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Revision History for A035185

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A035185

Number of divisors of n == 1 or 7 (mod 8) minus number of divisors of n == 3 or 5 (mod 8).
(history; published version)

#40 by Michael De Vlieger at Tue Oct 11 07:59:19 EDT 2022

STATUS

proposed

approved

#39 by Hugo Pfoertner at Tue Oct 11 06:26:50 EDT 2022

STATUS

editing

proposed

#38 by Hugo Pfoertner at Tue Oct 11 06:24:23 EDT 2022

COMMENTS

Let zetaQ(sqrt(2))(s) = Sum (1/(Z(sqrt(2)):A)^s), a Dedekind zeta function, where A runs through the nonzero ideals of Z(sqrt(2)) and where (Z(sqrt(2)):A) is the norm of A; then zetaQ(sqrt(2))(s) = Sum_{n>=1}, a(n)/n^s); Sum{k=1..n} a(k) is asymptotic to c*n where c = log(1 + sqrt(2))/sqrt(2). - _Benoit Cloitre_, Jan 01 2003

EXTENSIONS

Additional comments from Benoit Cloitre, Jan 01 2003

STATUS

reviewed

editing

#37 by Joerg Arndt at Tue Oct 11 05:16:51 EDT 2022

STATUS

proposed

reviewed

#36 by Amiram Eldar at Tue Oct 11 02:07:34 EDT 2022

STATUS

editing

proposed

#35 by Amiram Eldar at Tue Oct 11 01:56:51 EDT 2022

PROG

(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p%8>1 && p%8<7, !(e%2), e+1)))}; /* _\\ _Michael Somos_, Aug 17 2006 */

(PARI) {a(n) = if( n<1, 0, polcoeff( sum(k=1, n, x^k * (1 - x^(2*k)) / (1 + x^(4*k)), x * O(x^n)), n))}; /* _\\ _Michael Somos_, Jul 06 2015 */

#34 by Amiram Eldar at Tue Oct 11 01:56:13 EDT 2022

FORMULA

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(sqrt(2)+1)/sqrt(2) = A091648/A002193 = 0.623225... . - Amiram Eldar, Oct 11 2022

CROSSREFS

Cf. A002193, A091648.

STATUS

approved

editing

#33 by N. J. A. Sloane at Tue Mar 22 16:22:26 EDT 2022

CROSSREFS

Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

Discussion

Tue Mar 22

16:22

OEIS Server: https://oeis.org/edit/global/2934

#32 by N. J. A. Sloane at Tue Mar 22 16:17:11 EDT 2022

STATUS

editing

approved

#31 by N. J. A. Sloane at Tue Mar 22 16:17:10 EDT 2022

COMMENTS

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 8. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

CROSSREFS

Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.

Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

STATUS

approved

editing