OFFSET
1,2
COMMENTS
From Antti Karttunen, Aug 21 2018: (Start)
a(n) is the denominator of any rational-valued sequence f(n) which has been defined as f(n) = (1/2) * (b(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)), with f(1) = 1, where b(n) is any integer-valued sequence such that b(1) = 1 and b(p) = odd for all primes p. In other words, integer sequence b is obtained as the Dirichlet Convolution of rational sequence f (the latter is the "Dirichlet Square Root" of the former).
Proof:
Proof is by induction. We assume as our induction hypothesis that the given multiplicative formula for A046644 (resp. additive formula for A046645) holds for all proper divisors d|n, d<n. As base cases, we have a(1) = 1 and for primes p, as f(p) = b(p)/2 = odd/2, a(p) = 2 and A046645(p) = 1. [Remark: for squares of primes, f(p^2) = (4*b(p^2) - 1)/8, thus a(p^2) = 8.]
First we note that A005187(x+y) <= A005187(x) + A005187(y), with equivalence attained only when A004198(x,y) = 0, that is, when x and y do not have any 1-bits in the shared positions. Let m = Sum_{e} A005187(e), with e ranging over the exponents in prime factorization of n.
For [case A] any n in A268388 it happens that only when d (and thus also n/d) are infinitary divisors of n will Sum_{e} A005187(e) [where e now ranges over the union of multisets of exponents in the prime factorizations of d and n/d] attain value m, which is the maximum possible for such sums computed for all divisor pairs d and n/d. For any n in A268388, A037445(n) = 2^k, k >= 2, thus A037445(n) - 2 = 2 mod 4 (excluding 1 and n from the count, thus -2). Thus, in the recursive formula above, the maximal denominator that occurs in the sum is 2^m which occurs k times, with k being an even number, but not a multiple of 4, thus the factor (1/2) in the front of the whole sum will ensure that the denominator of the whole expression is 2^m [which thus is equal to 2^A046645(n) = a(n)].
On the other hand [case B], for squares in A050376 (A082522, numbers of the form p^(2^k) with p prime and k>0), all the sums A005187(x)+A005187(y), where x+y = 2^k, 0 < x <= y < 2^k are less than A005187(2^k), thus it is the lonely "middle pair" f(p^(2^(k-1))) * f(p^(2^(k-1))) among all the pairs f(d)*f(n/d), 1 < d < n = p^(2^k) which yields the maximal denominator. Furthermore, as it occurs an odd number of times (only once), the common factor (1/2) for the whole sum will increase the exponent of 2 in denominator by one, which will be (2*A005187(2^(k-1))) + 1 = A005187(2^k) = A046645(p^(2^k)).
(End)
From Antti Karttunen, Aug 21 2018: (Start)
The following list gives a few such pairs num(n), b(n) for which b(n) is Dirichlet convolution of num(n)/a(n). Here ε stands for sequence A063524 (1, 0, 0, ...).
Numerators Dirichlet convolution of numerator(n)/a(n) yields
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(End)
This sequence gives an upper bound for the denominators of any rational-valued sequence obtained as the "Dirichlet Square Root" of any integer-valued sequence. - Andrew Howroyd, Aug 23 2018
LINKS
FORMULA
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_] := Denominator[b[n]]; a /@ Range[78] (* Jean-François Alcover, Apr 04 2011, after Maple code in A046643 *)
a18804[n_] := Sum[n EulerPhi[d]/d, {d, Divisors[n]}];
f[1] = 1; f[n_] := f[n] = 1/2 (a18804[n] - Sum[f[d] f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
a[n_] := f[n] // Denominator;
Array[a, 78] (* Jean-François Alcover, Sep 13 2018, after A318443 *)
PROG
(PARI)
A046643perA046644(n) = { my(c=1); if(1==n, c, fordiv(n, d, if((d>1)&&(d<n), c -= (A046643perA046644(d)*A046643perA046644(n/d)))); (c/2)); }
A046644(n) = denominator(A046643perA046644(n)); \\ After the Maple-program given in A046643, Antti Karttunen, Jul 08 2017
(PARI)
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
(Scheme, with memoization-macro definec)
CROSSREFS
KEYWORD
nonn,easy,frac,nice,mult
AUTHOR
STATUS
approved