A373991 a(n) = 1 if A328768(n) is a multiple of 3, otherwise 0, where A328768 is the first primorial based variant of arithmetic derivative.

1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1

Offset

0

Links

Antti Karttunen, Table of n, a(n) for n = 0..100000

Index entries for characteristic functions.

Index entries for sequences related to primorial numbers.

Formula

a(n) = A079978(A328768(n)).

For all n >= 0, a(9*n) = 1, a(3*(3n+1)) = a(3*(3n+2)) = 0, a(8*n) = a(n), a(8n+4) = a(4n+2) = 0, and a(2n+1) = 1 when A007949(2n+1) != 1.

a(n) = [A007949(n) > 1] + [A007949(n) = 0]*[A007814(n) == 0 (mod 3)], where [ ] is the Iverson bracket.

a(n) = A267142(n) + A374043(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 31/63. - Amiram Eldar, Jun 28 2024

Mathematica

a[n_] := If[Divisible[n, 9] || (!Divisible[n, 3] && Divisible[IntegerExponent[n, 2], 3]), 1, 0]; Array[a, 100, 0] (* Amiram Eldar, Jun 28 2024 *)

Prog

(PARI)
A002110(n) = prod(i=1, n, prime(i));
A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i, 1])-1)/f[i, 1]));
A373991(n) = !(A328768(n)%3);
(PARI) A373991(n) = if(!(n%9), 1, if(!(n%3), 0, if(!(n%8), A373991(n/8), (n%2))));
(PARI) A373991(n) = { my(v2 = valuation(n, 2), v3 = valuation(n, 3)); ((v3 > 1) || (0==v3 && 0==(v2%3))); };

Crossrefs

Characteristic function of A373992.

Cf. A002110, A007814, A007949, A079978, A267142, A328768, A374043.

Sequence in context: A266837 A321081 A267126 * A266892 A267152 A151667

Adjacent sequences: A373988 A373989 A373990 * A373992 A373993 A373994

Keyword

nonn,easy

Author

Antti Karttunen, Jun 26 2024

Status

approved