A359577 -id:A359577 - cp's OEIS frontend

A359588 Dirichlet inverse of A083346.

1, -2, -3, 3, -5, 6, -7, -6, 6, 10, -11, -9, -13, 14, 15, 12, -17, -12, -19, -15, 21, 22, -23, 18, 20, 26, -10, -21, -29, -30, -31, -24, 33, 34, 35, 18, -37, 38, 39, 30, -41, -42, -43, -33, -30, 46, -47, -36, 42, -40, 51, -39, -53, 20, 55, 42, 57, 58, -59, 45, -61, 62, -42, 48, 65, -66, -67, -51, 69

Offset: 1

Antti Karttunen, Jan 09 2023

Multiplicative because A083346 is.

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

Cf. A083346, A091428 (positions of odd terms), A359592 (parity of terms).

Cf. also A359577.

f[p_, e_] := If[Divisible[e, p], 1, p]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, s[n/#]*a[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, May 18 2023 *)

A083346(n) = { my(f=factor(n)); denominator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
memoA359588 = Map();
A359588(n) = if(1==n,1,my(v); if(mapisdefined(memoA359588,n,&v), v, v = -sumdiv(n,d,if(dA083346(n/d)*A359588(d),0)); mapput(memoA359588,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA083346(n/d) * a(d).

A366292 Dirichlet inverse of A353271, where A353271(n) is the numerator of n / A005940(1+(3*A156552(n))).

Original entry on oeis.org

1, -1, -1, -1, -1, -1, -1, -1, -2, -3, -1, 1, -1, -5, -3, -1, -1, 0, -1, 9, -5, -9, -1, 11, -4, -11, -4, 13, -1, 5, -1, -1, -9, -15, -5, 6, -1, -17, -11, 5, -1, 21, -1, 21, -2, -21, -1, 5, -6, -8, -15, 25, -1, 22, -9, 7, -17, -27, -1, 3, -1, -29, 14, -1, -11, 11, -1, 33, -21, -3, -1, 16, -1, -35, -8, 37, -9, 13

Offset: 1

Antti Karttunen, Oct 06 2023

sign

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

Cf. A332449, A353271.
Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms).
Cf. also A209635, A342417, A354347, A354823, A359432, A359433, A359577 for other sequences that are equal modulo 2.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A332449(n) = A005940(1+(3*A156552(n)));
A353271(n) = (n / gcd(n, A332449(n)));
memoA366292 = Map();
A366292(n) = if(1==n,1,my(v); if(mapisdefined(memoA366292,n,&v), v, v = -sumdiv(n,d,if(dA353271(n/d)*A366292(d),0)); mapput(memoA366292,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA353271(n/d) * a(d).

cp's OEIS Frontend

A359588 Dirichlet inverse of A083346.

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