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A031359 Bisection of A001615. 1
1, 4, 6, 8, 12, 12, 14, 24, 18, 20, 32, 24, 30, 36, 30, 32, 48, 48, 38, 56, 42, 44, 72, 48, 56, 72, 54, 72, 80, 60, 62, 96, 84, 68, 96, 72, 74, 120, 96, 80, 108, 84, 108, 120, 90, 112, 128, 120, 98, 144, 102, 104, 192, 108, 110, 152, 114, 144, 168, 144, 132, 168 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Number of coincidence site lattices of index 2n-1 in lattice Z^3.

REFERENCES

M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

M. Baake, Solution of coincidence problem in dimensions d<=4, arXiv:math/0605222 [math.MG], 2006.

FORMULA

a(n) = b(2*n - 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = p^(e-1) * (p+1) if p > 2. - Michael Somos, Nov 22 2013

Dirichlet series: Product (1+p^(-s))/(1-p^(1-s)); p != 2.

a(n) = A001615(2*n - 1).

From Peter Bala, Mar 19 2019: (Start)

a(n) = (2*n - 1)*Product_{p|(2*n-1), p prime} (1 + 1/p).

a(n) = Sum_{ d|(2*n-1) } mu(d)^2*(2*n-1)/d, where mu(n) = A008683(n) is the Möbius function.

a(n) = Sum_{ d^2|(2*n-1) } mu(d)*sigma((2*n-1)/d^2), where sigma(n) = A000203(n) is the sum of the divisors of n, and also

a(n) = Sum_{ d|(2*n-1) } 2^omega(d)*phi((2*n-1)/d), where omega(n) = A001221(n) is the number of different primes dividing n and phi(n) = A000010 is the Euler totient function.

O.g.f.: Sum_{n >= 1} mu(2*n-1)^2*x^n*(1 + x^(2*n-1))/(1 - x^(2*n-1))^2.

Bisection of A159634. (End)

EXAMPLE

G.f. = x + 4*x^2 + 6*x^3 + 8*x^4 + 12*x^5 + 12*x^6 + 14*x^7 + 24*x^8 + ...

G.f. = q + 4*q^3 + 6*q^5 + 8*q^7 + 12*q^9 + 12*q^11 + 14*q^13 + 24*q^15 + ...

MAPLE

A001615 := n -> mul((op(1, i)+1)*op(1, i)^(op(2, i)-1), i=op(2, numtheory[ifactors](n)));

A031359 := n -> A001615(2*n-1); # Peter Luschny, Oct 23 2010

MATHEMATICA

a[n_] := (2n-1)*Sum[ MoebiusMu[d]^2/d, {d, Divisors[2n-1]}]; Table[a[n], {n, 1, 62}] (* Jean-François Alcover, Jan 18 2012, after Michael Somos *)

a[ n_] := If[ n < 1, 0, With[{m = 2 n - 1}, m Sum[ MoebiusMu[ d]^2 / d, {d, Divisors[m]}]]] (* Michael Somos, Nov 22 2013 *)

PROG

(Haskell)

a031359 = a001615 . (subtract 1) . (* 2)

-- Reinhard Zumkeller, Jun 03 2013

(PARI) {a(n) = my(m); if( n<1, 0, m = 2*n - 1; m * sumdiv( m, d, moebius(d)^2 / d))} /* Michael Somos, Nov 22 2013 */

(PARI) {a(n) = my(m); if( n<1, 0, m = 2*n - 1; direuler( p=2, m, (1 + X) / (1 - p*X))[ m])} /* Michael Somos, Nov 22 2013 */

(PARI) {a(n) = my(A, p, e); if( n<1, 0, n = 2*n - 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, p^(e-1) * (p + 1)))))} /* Michael Somos, Nov 22 2013 */

CROSSREFS

Cf. A001615, A159634.

Sequence in context: A320126 A234520 A275789 * A274790 A179852 A281020

Adjacent sequences:  A031356 A031357 A031358 * A031360 A031361 A031362

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Better description from Vladeta Jovovic, Jan 25 2002

More terms from Sascha Kurz, Mar 24 2002

STATUS

approved

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Last modified November 13 21:26 EST 2019. Contains 329106 sequences. (Running on oeis4.)