A061397 Characteristic function sequence of primes multiplied componentwise by N, the natural numbers.

0, 2, 3, 0, 5, 0, 7, 0, 0, 0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 0, 0, 0, 23, 0, 0, 0, 0, 0, 29, 0, 31, 0, 0, 0, 0, 0, 37, 0, 0, 0, 41, 0, 43, 0, 0, 0, 47, 0, 0, 0, 0, 0, 53, 0, 0, 0, 0, 0, 59, 0, 61, 0, 0, 0, 0, 0, 67, 0, 0, 0, 71, 0, 73, 0, 0, 0, 0, 0, 79, 0, 0, 0, 83, 0, 0, 0, 0, 0, 89, 0, 0, 0, 0, 0

Offset

1,2

Comments

Frequently, holes in a sequence are filled with zeros. This is a canonical way to do this and applied here to primes(A000040). A pre-scalar product when summation is omitted.

Equals row sums of triangle A143536. - Gary W. Adamson, Aug 23 2008

Mobius transform of sum of the distinct primes dividing n (A008472). - Steven Foster Clark, Jun 26 2020

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Prime zeta function.

Formula

a(n) = A010051(n)*A000027(n).

Dirichlet generating function: primezeta(s-1). - Franklin T. Adams-Watters, Sep 11 2005

a(1)=0; for n>=1, a(n)=0, if either p_1|n or p_2|n or...or p_i|n, when n is in [p_i^2,p_(i+1)^2), i=1,2,..], where p_i is the i-th prime; otherwise a(n)=n. - Vladimir Shevelev, Apr 24 2010

a(n) = n*floor(gcd(((n-1)! + 1)/n,2)). - José de Jesús Camacho Medina, Apr 30 2016

a(n) = n*floor(1/A001065(n)); for n>1. - José de Jesús Camacho Medina, Aug 07 2016

G.f.: x*f'(x), where f(x) = Sum_{k>=1} x^prime(k). - Ilya Gutkovskiy, Apr 10 2017

a(n) = (2*n-1)! mod n^2, by Wilson's theorem. - Thomas Ordowski, Dec 27 2017

Example

If 1<n<=8, a(n)=0 iff it is even on interval [4,9); if 9<=n<=25, then a(n)=0 iff n is either even or multiple of 3 on interval [9,25) etc. - Vladimir Shevelev, Apr 24 2010

Maple

seq(`if`(isprime(n), n, 0), n=1..100); # Robert Israel, May 02 2016

Mathematica

If[PrimeQ@ #, #, 0] & /@ Range@ 94 (* or *)
Replace[#, n_ /; ! PrimeQ@ n -> 0] & /@ Range@ 94 (* Michael De Vlieger, May 02 2016 *)
Table[n*Floor[GCD[((n-1)! + 1)/n, 2]], {n, 2, 100}] (* José de Jesús Camacho Medina, Apr 30 2016 *)

Prog

(PARI) a(n)=if(isprime(n), n) \\ Charles R Greathouse IV, Oct 29 2011
(Haskell)
a061397 n = (fromIntegral $ a010051 n) * n  -- Reinhard Zumkeller, Mar 21 2014

Crossrefs

Cf. A000040, A010051, A143536.

Cf. A034387 (partial sums).

Sequence in context: A085563 A071375 A269668 * A331045 A331044 A377777

Adjacent sequences: A061394 A061395 A061396 * A061398 A061399 A061400

Keyword

nonn,easy

Author

Labos Elemer, Jun 07 2001

Status

approved