A317932 Denominators of certain "Dirichlet Square Root" sequences: a(n) = A046644(n)/(2^A007949(n)).

1, 2, 1, 8, 2, 2, 2, 16, 2, 4, 2, 8, 2, 4, 2, 128, 2, 4, 2, 16, 2, 4, 2, 16, 8, 4, 2, 16, 2, 4, 2, 256, 2, 4, 4, 16, 2, 4, 2, 32, 2, 4, 2, 16, 4, 4, 2, 128, 8, 16, 2, 16, 2, 4, 4, 32, 2, 4, 2, 16, 2, 4, 4, 1024, 4, 4, 2, 16, 2, 8, 2, 32, 2, 4, 8, 16, 4, 4, 2, 256, 8, 4, 2, 16, 4, 4, 2, 32, 2, 8, 4, 16, 2, 4, 4, 256, 2, 16, 4, 64, 2, 4, 2, 32, 4

Offset

1,2

Comments

These are denominators for rational valued sequences that are obtained as "Dirichlet Square Roots" of sequences b that satisfy the condition b(3) = 2, and b(p) = odd number for any other primes p. For example, A064989, A065769 and A234840. - Antti Karttunen, Aug 31 2018

The original definition was: Denominators of the rational valued sequence whose Dirichlet convolution with itself yields A002487, Stern's Diatomic sequence. However, this definition depends on the conjecture given in A261179.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences related to Stern's sequences

Formula

a(n) = A046644(n)/A318666(n) = 2^A305439(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 can be A064989, A065769 or A234840 for example, conjecturally also A002487.

Multiplicative with a(3^e) = 2^A011371(e), a(p^e) = 2^A005187(e) for any other primes. - Antti Karttunen, Sep 03 2018

Prog

(PARI)
\\ Original program, based on conjectural formula:
A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317931perA317932(n) = if(1==n, n, (A002487(n)-sumdiv(n, d, if((d>1)&&(d<n), A317931perA317932(d)*A317931perA317932(n/d), 0)))/2);
A317932(n) = denominator(A317931perA317932(n));
(PARI)
\\ New fast program implementing the new definition:
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A046644(n) = factorback(apply(e -> 2^A005187(e), factor(n)[, 2]));
A317932(n) = (A046644(n)/2^valuation(n, 3)); \\ Antti Karttunen, Aug 31 2018
(PARI)
A011371(n) = (A005187(n)-n);
A317932(n) = { my(f = factor(n), m=1); for(i=1, #f~, if(3 == f[i, 1], m *= 2^(A011371(f[i, 2])), m *= 2^A005187(f[i, 2]))); (m); }; \\ Antti Karttunen, Sep 03 2018

Crossrefs

Cf. A317930, A318319, A318669 (some of the numerator sequences), A317931 (conjectured, for A002487).

Cf. A305439 (the 2-adic valuation), A318666.

Cf. also A046644, A261179, A299150, A237649, A299150, A317928, A317839, A317843, A317934, A318319.

Sequence in context: A367024 A114706 A046740 * A253583 A130562 A363894

Adjacent sequences: A317929 A317930 A317931 * A317933 A317934 A317935

Keyword

nonn,frac,mult

Author

Antti Karttunen, Aug 11 2018

Extensions

Definition changed, the original (now conjectured alternative definition) moved to the comments section by Antti Karttunen, Aug 31 2018

Keyword:mult added by Antti Karttunen, Sep 03 2018

Status

approved