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
LINKS
FORMULA
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; };
(PARI)
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
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