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Bisection of A001615.
2

%I #39 Nov 24 2022 03:06:27

%S 1,4,6,8,12,12,14,24,18,20,32,24,30,36,30,32,48,48,38,56,42,44,72,48,

%T 56,72,54,72,80,60,62,96,84,68,96,72,74,120,96,80,108,84,108,120,90,

%U 112,128,120,98,144,102,104,192,108,110,152,114,144,168,144,132,168

%N Bisection of A001615.

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

%D Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Mathematics of Long-Range Aperiodic Order, Kluwer, 1997, pp. 9-44.

%H Reinhard Zumkeller, <a href="/A031359/b031359.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael Baake, <a href="https://arxiv.org/abs/math/0605222">Solution of coincidence problem in dimensions d<=4</a>, arXiv:math/0605222 [math.MG], 2006.

%H Michael Baake and Peter A. B. Pleasants, <a href="https://doi.org/10.1515/zna-1995-0802">Algebraic solution of the coincidence problem in two and three dimensions</a>, Zeitschrift für Naturforschung A 50.8 (1995): 711-717. See page 715, the Dirichlet g.f. following Eq. (18).

%F 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

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

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

%F From _Peter Bala_, Mar 19 2019: (Start)

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

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

%F 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

%F 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.

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

%F Bisection of A159634. (End)

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 12/Pi^2 = 1.215854... . - _Amiram Eldar_, Nov 24 2022

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

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

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

%p A031359 := n -> A001615(2*n-1); # _Peter Luschny_, Oct 23 2010

%t 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_ *)

%t 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 *)

%o (Haskell)

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

%o -- _Reinhard Zumkeller_, Jun 03 2013

%o (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 */

%o (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 */

%o (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 */

%Y Cf. A000010, A000203, A001615, A008683, A159634.

%K nonn,easy,nice

%O 1,2

%A _N. J. A. Sloane_

%E Better description from _Vladeta Jovovic_, Jan 25 2002

%E More terms from _Sascha Kurz_, Mar 24 2002