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 A046644 From square root of Riemann zeta function: form Dirichlet series Sum b_n/n^s whose square is zeta function; sequence gives denominator of b_n. 51
 1, 2, 2, 8, 2, 4, 2, 16, 8, 4, 2, 16, 2, 4, 4, 128, 2, 16, 2, 16, 4, 4, 2, 32, 8, 4, 16, 16, 2, 8, 2, 256, 4, 4, 4, 64, 2, 4, 4, 32, 2, 8, 2, 16, 16, 4, 2, 256, 8, 16, 4, 16, 2, 32, 4, 32, 4, 4, 2, 32, 2, 4, 16, 1024, 4, 8, 2, 16, 4, 8, 2, 128, 2, 4, 16, 16, 4, 8 (list; graph; refs; listen; history; text; internal format)
 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= 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   -------      -----------   A046643      A000012   A257098      A008683   A317935      A003557   A318321      A003961   A317830      A175851   A317833      A078898   A317834      A078899   A317847      A303757   A317835      ε + A003415   A317845      ε + A001065   A317846      ε + A051953   A317936      ε + A100995   A317937      ε + A001221   A317938      ε + A001222   A317939      ε + A010051 (= A080339) (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 Antti Karttunen, Table of n, a(n) for n = 1..8192 Wikipedia, Dirichlet convolution FORMULA From Antti Karttunen, Jul 08 2017: (Start) Multiplicative with a(p^n) = 2^A005187(n). a(1) = 1; for n > 1, a(n) = A000079(A005187(A067029(n))) * a(A028234(n)). a(n) = A000079(A046645(n)). (End) MATHEMATICA b = 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 (* Jean-François Alcover, Apr 04 2011, after Maple code in A046643 *) a18804[n_] := Sum[n EulerPhi[d]/d, {d, Divisors[n]}]; f = 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>=1, s+=n); s; }; A046644(n) = factorback(apply(e -> 2^A005187(e), factor(n)[, 2])); \\ Antti Karttunen, Aug 12 2018 (Scheme, with memoization-macro definec) (definec (A046644 n) (if (= 1 n) n (* (A000079 (A005187 (A067029 n))) (A046644 (A028234 n))))) ;; Antti Karttunen, Jul 08 2017 CROSSREFS Cf. A000079, A005187, A028234, A037445, A050376, A067029, A082522, A268388. See A046643 for more details. See also A046645, A317940. Cf. A299150, A299152, A317832, A317926, A317932, A317934 (for denominator sequences of other similar constructions). Sequence in context: A029623 A325753 A208133 * A161915 A174354 A011147 Adjacent sequences:  A046641 A046642 A046643 * A046645 A046646 A046647 KEYWORD nonn,easy,frac,nice,mult AUTHOR STATUS approved

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Last modified October 14 14:06 EDT 2019. Contains 328017 sequences. (Running on oeis4.)