A241122 -id:A241122 - cp's OEIS frontend

A005134 Number of n-dimensional unimodular lattices (or quadratic forms).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 8, 9, 13, 16, 28, 40, 68, 117, 297, 665, 2566, 17059, 374062

Offset: 0

N. J. A. Sloane

King gives the lower bounds a(29) >= 37938009 and a(30) >= 20169641025. - Robin Visser, Feb 08 2025

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 49.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Bill Allombert and Gaëtan Chenevier, Unimodular Hunting II, arXiv:2410.19569 [math.NT], 2024.
Gaëtan Chenevier, Unimodular Hunting, arXiv:2410.18788 [math.NT], 2024.
Oliver D. King, A mass formula for unimodular lattices with no roots, Math. Comp., 72 (2003), no. 242, 839-863. See page 854.

Cf. A054907, A054908, A054909, A054911.

If 8 divides n, then a(n) = A054911(n) + A054909(n/8), otherwise a(n) = A054911(n). - Robin Visser, Jan 24 2025

a(n) >= 2*A241121(n)/A241122(n). - Robin Visser, Feb 08 2025

a(26)-a(28) added from Bill Allombert's and Gaëtan Chenevier's computations by Robin Visser, Jan 24 2025

A054911 Number of n-dimensional odd unimodular lattices (or quadratic forms).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 9, 13, 16, 28, 40, 68, 117, 273, 665, 2566, 17059, 374062

Offset: 0

N. J. A. Sloane, May 23 2000

a(n) is also the class number of Z^n (the standard lattice with the identity as the basis), as every n-dimensional odd unimodular lattice lies in the same genus as Z^n. - Robin Visser, Jan 24 2025

King gives the lower bounds a(29) >= 37938009 and a(30) >= 20169641025. - Robin Visser, Feb 08 2025

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 49.

Bill Allombert and Gaëtan Chenevier, Unimodular Hunting II, arXiv:2410.19569 [math.NT], 2024.
Gaëtan Chenevier, Unimodular hunting, Cogent Seminar, Jul 05 2021.
Gaëtan Chenevier, Unimodular hunting, Modular Forms Workshop, Oberwolfach online, Feb 2021.
Gaëtan Chenevier, Unimodular Hunting, arXiv:2410.18788 [math.NT], 2024.
Steven R. Finch, Minkowski-Siegel mass constants [Broken link]
Steven R. Finch, Minkowski-Siegel mass constants
Oliver D. King, A mass formula for unimodular lattices with no roots, Math. Comp., 72 (2003), no. 242, 839-863. See Table 3 page 854.

Cf. A005134, A054907, A054908, A054909, A241121, A241122.

function a(n)
    if n lt 3 then return Min(1,n); end if;
    L := NumberFieldLattice(QNF(), n);
    return #GenusRepresentatives(L);
end function;  // Robin Visser, Jan 24 2025

If 8 divides n, then a(n) = A005134(n) - A054909(n/8), otherwise a(n) = A005134(n). - Robin Visser, Jan 24 2025

a(n) >= 2*A241121(n)/A241122(n). - Robin Visser, Feb 08 2025

A241121 Type I Minkowski-Siegel mass constants (numerators).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 31, 31, 691, 42151, 29713, 505121, 642332179, 692319119, 8003636403977, 248112728523287, 593468652605200909, 50904295073459007001, 1015740532498234470066371, 701876707956280018815862361

Offset: 1

Jean-François Alcover, Apr 16 2014

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, Chapter 16.

Robin Visser, Table of n, a(n) for n = 1..80
Steven R. Finch, Minkowski-Siegel mass constants, January 9, 2005. [Cached copy, with permission of the author]
Bernd C. Kellner, On asymptotic constants related to products of Bernoulli numbers and factorials, Integers 9 (2009), Article #A08, 83-106.
John Milnor and Dale Husemoller, Symmetric Bilinear Forms, Ergeb. Math. Grenzgeb., Band 73, Springer-Verlag, New York-Heidelberg, 1973. See page 50.
Wikipedia, Smith-Minkowski-Siegel mass formula

Cf. A005134, A054909, A054911, A241122.

a[n_ /; 1 <= n <= 8] = 1/(n!*2^n); a[n_ /; n > 8] := (k = Quotient[n, 2]; r = Mod[n, 8]; Switch[r, 0, (1-2^-k)*(1 + 2^(1-k))/(k!*2)*BernoulliB[k]*Product[BernoulliB[j], {j, 2, 2k-2, 2}], 1|7, (2^k+1)/(k!*2^(2k+1))*Product[BernoulliB[j], {j, 2, 2k, 2}], 2|6, 1/((k-1)!*2^(2k+1))*EulerE[k-1]*Product[BernoulliB[j], {j, 2, 2k-2, 2}], 3|5, (2^k-1)/(k!*2^(2k+1))*Product[BernoulliB[j], {j, 2, 2k, 2}], 4, (1-2^-k)*(1-2^(1-k))/(k!*2)*BernoulliB[k]*Product[BernoulliB[j], {j, 2, 2k-2, 2}], _, Print["error n = ", n]; 0] // Abs); Table[a[n] // Numerator, {n, 1, 30}]

def a(n):
    if n==1: M = 1/2
    elif n%8 == 0: M = (1-2^(-n/2))*(1+2^(1-n/2))*bernoulli(n/2)/(2*factorial(n/2))
    elif n%8 in [1,7]: M = (2^((n-1)/2) + 1)/(2^n*factorial((n-1)/2))
    elif n%8 in [2,6]: M = euler_number(n/2-1)/(factorial(n/2-1)*(2^(n+1)))
    elif n%8 in [3,5]: M = (2^((n-1)/2) - 1)/(2^n*factorial((n-1)/2))
    elif n%8 == 4: M = (1-2^(-n/2))*(1-2^(1-n/2))*bernoulli(n/2)/(2*factorial(n/2))
    M *= product([abs(bernoulli(i)) for i in range(2, n, 2)])
    return abs(M).numerator()  # Robin Visser, Feb 08 2025

a(n)/A241122(n) ~ C * (n/(2*Pi*e*sqrt(e)))^(n^2/4) * (8*Pi*e/n)^(n/4) * (1/n)^(1/24), where C = 2^(-5/4) * e^(1/24) * exp(1/12 - zeta'(-1))^(-1/2) * Product_{i>=1} zeta(2*i) = 0.7048648734... [Kellner and Milnor--Husemoller]. - Robin Visser, Feb 08 2025

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A005134 Number of n-dimensional unimodular lattices (or quadratic forms).

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A054911 Number of n-dimensional odd unimodular lattices (or quadratic forms).

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A241121 Type I Minkowski-Siegel mass constants (numerators).

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