A317934 - OEIS

%I #22 Aug 26 2018 12:26:08

%S 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,8,1,2,1,2,1,1,1,2,2,1,2,2,1,1,1,8,1,1,

%T 1,4,1,1,1,2,1,1,1,2,2,1,1,8,2,2,1,2,1,2,1,2,1,1,1,2,1,1,2,16,1,1,1,2,

%U 1,1,1,4,1,1,2,2,1,1,1,8,8,1,1,2,1,1,1,2,1,2,1,2,1,1,1,8,1,2,2,4,1,1,1,2,1

%N Multiplicative with a(p^n) = 2^A011371(n); denominators for certain "Dirichlet Square Roots" sequences.

%C a(n) is the denominator of certain rational valued sequences f(n), that have 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 a sequence like A034444 and A037445.

%C Many of the same observations as given in A046644 apply also here. Note that A011371 shares with A005187 the property that A011371(x+y) <= A011371(x) + A011371(y), with equivalence attained only when A004198(x,y) = 0, and also the property that A011371(2^(k+1)) = 1 + 2*A011371(2^k).

%C The following list gives such pairs num(n), b(n) for which b(n) is Dirichlet convolution of num(n)/a(n).

%C   Numerators   Dirichlet convolution of numerator(n)/a(n) yields

%C   -------      -----------

%C   A317933      A034444

%C   A317941      A037445

%C   A317940      A046644

%H Antti Karttunen, <a href="/A317934/b317934.txt">Table of n, a(n) for n = 1..65537</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_convolution">Dirichlet convolution</a>

%F a(n) = 2^A317946(n).

%F a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (b(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1, where b is A034444, A037445 or A046644 for example.

%o (PARI)

%o A011371(n) = (n - hammingweight(n));

%o A317934(n) = factorback(apply(e -> 2^A011371(e),factor(n)[,2]));

%Y Cf. A011371, A034444, A317940, A317941, A317946.

%Y Cf. A317933, A317940, A317941 (numerator-sequences).

%Y Cf. also A046644, A299150, A299152, A317832, A317932, A317926 (for denominator sequences of other similar constructions).

%K nonn,frac,mult

%O 1,4

%A _Antti Karttunen_, Aug 12 2018