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%I A323910 #22 Nov 14 2024 17:10:39
%S A323910 1,-1,-2,0,-4,4,-6,0,-1,6,-10,2,-12,8,10,0,-16,1,-18,2,14,12,-22,4,-3,
%T A323910 14,-2,2,-28,-16,-30,0,22,18,26,4,-36,20,26,4,-40,-24,-42,2,4,24,-46,
%U A323910 8,-5,-1,34,2,-52,0,42,4,38,30,-58,2,-60,32,6,0,50,-40,-66,2,46,-40,-70,12,-72,38,2,2,62,-48,-78,8,-4,42,-82,-2,66,44,58,4,-88,2,74,2
%N A323910 Dirichlet inverse of the deficiency of n, A033879.
%H A323910 Antti Karttunen, <a href="/A323910/b323910.txt">Table of n, a(n) for n = 1..20000</a>
%H A323910 Jon Maiga, <a href="http://sequencedb.net/s/A323910">Computer-generated formulas for A323910</a>, Sequence Machine.
%F A323910 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A033879(n/d) * a(d).
%F A323910 From _Antti Karttunen_, Nov 14 2024: (Start)
%F A323910 Following convolution formulas have been conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly:
%F A323910 a(n) = Sum_{d|n} A046692(d)*A067824(n/d).
%F A323910 a(n) = Sum_{d|n} A055615(d)*A074206(n/d).
%F A323910 a(n) = Sum_{d|n} A023900(d)*A174725(n/d).
%F A323910 a(n) = Sum_{d|n} A008683(d)*A323912(n/d).
%F A323910 a(n) = Sum_{d|n} A191161(d)*A327960(n/d).
%F A323910 a(n) = Sum_{d|n} A328722(d)*A330575(n/d).
%F A323910 a(n) = Sum_{d|n} A345182(d)*A349341(n/d).
%F A323910 a(n) = Sum_{d|n} A346246(d)*A349387(n/d).
%F A323910 a(n) = Sum_{d|n} A002033(d-1)*A055615(n/d).
%F A323910 (End)
%t A323910 b[n_] := 2 n - DivisorSigma[1, n];
%t A323910 a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
%t A323910 Array[a, 100] (* _Jean-François Alcover_, Feb 17 2020 *)
%o A323910 (PARI)
%o A323910 up_to = 16384;
%o A323910 DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -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 A323910 A033879(n) = (2*n-sigma(n));
%o A323910 v323910 = DirInverse(vector(up_to,n,A033879(n)));
%o A323910 A323910(n) = v323910[n];
%Y A323910 Cf. A033879, A323911, A323912, A359549 (parity of terms).
%Y A323910 Sequences that appear in the convolution formulas: A002033, A008683, A023900, A055615, A046692, A067824, A074206, A174725, A191161, A327960, A328722, A330575, A345182, A349341, A346246, A349387.
%K A323910 sign
%O A323910 1,3
%A A323910 _Antti Karttunen_, Feb 12 2019