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

Author

Labos Elemer, Jun 07 2001

Keywords

• nonn
• easy

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

Examples

			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
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Prime zeta function.

Crossrefs

Cf. A000040, A010051, A143536.

Cf. A034387 (partial sums).

Programs

Haskell

a061397 n = (fromIntegral $ a010051 n) * n  -- Reinhard Zumkeller, Mar 21 2014

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 *)

PARI

a(n)=if(isprime(n),n) \\ Charles R Greathouse IV, Oct 29 2011

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