A317941 -id:A317941

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A317934

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

+10
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).

KEYWORD

nonn,frac,mult

AUTHOR

Antti Karttunen, Aug 12 2018

STATUS

approved

A317940

Numerators of sequence whose Dirichlet convolution with itself yields A046644.

+10
5

1, 1, 1, 7, 1, 1, 1, 9, 7, 1, 1, 7, 1, 1, 1, 427, 1, 7, 1, 7, 1, 1, 1, 9, 7, 1, 9, 7, 1, 1, 1, 471, 1, 1, 1, 49, 1, 1, 1, 9, 1, 1, 1, 7, 7, 1, 1, 427, 7, 7, 1, 7, 1, 9, 1, 9, 1, 1, 1, 7, 1, 1, 7, 4099, 1, 1, 1, 7, 1, 1, 1, 63, 1, 1, 7, 7, 1, 1, 1, 427, 427, 1, 1, 7, 1, 1, 1, 9, 1, 7, 1, 7, 1, 1, 1, 471, 1, 7, 7, 49, 1, 1, 1, 9, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

Multiplicative because A046644 is.

No negative terms among the first 2^20 terms. Is the sequence nonnegative?

LINKS

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

FORMULA

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A046644(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

PROG

(PARI)

up_to = 65537;

DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };

A046644(n) = factorback(apply(e -> 2^A005187(e), factor(n)[, 2]));

v317940aux = DirSqrt(vector(up_to, n, A046644(n)));

A317940(n) = numerator(v317940aux[n]);