Numerator[f[n]], {n, 1, 100}] (* Vaclav Kotesovec, May 10 2025 *)

up_to = 16384;
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
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.
v318317_18 = DirSqrt(vector(up_to, n, A173557(n)));
A318317(n) = numerator(v318317_18[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A173557(n) - Sum_{d|n, d>1, d 1.
From Vaclav Kotesovec, May 10 2025: (Start)
Let f(s) = Product_{p prime} (1 - 2/(p + p^s)).
Sum_{k=1..n} A318317(k) / A318318(k) ~ n^2 * sqrt(f(2)/(4*Pi*log(n))) * (1 + (1 - gamma - f'(2)/f(2) + 6*zeta'(2)/Pi^2) / (4*log(n))), where
f(2) = A307868 = Product_{p prime} (1 - 2/(p*(p+1))) = 0.471680613612997868...
f'(2)/f(2) = Sum_{p prime} 2*p*log(p) / ((p+1)*(p^2+p-2)) = 0.7254208328519472161058521308839896283514823... and gamma is the Euler-Mascheroni constant A001620. (End)

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A317937 Numerators of sequence whose Dirichlet convolution with itself yields sequence A001221 (omega n) + A063524 (1, 0, 0, 0, ...).

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A317846 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A051953 (cototient of n) + A063524 (1, 0, 0, 0, ...).

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A317926 Denominators of rational valued sequence whose Dirichlet convolution with itself yields Euler's phi (A000010).

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A317933 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A034444 (number of unitary divisors of n).

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A317938 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A001222 (bigomega n) + A063524 (1, 0, 0, 0, ...).

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A318317 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A173557.

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