A347237 - OEIS

%I #39 Sep 18 2021 22:03:13

%S 1,0,1,6,1,0,3,6,17,0,1,6,3,0,1,42,1,0,3,6,3,0,5,6,37,0,49,18,1,0,5,

%T 78,1,0,3,102,3,0,3,6,1,0,3,6,17,0,5,42,89,0,1,18,5,0,1,18,3,0,1,6,5,

%U 0,51,330,3,0,3,6,5,0,1,102,5,0,37,18,3,0,3,42,353,0,5,18,1,0,1,6,7,0,9,30,5,0,3,78,3,0,17

%N Möbius transform of A347236.

%C Dirichlet convolution of A003972 (prime shifted phi) with A061019.

%C Dirichlet convolution of A003961 with A158523.

%C Multiplicative because A003972 and A061019 are, and also because A347236 is.

%C From _Antti Karttunen_, Aug 25 2021: (Start)

%C All terms are nonnegative because sequence is multiplicative and a(p^k) >= 0 for all primes p and k >= 0.

%C Proof: For any prime p, sequence a(p^k), k>=0, is obtained as an ordinary convolution of sequences (-p)^k and the first differences of q^k, where q = A151800(p). (E.g., for powers of 2, the sequences convolved are A122803 and A025192, giving A102901.) This convolution is an alternating sum, with the terms 1*(q-1)*q^(k-1), -(p)*(q-1)*q^(k-2), (p^2)*(q-1)*q^(k-3), -(p^3)*(q-1)*q^(k-4), ..., (p^(k-1))*(q-1), -(p^k), for odd k, with sum of each consecutive pair being nonnegative because q >= p+1, while with an even exponent k, the leftover term p^k at the end is also positive, thus the whole sum is nonnegative also in that case.

%C (End)

%H Antti Karttunen, <a href="/A347237/b347237.txt">Table of n, a(n) for n = 1..16383</a>

%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F a(n) = Sum_{d|n} A008683(n/d) * A347236(d).

%F a(n) = Sum_{d|n} A003972(n/d) * A061019(d).

%F a(n) = Sum_{d|n} A003961(n/d) * A158523(d).

%F For all n >= 1, a(A000040(n)) = A001223(n) - 1.

%F For all n >= 0, a(2^n) = A102901(n).

%F For all n >= 0, a(3^n) = A120612(n).

%F Multiplicative with a(p^e) = (-p)^e + (A151800(p)-1)*(A151800(p)^e-(-p)^e)/(A151800(p)+p). - _Sebastian Karlsson_, Sep 02 2021

%o (PARI)

%o A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o A061019(n) = (((-1)^bigomega(n))*n);

%o A347237(n) = sumdiv(n,d,A061019(d)*eulerphi(A003961(n/d)));

%o \\ Or alternatively as:

%o A158523(n) = { my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); ((-1)^e)*(p+1)*(p^(e-1))); };

%o A347237(n) = sumdiv(n,d,A003961(n/d)*A158523(d));

%Y Cf. A000040, A001223, A003961, A003972, A008683, A008836, A016825 (positions of zeros), A061019, A102901, A120612, A158523, A347236.

%Y Cf. also A347137.

%Y Cf. A151800.

%K nonn,mult

%O 1,4

%A _Antti Karttunen_, Aug 24 2021