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

%I A061397 #108 Mar 16 2025 08:55:26
%S A061397 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,
%T A061397 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,
%U A061397 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
%N A061397 Characteristic function sequence of primes multiplied componentwise by N, the natural numbers.
%C A061397 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.
%C A061397 Equals row sums of triangle A143536. - _Gary W. Adamson_, Aug 23 2008
%C A061397 Mobius transform of sum of the distinct primes dividing n (A008472). - _Steven Foster Clark_, Jun 26 2020
%H A061397 Reinhard Zumkeller, <a href="/A061397/b061397.txt">Table of n, a(n) for n = 1..10000</a>
%H A061397 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeZetaFunction.html">Prime zeta function</a>.
%F A061397 a(n) = A010051(n)*A000027(n).
%F A061397 Dirichlet generating function: primezeta(s-1). - _Franklin T. Adams-Watters_, Sep 11 2005
%F A061397 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
%F A061397 a(n) = n*floor(gcd(((n-1)! + 1)/n,2)). - _José de Jesús Camacho Medina_, Apr 30 2016
%F A061397 a(n) = n*floor(1/A001065(n)); for n>1. - _José de Jesús Camacho Medina_, Aug 07 2016
%F A061397 G.f.: x*f'(x), where f(x) = Sum_{k>=1} x^prime(k). - _Ilya Gutkovskiy_, Apr 10 2017
%F A061397 a(n) = (2*n-1)! mod n^2, by Wilson's theorem. - _Thomas Ordowski_, Dec 27 2017
%e A061397 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
%p A061397 seq(`if`(isprime(n),n,0), n=1..100); # _Robert Israel_, May 02 2016
%t A061397 If[PrimeQ@ #, #, 0] & /@ Range@ 94 (* or *)
%t A061397 Replace[#, n_ /; ! PrimeQ@ n -> 0] & /@ Range@ 94 (* _Michael De Vlieger_, May 02 2016 *)
%t A061397 Table[n*Floor[GCD[((n-1)! + 1)/n, 2]], {n, 2, 100}] (* _José de Jesús Camacho Medina_, Apr 30 2016 *)
%o A061397 (PARI) a(n)=if(isprime(n),n) \\ _Charles R Greathouse IV_, Oct 29 2011
%o A061397 (Haskell)
%o A061397 a061397 n = (fromIntegral $ a010051 n) * n  -- _Reinhard Zumkeller_, Mar 21 2014
%Y A061397 Cf. A000040, A010051, A143536.
%Y A061397 Cf. A034387 (partial sums).
%K A061397 nonn,easy
%O A061397 1,2
%A A061397 _Labos Elemer_, Jun 07 2001