A046644 -id:A046644 - cp's OEIS frontend

A317845 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A001065 (sum of proper divisors) + A063524 (1, 0, 0, 0, ...).

1, 1, 1, 11, 1, 11, 1, 45, 15, 15, 1, 95, 1, 19, 17, 659, 1, 131, 1, 135, 21, 27, 1, 315, 23, 31, 89, 175, 1, 125, 1, 2319, 29, 39, 25, 901, 1, 43, 33, 455, 1, 165, 1, 255, 215, 51, 1, 3739, 31, 291, 41, 295, 1, 671, 33, 595, 45, 63, 1, 731, 1, 67, 271, 16319, 37, 245, 1, 375, 53, 237, 1, 2135, 1, 79, 335, 415, 37, 285, 1, 5419, 1979, 87, 1

Offset: 1

Antti Karttunen, Aug 12 2018

The first negative term is a(360) = -12947.

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

Cf. A001065, A063524, A046644 (denominators).

Cf. also A317831, A317846.

A317845aux(n) = if(1==n,n,((sigma(n)-n)-sumdiv(n,d,if((d>1)&&(dA317845aux(d)*A317845aux(n/d),0)))/2);
A317845(n) = numerator(A317845aux(n));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A001065(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.

A318321 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A003961.

Original entry on oeis.org

1, 3, 5, 27, 7, 15, 11, 135, 75, 21, 13, 135, 17, 33, 35, 2835, 19, 225, 23, 189, 55, 39, 29, 675, 147, 51, 625, 297, 31, 105, 37, 15309, 65, 57, 77, 2025, 41, 69, 85, 945, 43, 165, 47, 351, 525, 87, 53, 14175, 363, 441, 95, 459, 59, 1875, 91, 1485, 115, 93, 61, 945, 67, 111, 825, 168399, 119, 195, 71, 513, 145, 231, 73

Offset: 1

Antti Karttunen, Aug 24 2018

Multiplicative because A003961 is.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A046644 (denominators).

Cf. also A318319.

up_to = 16384;
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
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.
v318321aux = DirSqrt(vector(up_to, n, A003961(n)));
A318321(n) = numerator(v318321aux[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A003961(n) - Sum_{d|n, d>1, d 1.

A318662 Denominators of the sequence whose Dirichlet convolution with itself yields A055653, sum of phi(d) over all unitary divisors d of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 2, 8, 2, 2, 2, 2, 2, 4, 2, 2, 8, 2, 2, 4, 2, 2, 4, 8, 2, 16, 2, 2, 4, 2, 2, 4, 2, 4, 8, 2, 2, 4, 4, 2, 4, 2, 2, 16, 2, 2, 4, 8, 8, 4, 2, 2, 16, 4, 4, 4, 2, 2, 4, 2, 2, 16, 8, 4, 4, 2, 2, 4, 4, 2, 16, 2, 2, 16, 2, 4, 4, 2, 4, 128, 2, 2, 4, 4, 2, 4, 4, 2, 16, 4, 2, 4, 2, 4, 4, 2, 8, 16, 8, 2, 4, 2, 4, 8

Offset: 1

Antti Karttunen, Sep 03 2018

The sequence seems to give the denominators of several other similarly constructed "Dirichlet Square Roots".

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

Cf. A055653, A318661 (numerators), A318663.

Cf. also A046644, A299150, A317932, A317934, A318314.

up_to = 1+(2^16);
A055653(n) = sumdiv(n, d, if(gcd(n/d, d)==1, eulerphi(d))); \\ From A055653
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&&dA055653(n)));
A318661(n) = numerator(v318661_62[n]);
A318662(n) = denominator(v318661_62[n]);
A318663(n) = valuation(A318662(n),2);

for(n=1, 100, print1(denominator(direuler(p=2, n, ((1 + X^2 - p*X^2 - X)/((1-X)*(1-p*X)))^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 10 2025

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A055653(n) - Sum_{d|n, d>1, d 1.

A318672 Denominators of the sequence whose Dirichlet convolution with itself yields A049599, number of (1+e)-divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Sep 03 2018

The sequence seems to give the denominators of a few other similarly constructed rational valued sequences obtained as "Dirichlet Square Roots" (possibly of A282446 and A318469).

Antti Karttunen, Table of n, a(n) for n = 1..16385

Cf. A049599, A318671 (numerators), A318673.

Cf. also A046644, A299150, A317932, A317934, A318498.

up_to = (2^16)+1;
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&&dA049599(n) = factorback(apply(e -> (1+numdiv(e)),factor(n)[,2]));
v318671_62 = DirSqrt(vector(up_to, n, A049599(n)));
A318671(n) = numerator(v318671_62[n]);
A318672(n) = denominator(v318671_62[n]);
A318673(n) = valuation(A318672(n),2);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A049599(n) - Sum_{d|n, d>1, d 1.

a(n) = 2^A318673(n).

A257100 From fourth root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose fourth power is 1/zeta; sequence gives numerator of b(n).

Original entry on oeis.org

1, -1, -1, -3, -1, 1, -1, -7, -3, 1, -1, 3, -1, 1, 1, -77, -1, 3, -1, 3, 1, 1, -1, 7, -3, 1, -7, 3, -1, -1, -1, -231, 1, 1, 1, 9, -1, 1, 1, 7, -1, -1, -1, 3, 3, 1, -1, 77, -3, 3, 1, 3, -1, 7, 1, 7, 1, 1, -1, -3, -1, 1, 3, -1463, 1, -1, -1, 3, 1, -1, -1, 21, -1, 1, 3, 3, 1, -1, -1, 77, -77, 1, -1, -3, 1, 1, 1, 7, -1, -3, 1, 3, 1, 1, 1, 231, -1, 3, 3, 9

Offset: 1

Wolfgang Hintze, Apr 16 2015

Dirichlet g.f. of b(n) = a(n)/A256691(n) is (zeta(x))^(-1/4).
Denominator is the same as for Dirichlet g.f. (zeta(x))^(+1/4).
Formula holds for general Dirichlet g.f. zeta(x)^(-1/k) with k = 1, 2, ...

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..500 from Wolfgang Hintze)

Cf. family zeta^(-1/k): A257098/A046644 (k=2), A257099/A256689 (k=3), A257100/A256691 (k=4), A257101/A256693 (k=5).

Cf. family zeta^(+1/k): A046643/A046644 (k=2), A256688/A256689 (k=3), A256690/A256691 (k=4), A256692/A256693 (k=5).

k = 4;
c[1, n_] = b[n];
c[k_, n_] := DivisorSum[n, c[1, #1]*c[k - 1, n/#1] & ]
nn = 100; eqs = Table[c[k, n]==MoebiusMu[n], {n, 1, nn}];
sol = Solve[Join[{b[1] == 1}, eqs], Table[b[i], {i, 1, nn}], Reals];
t = Table[b[n], {n, 1, nn}] /. sol[[1]];
num = Numerator[t] (* A257100 *)
den = Denominator[t] (* A256691 *)

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^(-1/4))[n]), ", ")) \\ Vaclav Kotesovec, May 04 2025

with k = 4;
zeta(x)^(-1/k) = Sum_{n>=1} b(n)/n^x;
c(1,n)=b(n); c(k,n) = Sum_{d|n} c(1,d)*c(k-1,n/d), k>1;
Then solve c(k,n) = mu(n) for b(m);
a(n) = numerator(b(n)).
Sum_{j=1..n} A257100(j)/A256691(j) ~ n / (Gamma(-1/4) * log(n)^(5/4)) * (1 + 5*(gamma/4 + 1)/(4*log(n))), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the gamma function. - Vaclav Kotesovec, May 05 2025

A257101 From fifth root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose fifth power is 1/zeta; sequence gives numerator of b(n).

Original entry on oeis.org

1, -1, -1, -2, -1, 1, -1, -6, -2, 1, -1, 2, -1, 1, 1, -21, -1, 2, -1, 2, 1, 1, -1, 6, -2, 1, -6, 2, -1, -1, -1, -399, 1, 1, 1, 4, -1, 1, 1, 6, -1, -1, -1, 2, 2, 1, -1, 21, -2, 2, 1, 2, -1, 6, 1, 6, 1, 1, -1, -2, -1, 1, 2, -1596, 1, -1, -1, 2, 1, -1, -1, 12, -1, 1, 2, 2, 1, -1, -1, 21, -21, 1, -1, -2, 1, 1, 1, 6, -1, -2, 1, 2, 1, 1, 1, 399, -1, 2, 2, 4

Offset: 1

Wolfgang Hintze, Apr 16 2015

Dirichlet g.f. of b(n) = A257101(n)/A256693(n) is (zeta(x))^(-1/5).
Denominator is the same as for Dirichlet g.f. (zeta(x))^(+1/5).
Formula holds for general Dirichlet g.f. zeta(x)^(-1/k) with k = 1, 2, ...

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..500 from Wolfgang Hintze)
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. family zeta^(-1/k): A257098/A046644 (k=2), A257099/A256689 (k=3), A257100/A256691 (k=4), A257101/A256693 (k=5).

Cf. family zeta^(+1/k): A046643/A046644 (k=2), A256688/A256689 (k=3), A256690/A256691 (k=4), A256692/A256693 (k=5).

k = 5;
c[1, n_] = b[n];
c[k_, n_] := DivisorSum[n, c[1, #1]*c[k - 1, n/#1] & ]
nn = 100; eqs = Table[c[k, n]==MoebiusMu[n], {n, 1, nn}];
sol = Solve[Join[{b[1] == 1}, eqs], Table[b[i], {i, 1, nn}], Reals];
t = Table[b[n], {n, 1, nn}] /. sol[[1]];
num = Numerator[t] (* A257101 *)
den = Denominator[t] (* A256693 *)

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^(-1/5))[n]), ", ")) \\ Vaclav Kotesovec, May 04 2025

with k = 5;
zeta(x)^(-1/k) = Sum_{n>=1} b(n)/n^x;
c(1,n)=b(n); c(k,n) = Sum_{d|n} c(1,d)*c(k-1,n/d), k>1;
Then solve c(k,n) = mu(n) for b(m);
a(n) = numerator(b(n)).
Sum_{j=1..n} A257101(j)/A256693(j) ~ n / (Gamma(-1/5) * log(n)^(6/5)) * (1 + 6*(gamma/5 + 1)/(5*log(n))), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the gamma function. - Vaclav Kotesovec, May 05 2025

A317936 Numerators of sequence whose Dirichlet convolution with itself yields A100995 + A063524, that is, the characteristic function of A000961 (prime powers).

Original entry on oeis.org

1, 1, 1, 7, 1, -1, 1, 17, 7, -1, 1, -5, 1, -1, -1, 139, 1, -5, 1, -5, -1, -1, 1, -5, 7, -1, 17, -5, 1, 3, 1, 263, -1, -1, -1, -31, 1, -1, -1, -5, 1, 3, 1, -5, -5, -1, 1, 19, 7, -5, -1, -5, 1, -5, -1, -5, -1, -1, 1, 9, 1, -1, -5, 995, -1, 3, 1, -5, -1, 3, 1, -53, 1, -1, -5, -5, -1, 3, 1, 19, 139, -1, 1, 9, -1, -1, -1, -5, 1, 9

Offset: 1

Antti Karttunen, Aug 14 2018

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

Cf. A000961, A100995, A046644 (denominators).

Cf. also A317939.

up_to = 65537;
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.
v317936aux = DirSqrt(vector(up_to, n, if(1==n,1,isprimepower(n))));
A317936(n) = numerator(v317936aux[n]);
for(n=1,up_to,write("b317936.txt", n, " ", A317936(n)));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A100995(n) - Sum_{d|n, d>1, d 1.

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.

A317939 Numerators of sequence whose Dirichlet convolution with itself yields A080339 = A010051 (characteristic function of primes) + A063524 (1, 0, 0, 0, ...).

Original entry on oeis.org

1, 1, 1, -1, 1, -1, 1, 1, -1, -1, 1, 3, 1, -1, -1, -5, 1, 3, 1, 3, -1, -1, 1, -5, -1, -1, 1, 3, 1, 3, 1, 7, -1, -1, -1, -15, 1, -1, -1, -5, 1, 3, 1, 3, 3, -1, 1, 35, -1, 3, -1, 3, 1, -5, -1, -5, -1, -1, 1, -15, 1, -1, 3, -21, -1, 3, 1, 3, -1, 3, 1, 35, 1, -1, 3, 3, -1, 3, 1, 35, -5, -1, 1, -15, -1, -1, -1, -5, 1, -15, -1, 3, -1, -1, -1, -63, 1, 3, 3

Offset: 1

Antti Karttunen, Aug 14 2018

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

Cf. A010051, A080339, A046644 (denominators).

Cf. also A257098, A317830, A317936, A317937.

up_to = 65537;
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.
v317939aux = DirSqrt(vector(up_to, n, if(1==n,1,isprime(n))));
A317939(n) = numerator(v317939aux[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A010051(n) - Sum_{d|n, d>1, d 1.

cp's OEIS Frontend

A317845 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A001065 (sum of proper divisors) + 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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A318321 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A003961.

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A318662 Denominators of the sequence whose Dirichlet convolution with itself yields A055653, sum of phi(d) over all unitary divisors d of n.

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A318672 Denominators of the sequence whose Dirichlet convolution with itself yields A049599, number of (1+e)-divisors of n.

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A257100 From fourth root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose fourth power is 1/zeta; sequence gives numerator of b(n).

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A257101 From fifth root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose fifth power is 1/zeta; sequence gives numerator of b(n).

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A317936 Numerators of sequence whose Dirichlet convolution with itself yields A100995 + A063524, that is, the characteristic function of A000961 (prime powers).

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