A046643 From square root of Riemann zeta function: form Dirichlet series Sum b_n/n^s whose square is zeta function; sequence gives numerator of b_n.

1, 1, 1, 3, 1, 1, 1, 5, 3, 1, 1, 3, 1, 1, 1, 35, 1, 3, 1, 3, 1, 1, 1, 5, 3, 1, 5, 3, 1, 1, 1, 63, 1, 1, 1, 9, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 35, 3, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 3, 1, 1, 3, 231, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 3, 3, 1, 1, 1, 35, 35, 1, 1, 3, 1, 1, 1, 5, 1, 3

Offset: 1

Author

N. J. A. Sloane, Dec 11 1999

Keywords

• nonn
• easy
• frac
• nice
• mult

Comments

b(n) = A046643(n)/A046644(n) is multiplicative with b(p^n) = (2n-1)!!/2^n/n!. Dirichlet g.f. of A046643(n)/A046644(n) is sqrt(zeta(x)). - Christian G. Bower, May 16 2005

That is, b(p^n) = A001147(n) / (A000079(n)*A000142(n)) = A010050(n)/A000290(A000165(n)) = (2n)!/((2^n*n!)^2). - Antti Karttunen, Jul 08 2017

Examples

			b_1, b_2, ... = 1, 1/2, 1/2, 3/8, 1/2, 1/4, 1/2, 5/16, 3/8, 1/4, 1/2, 3/16, ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000079, A000165, A001147, A001790, A005187, A010050, A028234, A067029.

Cf. A046644, A046645.

Programs

Maple

b := proc(n) option remember; local c,i,t1; if n = 1 then 1 else c := 1; t1 := divisors(n);
for i from 2 to nops(t1)-1 do c := c-b(t1[ i ])*b(n/t1[ i ]); od; c/2; fi; end;

Mathematica

b[1] = 1; b[n_] := b[n] = (dn = Divisors[n]; c = 1;
Do[c = c - b[dn[[i]]]*b[n/dn[[i]]], {i, 2, Length[dn] - 1}]; c/2); a[n_] := Numerator[b[n]]; a /@ Range[90] (* Jean-François Alcover, Apr 04 2011, after Maple version *)

PARI

A046643perA046644(n) = { my(c=1); if(1==n,c,fordiv(n,d, if((d>1)&&(dA046643perA046644(d)*A046643perA046644(n/d)))); (c/2)); }
A046643(n) = numerator(A046643perA046644(n)); \\ After Maple-program, Antti Karttunen, Jul 08 2017

PARI

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

Scheme

(define (A046643 n) (if (= 1 n) n (* (A001790 (A067029 n)) (A046643 (A028234 n)))))
;; Or, after Christian G. Bower's May 16 2005 comment:
(definec (A046643perA046644 n) (if (= 1 n) n (* (/ (A010050 (A067029 n)) (A000290 (A000165 (A067029 n)))) (A046643perA046644 (A028234 n)))))
(define (A046643 n) (numerator (A046643perA046644 n)))
(define (A046644 n) (denominator (A046643perA046644 n)))
;; Antti Karttunen, Jul 08 2017

Formula

Sum_{b|d} b(d)b(n/d) = 1. Also b_{2^j} = A001790[ j ]/2^A005187[ j ].

From Antti Karttunen, Jul 08 2017: (Start)

Multiplicative with a(p^n) = A001790(n).

a(1) = 1; for n > 1, a(n) = A001790(A067029(n)) * a(A028234(n)).

(End)

Sum_{j=1..n} A046643(j)/A046644(j) ~ n / sqrt(Pi*log(n)) * (1 + (1 - gamma/2)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 04 2025