A317937 -id:A317937 - cp's OEIS frontend

A318319 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A064989.

1, 1, 1, 3, 3, 1, 5, 5, 3, 3, 7, 3, 11, 5, 3, 35, 13, 3, 17, 9, 5, 7, 19, 5, 27, 11, 5, 15, 23, 3, 29, 63, 7, 13, 15, 9, 31, 17, 11, 15, 37, 5, 41, 21, 9, 19, 43, 35, 75, 27, 13, 33, 47, 5, 21, 25, 17, 23, 53, 9, 59, 29, 15, 231, 33, 7, 61, 39, 19, 15, 67, 15, 71, 31, 27, 51, 35, 11, 73, 105, 35, 37, 79, 15, 39, 41, 23, 35, 83, 9, 55, 57

Offset: 1

Antti Karttunen, Aug 24 2018

Multiplicative because A064989 is.

No negative terms among the first 2^20 terms.

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

Cf. A064989, A317932 (seems to give denominators, see A261179).

Cf. also A318321.

up_to = 16384;
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
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.
v318319aux = DirSqrt(vector(up_to, n, A064989(n)));
A318319(n) = numerator(v318319aux[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A064989(n) - Sum_{d|n, d>1, d 1.

A318450 Denominators of the sequence whose Dirichlet convolution with itself yields A001511, the 2-adic valuation of 2n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 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. A001511, A318449 (numerators), A318451.

a1511[n_] := IntegerExponent[2n, 2];
f[1] = 1; f[n_] := f[n] = 1/2 (a1511[n] - Sum[f[d] f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
Table[f[n] // Denominator, {n, 1, 105}] (* Jean-François Alcover, Sep 13 2018 *)

up_to = 65537;
A001511(n) = 1+valuation(n,2);
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.
v318449_51 = DirSqrt(vector(up_to, n, A001511(n)));
A318450(n) = denominator(v318449_51[n]);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A001511(n) - Sum_{d|n, d>1, d 1.

a(n) = 2^A318451(n).

A318453 Numerators of the sequence whose Dirichlet convolution with itself yields A001227, number of odd divisors of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 63, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 231, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 63, 1, 1, 1, 3, 1, 1, 1, 5, 1

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (16384 terms)

Cf. A001227.
Cf. A318454 (gives the denominators).
Differs from A318313 for the first time at n=81, where a(81) = 1, while A318313(81) = 3.

f[1] = 1; f[n_] := f[n] = 1/2 (Sum[Mod[d, 2], {d, Divisors[n]}] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]);
Table[f[n] // Numerator, {n, 1, 105}] (* Jean-François Alcover, Sep 13 2018 *)

up_to = 16384;
A001227(n) = numdiv(n>>valuation(n, 2)); \\ From A001227
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.
v318453_54 = DirSqrt(vector(up_to, n, A001227(n)));
A318453(n) = numerator(v318453_54[n]);
A318454(n) = denominator(v318453_54[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A001227(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A318453(k) / A318454(k) ~ n/sqrt(2). - Vaclav Kotesovec, May 09 2025

A318454 Denominators of the sequence whose Dirichlet convolution with itself yields A001227, number of odd divisors of n.

Original entry on oeis.org

1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 128, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 256, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 128, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 1024, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 128, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 256, 1, 2, 1, 8, 1, 2, 1, 16, 1

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 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. A001227.

Cf. A318453 (numerators), A318455.

f[1] = 1; f[n_] := f[n] = 1/2 (Sum[Mod[d, 2], {d, Divisors[n]}] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]);
Table[f[n] // Denominator, {n, 1, 105}] (* Jean-François Alcover, Sep 13 2018 *)

up_to = 16384;
A001227(n) = numdiv(n>>valuation(n, 2)); \\ From A001227
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.
v318453_54 = DirSqrt(vector(up_to, n, A001227(n)));
A318454(n) = denominator(v318453_54[n]);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A001227(n) - Sum_{d|n, d>1, d 1.
a(n) = 2^A318455(n).
Sum_{k=1..n} A318453(k) / a(k) ~ n/sqrt(2). - Vaclav Kotesovec, May 09 2025

A318498 Denominators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 30 2018

The sequence seems to give the denominators of a few other similarly constructed rational valued sequences obtained as "Dirichlet Square Roots" (of possibly A092520 and A293443).

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

Cf. A061389, A318497 (numerators), A318499.

Cf. also A299150, A046644.

up_to = 65537;
A061389(n) = factorback(apply(e -> (1+eulerphi(e)),factor(n)[,2]));
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.
v318497_98 = DirSqrt(vector(up_to, n, A061389(n)));
A318497(n) = numerator(v318497_98[n]);
A318498(n) = denominator(v318497_98[n]);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A061389(n) - Sum_{d|n, d>1, d 1.

a(n) = 2^A318499(n).

A317847 Numerators of sequence whose Dirichlet convolution with itself yields A303757, the ordinal transform of function a(1) = 0; a(n) = phi(n) for n > 1, where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 1, 1, 7, 1, 5, 1, 9, 7, 5, 1, 15, 1, 5, 1, 43, 1, 15, 1, 7, 3, 3, 1, 5, 3, 5, 9, 15, 1, 9, 1, 87, 3, 5, 1, 1, 1, 5, 3, 13, 1, 11, 1, 11, 15, 3, 1, 187, 7, 19, 1, 15, 1, 5, 3, 21, 3, 3, 1, -1, 1, 3, 11, 387, 1, 9, 1, 7, 1, 13, 1, 119, 1, 7, 19, 23, 3, 19, 1, 139, -21, 7, 1, 21, 1, 5, 1, 39, 1, 67, 3, 3, 5, 3, 5, 451, 1, 15, 19, 69, 1, 13, 1, -27, 7

Offset: 1

Antti Karttunen, Aug 14 2018

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

Cf. A000010, A303757, A046644 (denominators).

Cf. also A317830, A317833, A317834, A317937.

A303757[n_] := If[n == 2, 1, Count[EulerPhi[Range[n]] - EulerPhi[n], 0]];
f[n_] := f[n] = If[n == 1, 1, (1/2)(A303757[n] -
     Sum[If[1Jean-François Alcover, Dec 20 2021 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
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.
v303757 = ordinal_transform(vector(up_to,n,if(1==n,0,eulerphi(n))));
v317847 = DirSqrt(vector(up_to, n, v303757[n]));
A317847(n) = numerator(v317847[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A303757(n) - Sum_{d|n, d>1, d 1.

A317941 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A037445, number of infinitary divisors (or i-divisors) of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, -11, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, -5, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Antti Karttunen, Aug 22 2018

Multiplicative because A037445 is.

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

Cf. A037445, A317934 (denominators).

Cf. also A317933, A317940.

up_to = 1+(2^16);
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.
A037445(n) = factorback(apply(a -> 2^hammingweight(a), factorint(n)[, 2])) \\ From A037445
v317941aux = DirSqrt(vector(up_to, n, A037445(n)));
A317941(n) = numerator(v317941aux[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A037445(n) - Sum_{d|n, d>1, d 1.

A318313 Numerators of the sequence whose Dirichlet convolution with itself yields A068068, number of odd unitary divisors of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 63, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 231, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 3, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 63, 1, 1, 1, 3, 1, 1, 1, 5, 1

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (16384 terms)

Cf. A068068, A318314 (denominators).

Differs from A318453 for the first time at n=81, where a(81) = 3, while A318453(81) = 1.

up_to = 16384;
A068068(n) = (2^omega(n>>valuation(n, 2))); \\ From A068068
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.
v318313_15 = DirSqrt(vector(up_to, n, A068068(n)));
A318313(n) = numerator(v318313_15[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A068068(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A318313(k) / A318314(k) ~ 2*n/Pi. - Vaclav Kotesovec, May 10 2025

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)

A318443 Numerators of the sequence whose Dirichlet convolution with itself yields A018804, Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n).

Original entry on oeis.org

1, 3, 5, 23, 9, 15, 13, 91, 59, 27, 21, 115, 25, 39, 45, 1451, 33, 177, 37, 207, 65, 63, 45, 455, 179, 75, 353, 299, 57, 135, 61, 5797, 105, 99, 117, 1357, 73, 111, 125, 819, 81, 195, 85, 483, 531, 135, 93, 7255, 363, 537, 165, 575, 105, 1059, 189, 1183, 185, 171, 117, 1035, 121, 183, 767, 46355, 225, 315, 133, 759, 225, 351, 141, 5369

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Because A018804 gets only odd values on primes, A046644 gives the sequence of denominators. Because both of those sequences are multiplicative, this is also.

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

Cf. A018804, A046644 (denominators).

Cf. also A318444.

a18804[n_] := Sum[n EulerPhi[d]/d, {d, Divisors[n]}];
f[1] = 1; f[n_] := f[n] = 1/2 (a18804[n] - Sum[f[d] f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
a[n_] := f[n] // Numerator;
Array[a, 72] (* Jean-François Alcover, Sep 13 2018 *)

up_to = 16384;
A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
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.
v318443aux = DirSqrt(vector(up_to, n, A018804(n)));
A318443(n) = numerator(v318443aux[n]);

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

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A018804(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A318443(k) / A046644(k) ~ sqrt(3/2)*n^2/Pi. - Vaclav Kotesovec, May 10 2025

cp's OEIS Frontend

A318319 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A064989.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A318450 Denominators of the sequence whose Dirichlet convolution with itself yields A001511, the 2-adic valuation of 2n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A318453 Numerators of the sequence whose Dirichlet convolution with itself yields A001227, number of odd divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A318454 Denominators of the sequence whose Dirichlet convolution with itself yields A001227, number of odd divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A318498 Denominators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A317847 Numerators of sequence whose Dirichlet convolution with itself yields A303757, the ordinal transform of function a(1) = 0; a(n) = phi(n) for n > 1, where phi is Euler's totient function (A000010).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A317941 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A037445, number of infinitary divisors (or i-divisors) of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A318313 Numerators of the sequence whose Dirichlet convolution with itself yields A068068, number of odd unitary divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs