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A317934 Multiplicative with a(p^n) = 2^A011371(n); denominators for certain "Dirichlet Square Roots" sequences. 12
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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.

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).

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

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

  -------      -----------

  A317933      A034444

  A317941      A037445

  A317940      A046644

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Wikipedia, Dirichlet convolution

FORMULA

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

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.

PROG

(PARI)

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

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

CROSSREFS

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

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

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

Sequence in context: A255231 A294874 A318324 * A279848 A001826 A003641

Adjacent sequences:  A317931 A317932 A317933 * A317935 A317936 A317937

KEYWORD

nonn,frac,mult

AUTHOR

Antti Karttunen, Aug 12 2018

STATUS

approved

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Last modified August 3 01:53 EDT 2021. Contains 346429 sequences. (Running on oeis4.)