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A349135 Sum of Kimberling's paraphrases (A003602) and its Dirichlet inverse.

%I A349135 #27 Nov 30 2024 12:56:35
%S A349135 2,0,0,1,0,4,0,1,4,6,0,2,0,8,12,1,0,6,0,3,16,12,0,2,9,14,12,4,0,4,0,1,
%T A349135 24,18,24,5,0,20,28,3,0,6,0,6,26,24,0,2,16,17,36,7,0,16,36,4,40,30,0,
%U A349135 8,0,32,36,1,42,10,0,9,48,12,0,5,0,38,46,10,48,12,0,3,37,42,0,11,54,44,60,6,0,20,56,12
%N A349135 Sum of Kimberling's paraphrases (A003602) and its Dirichlet inverse.
%C A349135 Question: Are all terms nonnegative?
%C A349135 The answer to the above question is no, because A323894 (which is a prime-shifted version of this sequence) also contains negative values. For example, for n=72747675, 88062975, 130945815, 111035925 we get here a(n) = -14126242, -17546656, -14460312, -22677277. The indices are obtained by prime-shifting with A003961 the four indices mentioned in the Apr 20 2022 comment of A323894. - _Antti Karttunen_, Nov 30 2024
%H A349135 Antti Karttunen, <a href="/A349135/b349135.txt">Table of n, a(n) for n = 1..16384</a>
%H A349135 Antti Karttunen, <a href="/A349135/a349135.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A349135 a(n) = A003602(n) + A349134(n).
%F A349135 a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1<d<n} A003602(d) * A349134(n/d).
%F A349135 For all n >= 1, a(4*n) = A003602(n). - _Antti Karttunen_, Dec 07 2021
%t A349135 k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; d[1] = 1; d[n_] := d[n] = -DivisorSum[n, d[#]*k[n/#] &, # < n &]; a[n_] := k[n] + d[n]; Array[a, 100] (* _Amiram Eldar_, Nov 13 2021 *)
%o A349135 (PARI)
%o A349135 up_to = 16384;
%o A349135 DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
%o A349135 A003602(n) = (1+(n>>valuation(n,2)))/2;
%o A349135 v349134 = DirInverseCorrect(vector(up_to,n,A003602(n)));
%o A349135 A349134(n) = v349134[n];
%o A349135 A349135(n) = (A003602(n)+A349134(n));
%o A349135 (PARI) A349135(n) = if(1==n,2,-sumdiv(n, d, if(1==d||n==d,0,A003602(d)*A349134(n/d)))); \\ (Demonstrates the "cut convolution" formula) - _Antti Karttunen_, Nov 13 2021
%o A349135 (PARI)
%o A349135 A003602(n) = (1+(n>>valuation(n,2)))/2;
%o A349135 memoA349134 = Map();
%o A349135 A349134(n) = if(1==n,1,my(v); if(mapisdefined(memoA349134,n,&v), v, v = -sumdiv(n,d,if(d<n,A003602(n/d)*A349134(d),0)); mapput(memoA349134,n,v); (v)));
%o A349135 A349135(n) = (A003602(n)+A349134(n)); \\ _Antti Karttunen_, Nov 30 2024
%Y A349135 Cf. A003602 (also quadrisection of this sequence), A349134, A323894 [= a(A003961(n))].
%Y A349135 Cf. also A323882, A349126.
%K A349135 sign
%O A349135 1,1
%A A349135 _Antti Karttunen_, Nov 13 2021