A359791 Dirichlet inverse of function f(n) = 1 + A349905(n), where A349905(n) is the arithmetic derivative of prime shifted n.

1, -2, -2, -3, -2, -1, -2, -8, -7, -3, -2, 0, -2, -7, -5, -16, -2, 0, -2, -4, -9, -9, -2, 23, -11, -13, -40, -12, -2, 12, -2, -16, -11, -15, -11, 42, -2, -19, -15, 21, -2, 12, -2, -16, -24, -25, -2, 128, -19, -12, -17, -24, -2, 67, -13, 17, -21, -27, -2, 105, -2, -33, -48, 48, -17, 12, -2, -28, -27, 0, -2, 224

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from indices in prime factorization

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} (1+A349905(n/d)) * a(d).

a(n) = A359790(A003961(n)).

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349905(n) = A003415(A003961(n));
memoA359791 = Map();
A359791(n) = if(1==n, 1, my(v); if(mapisdefined(memoA359791, n, &v), v, v = -sumdiv(n, d, if(d<n, (1+A349905(n/d))*A359791(d), 0)); mapput(memoA359791, n, v); (v)));

Crossrefs

Cf. A003415, A003961, A349905, A359790.

Cf. A359764 (parity of terms), A359765 (positions of odd terms), A359766 (of even terms).

Cf. also A359169.

Sequence in context: A116199 A369031 A162915 * A242266 A239617 A304737

Adjacent sequences: A359788 A359789 A359790 * A359792 A359793 A359794

Keyword

sign

Author

Antti Karttunen, Jan 13 2023

Status

approved