A317933 -id:A317933 - 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.

A317934 Multiplicative with a(p^n) = 2^A011371(n); denominators for certain "Dirichlet Square Roots" sequences.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 12 2018

a(n) is the denominator of certain rational valued sequences f(n), that have been defined as f(n) = (1/2) * (b(n) - Sum_{d|n, d>1, dA034444 and A037445.
Many of the same observations as given in A046644 apply also here. Note that A011371 shares with A005187 the property that A011371(x+y) <= A011371(x) + A011371(y), with equivalence attained only when A004198(x,y) = 0, and also the property that A011371(2^(k+1)) = 1 + 2*A011371(2^k).
The following list gives such pairs num(n), b(n) for which b(n) is Dirichlet convolution of num(n)/a(n).
  Numerators   Dirichlet convolution of numerator(n)/a(n) yields
  -------      -----------
  A317933      A034444
  A317941      A037445
  A317940      A046644
Expansion of Dirichlet g.f. Product_{prime} 1/(1 - 2/p^s)^(1/2) is A046643/A317934. - Vaclav Kotesovec, May 08 2025

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)
Wikipedia, Dirichlet convolution

Cf. A011371, A034444, A317940, A317941, A317946.
Cf. A317933, A317940, A317941 (numerator-sequences).
Cf. also A046644, A299150, A299152, A317832, A317932, A317926 (for denominator sequences of other similar constructions).
Cf. A046643, A167864, A347195.

A011371(n) = (n - hammingweight(n));
A317934(n) = factorback(apply(e -> 2^A011371(e),factor(n)[,2]));

for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-2*X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 07 2025

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

a(n) = 2^A317946(n).
a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (b(n) - Sum_{d|n, d>1, d 1, where b is A034444, A037445 or A046644 for example.
Sum_{k=1..n} A046643(k)/a(k) ~ n * sqrt(A167864*log(n)/(Pi*log(2))) * (1 + (4*(gamma - 1) + 5*log(2) - 4*A347195)/(8*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 08 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.

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.

cp's OEIS Frontend

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

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A317934 Multiplicative with a(p^n) = 2^A011371(n); denominators for certain "Dirichlet Square Roots" sequences.

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

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