A349358 -id:A349358 - cp's OEIS frontend

A349398 Dirichlet convolution of A048673 with the Dirichlet inverse of its inverse permutation.

1, 0, 0, 0, 0, 1, -5, 8, 0, -6, -3, 2, 0, 19, -5, -4, -4, 20, -19, 22, 6, -15, 3, -8, 0, 0, 16, 16, -18, 24, -40, 70, 9, -24, 21, -7, -50, 55, 8, -24, 6, -41, -15, 58, 20, -17, -31, 108, 27, 70, -37, -24, 0, -20, -49, -98, 6, 26, -13, 21, -15, 62, 158, 84, -22, 9, -49, 130, -67, 12, -49, 62, -29, 112, 4, -60, 103, 16

Offset: 1

Antti Karttunen, Nov 19 2021

sign

Dirichlet convolution of A048673 with A349358, which is the Dirichlet inverse of A064216 (inverse permutation of A048673). Therefore, convolving A064216 with this sequence gives A048673.

Note how for n = 1 .. 35, a(n) = -A349397(n).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A048673, A064216, A064989, A323893, A349397 (Dirichlet inverse), A349399 (sum with it).

Cf. also A349376, A349377, A349385.

A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
A064216(n) = { my(f = factor(n+n-1)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
memoA349358 = Map();
A349358(n) = if(1==n,1,my(v); if(mapisdefined(memoA349358,n,&v), v, v = -sumdiv(n,d,if(dA064216(n/d)*A349358(d),0)); mapput(memoA349358,n,v); (v)));
A349398(n) = sumdiv(n,d,A048673(n/d)*A349358(d));

a(n) = Sum_{d|n} A048673(n/d) * A349358(d).

A349359 Sum of A064216 and its Dirichlet inverse, where A064216 = A064989(2n-1), and A064989 is fully multiplicative with a(2) = 1 and a(p) = prevprime(p) for odd primes p.

Original entry on oeis.org

2, 0, 0, 4, 0, 12, 0, 12, 9, 16, 0, 22, 0, 44, 24, 5, 0, 40, 0, 60, 66, 40, 0, 14, 16, 36, 51, 10, 0, 106, 0, 82, 60, 56, 88, 26, 0, 124, 54, -10, 0, -46, 0, 144, 134, 48, 0, 235, 121, 140, 84, 86, 0, 19, 80, -108, 186, 136, 0, -44, 0, 236, 211, 29, 72, 158, 0, 216, 72, 62, 0, 152, 0, 284, 190, 10, 220, 98, 0, 260, 181

Offset: 1

Antti Karttunen, Nov 17 2021

sign

Compare to A323894 which in contrast to this sequence seems to have only nonnegative terms.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A064216, A064989, A349358.

Cf. also A323894, A349126.

A064989(n) = { my(f = factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A064216(n) = A064989((2*n)-1);
memoA349358 = Map();
A349358(n) = if(1==n,1,my(v); if(mapisdefined(memoA349358,n,&v), v, v = -sumdiv(n,d,if(dA064216(n/d)*A349358(d),0)); mapput(memoA349358,n,v); (v)));
A349359(n) = (A064216(n)+A349358(n));

a(n) = A064216(n) + A349358(n).

a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1A064216(d) * A349358(n/d).

cp's OEIS Frontend

A349398 Dirichlet convolution of A048673 with the Dirichlet inverse of its inverse permutation.

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