cp's OEIS Frontend

A349631 Dirichlet convolution of A003961 with A346479, which is Dirichlet inverse of A250469.

%I A349631 #19 Nov 29 2021 14:29:40
%S A349631 1,0,0,0,0,0,0,6,0,-6,0,12,0,-6,0,18,0,24,0,24,0,-24,0,0,0,-24,60,36,
%T A349631 0,48,0,42,-20,-42,0,-12,0,-42,-10,12,0,72,0,60,60,-48,0,-24,0,42,-30,
%U A349631 72,0,-84,0,12,-30,-78,0,-120,0,-72,120,126,0,180,0,96,-30,132,0,-48,0,-96,60,108,0,174,0,-84,120
%N A349631 Dirichlet convolution of A003961 with A346479, which is Dirichlet inverse of A250469.
%C A349631 Note that for n = 2..36, a(n) = -A349632(n).
%C A349631 Dirichlet convolution of this sequence with A347376 is A003972.
%H A349631 Antti Karttunen, <a href="/A349631/b349631.txt">Table of n, a(n) for n = 1..20000</a>
%H A349631 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A349631 <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%F A349631 a(n) = Sum_{d|n} A003961(d) * A346479(n/d).
%o A349631 (PARI)
%o A349631 up_to = 20000;
%o A349631 A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
%o A349631 ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
%o A349631 v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
%o A349631 A078898(n) = v078898[n];
%o A349631 A250469(n) = if(1==n,n,my(spn = nextprime(1+A020639(n)), c = A078898(n), k = 0); while(c, k++; if((1==k)||(A020639(k)>=spn),c -= 1)); (k*spn));
%o A349631 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 A349631 v346479 = DirInverseCorrect(vector(up_to,n,A250469(n)));
%o A349631 A346479(n) = v346479[n];
%o A349631 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A349631 A349631(n) = sumdiv(n,d,A003961(d)*A346479(n/d));
%Y A349631 Cf. A003961, A250469, A346479, A349632 (Dirichlet inverse).
%Y A349631 Cf. also A003972, A347376, A349381.
%Y A349631 Cf. also arrays A083221, A246278, A249821, A249822 and permutations A250245, A250246.
%K A349631 sign
%O A349631 1,8
%A A349631 _Antti Karttunen_, Nov 27 2021