A347237 Möbius transform of A347236.

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, 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, 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

Offset: 1

Author

Antti Karttunen, Aug 24 2021

Keywords

• nonn
• mult

Comments

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

Dirichlet convolution of A003961 with A158523.

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

From Antti Karttunen, Aug 25 2021: (Start)

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

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.

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16383
• Index entries for primes, gaps between
• Index entries for sequences computed from indices in prime factorization

Crossrefs

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

Cf. also A347137.

Cf. A151800.

Programs

PARI

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A061019(n) = (((-1)^bigomega(n))*n);
A347237(n) = sumdiv(n,d,A061019(d)*eulerphi(A003961(n/d)));
\\ Or alternatively as:
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))); };
A347237(n) = sumdiv(n,d,A003961(n/d)*A158523(d));

Formula

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

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

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

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

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

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

Multiplicative with a(p^e) = (-p)^e + (A151800(p)-1)*(A151800(p)^e-(-p)^e)/(A151800(p)+p). - Sebastian Karlsson, Sep 02 2021