A346482 -id:A346482 - cp's OEIS frontend

A346484 Composite numbers k for which A346482(k) = 0.

16, 81, 128, 360, 504, 540, 600, 625, 756, 792, 936, 1024, 1176, 1188, 1224, 1350, 1368, 1400, 1404, 1500, 1656, 1836, 1960, 2052, 2088, 2187, 2200, 2232, 2250, 2401, 2484, 2600, 2646, 2664, 2904, 2952, 3096, 3132, 3348, 3384, 3400, 3500, 3800, 3816, 3996, 4056, 4116, 4248, 4312, 4392, 4428, 4600, 4644, 4725, 4824

Offset: 1

Mats Granvik and Antti Karttunen, Aug 17 2021

nonn

Let p, q and r be distinct primes and m >= 1. If k = p^(3*m + 1) or k = p^3 * q^2 * r, then k is in the sequence. Do all terms have this form? - Sebastian Karlsson, Dec 09 2022

Cf. A346482.

isA346484(n) = (!isprime(n) && !A346482(n));

A355939 Dirichlet inverse of A080339, characteristic function of noncomposite numbers.

Original entry on oeis.org

1, -1, -1, 1, -1, 2, -1, -1, 1, 2, -1, -3, -1, 2, 2, 1, -1, -3, -1, -3, 2, 2, -1, 4, 1, 2, -1, -3, -1, -6, -1, -1, 2, 2, 2, 6, -1, 2, 2, 4, -1, -6, -1, -3, -3, 2, -1, -5, 1, -3, 2, -3, -1, 4, 2, 4, 2, 2, -1, 12, -1, 2, -3, 1, 2, -6, -1, -3, 2, -6, -1, -10, -1, 2, -3, -3, 2, -6, -1, -5, 1, 2, -1, 12, 2, 2, 2, 4, -1, 12, 2, -3, 2, 2, 2, 6, -1, -3, -3, 6, -1, -6, -1, 4, -6

Offset: 1

Antti Karttunen, Jul 21 2022

sign

The absolute values of this sequence are given by A008480. Compare also to A355817 and A335452.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A008480, A010051, A080339.

Cf. also A346482, A355817, A355827.

s[n_] := If[CompositeQ[n], 0, 1]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, s[n/#]*a[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 21 2022 *)

memoA355939 = Map();
A355939(n) = if(1==n,1,my(v); if(mapisdefined(memoA355939,n,&v), v, v = -sumdiv(n,d,if(dA355939(d),0)); mapput(memoA355939,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA010051(n/d) * a(d).

Dirichlet g.f.: 1/(1 + B(s)), where B(s) is d.g.f. of characteristic function of primes. - Vaclav Kotesovec, Jul 22 2022

A346483 Sum of A005171 (characteristic function of nonprimes) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 4, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 2, 0

Offset: 1

Mats Granvik and Antti Karttunen, Aug 17 2021

sign

The first negative term is a(192) = -1.

Positions of nonzero terms are given by A033987, except for positions n = 256, 512, 6561, 16384, 19683, 32768, 390625, 1048576, ..., at which a(n) = 0 also.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A005171, A010051, A033987, A346482.

nn = 87; b = Table[If[PrimeQ[n], 1, 0], {n, nn}]; a = 1 - b; A = Table[Table[If[Mod[n, k] == 0, a[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]; B = Inverse[A]; S = A[[Range[nn]]] + B[[Range[nn]]]; S[[All, 1]]

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA005171(n) = (1-isprime(n));
v346482 = DirInverseCorrect(vector(up_to,n,A005171(n)));
A346482(n) = v346482[n];
A346483(n) = (A005171(n)+A346482(n));

a(n) = A005171(n) + A346482(n).

For n > 1, a(n) = -Sum_{d|n, 1A005171(d) * A346482(n/d).

cp's OEIS Frontend

A346484 Composite numbers k for which A346482(k) = 0.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A355939 Dirichlet inverse of A080339, characteristic function of noncomposite numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A346483 Sum of A005171 (characteristic function of nonprimes) and its Dirichlet inverse.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula