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A061397 Characteristic function sequence of primes multiplied componentwise by N, the natural numbers. 36
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 (list; graph; refs; listen; history; text; internal format)
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
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. A034387 (partial sums).
Sequence in context: A085563 A071375 A269668 * A331045 A331044 A093438
KEYWORD
nonn,easy
AUTHOR
Labos Elemer, Jun 07 2001
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)