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%I #35 Sep 23 2018 21:28:38
%S 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,
%T 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,
%U 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
%N Denominators of certain "Dirichlet Square Root" sequences: a(n) = A046644(n)/(2^A007949(n)).
%C 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
%C 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.
%H Antti Karttunen, <a href="/A317932/b317932.txt">Table of n, a(n) for n = 1..16384</a>
%H <a href="/index/St#Stern">Index entries for sequences related to Stern's sequences</a>
%F a(n) = A046644(n)/A318666(n) = 2^A305439(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 can be A064989, A065769 or A234840 for example, conjecturally also A002487.
%F Multiplicative with a(3^e) = 2^A011371(e), a(p^e) = 2^A005187(e) for any other primes. - _Antti Karttunen_, Sep 03 2018
%o (PARI)
%o \\ Original program, based on conjectural formula:
%o A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
%o A317931perA317932(n) = if(1==n,n,(A002487(n)-sumdiv(n,d,if((d>1)&&(d<n),A317931perA317932(d)*A317931perA317932(n/d),0)))/2);
%o A317932(n) = denominator(A317931perA317932(n));
%o (PARI)
%o \\ New fast program implementing the new definition:
%o A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
%o A046644(n) = factorback(apply(e -> 2^A005187(e),factor(n)[,2]));
%o A317932(n) = (A046644(n)/2^valuation(n,3)); \\ _Antti Karttunen_, Aug 31 2018
%o (PARI)
%o A011371(n) = (A005187(n)-n);
%o 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
%Y Cf. A317930, A318319, A318669 (some of the numerator sequences), A317931 (conjectured, for A002487).
%Y Cf. A305439 (the 2-adic valuation), A318666.
%Y Cf. also A046644, A261179, A299150, A237649, A299150, A317928, A317839, A317843, A317934, A318319.
%K nonn,frac,mult
%O 1,2
%A _Antti Karttunen_, Aug 11 2018
%E Definition changed, the original (now conjectured alternative definition) moved to the comments section by _Antti Karttunen_, Aug 31 2018
%E Keyword:mult added by _Antti Karttunen_, Sep 03 2018
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