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

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

1,4

Comments

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, d<n} f(d) * f(n/d)), with f(1) = 1, where b(n) is a sequence like A034444 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

Links

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

Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Wikipedia, Dirichlet convolution

Formula

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<n} f(d) * f(n/d)) for n > 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

Prog

(PARI)
A011371(n) = (n - hammingweight(n));
A317934(n) = factorback(apply(e -> 2^A011371(e), factor(n)[, 2]));
(PARI) for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-2*X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 07 2025
(PARI) for(n=1, 100, print1(denominator(direuler(p=2, n, ((1+X)/(1-X))^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

Crossrefs

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.

Sequence in context: A347456 A294874 A318324 * A353376 A279848 A001826

Adjacent sequences: A317931 A317932 A317933 * A317935 A317936 A317937

Keyword

nonn,frac,mult

Author

Antti Karttunen, Aug 12 2018

Status

approved