A143536 -id:A143536 - cp's OEIS frontend

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

Labos Elemer, Jun 07 2001

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

			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

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

Cf. A000040, A010051, A143536.

Cf. A034387 (partial sums).

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

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

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 *)

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

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

A143537 Triangle read by rows: T(n,k) = number of primes in the interval [k..n], n >= 1, 1 <= k <= n.

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 2, 2, 1, 0, 3, 3, 2, 1, 1, 3, 3, 2, 1, 1, 0, 4, 4, 3, 2, 2, 1, 1, 4, 4, 3, 2, 2, 1, 1, 0, 4, 4, 3, 2, 2, 1, 1, 0, 0, 4, 4, 3, 2, 2, 1, 1, 0, 0, 0, 5, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1, 5, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1, 0, 6, 6, 5, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1

Offset: 1

Gary W. Adamson, Aug 23 2008

Old name: triangle read by rows, A000012 * A143536, 1<=k<=n.

			Triangle T(n,k) begins:
n\k 1  2  3  4  5  6  7  8 ...
1:  0;
2:  1, 1;
3:  2, 2, 1;
4:  2, 2, 1, 0;
5:  3, 3, 2, 1, 1;
6:  3, 3, 2, 1, 1, 0;
7:  4, 4, 3, 2, 2, 1, 1;
8:  4, 4, 3, 2, 2, 1, 1, 0;
...

Row sums are A034387.
Column k=1 gives A000720.
Main diagonal gives A010051.
T(2n,n) gives A035250.
Cf. A143536.

T(n,k) = pi(n) - pi(k-1), where pi = A000720. - Ilya Gutkovskiy, Mar 19 2020

New name and corrected by Ilya Gutkovskiy, Mar 19 2020

A143544 Triangle read by rows, T(n,k) = 2 if n is prime, 1 otherwise; 1<=k<=n.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Gary W. Adamson, Aug 23 2008

Row sums = A143545: (1, 4, 6, 4, 10, 6, 14,...) = componentwise addition of (1, 2, 3, 4, 5,...) and A061397: (0, 2, 3, 0, 5, 0, 7,...).

			First few rows of the triangle =
1;
2, 2;
2, 2, 2;
1, 1, 1, 1;
2, 2, 2, 2, 2;
1, 1, 1, 1, 1, 1;
2, 2, 2, 2, 2, 2, 2;
...

Michael De Vlieger, Table of n, a(n) for n = 1..11325

Cf. A010051, A061397, A143536, A143545.

Table[1 + Boole[PrimeQ[n]], {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Oct 31 2021 *)

Triangle read by rows, T(n,k) = 2 if n is prime, 1 otherwise; 1<=k<=n.

T(n, k) = A143536(n, k) + 1. - Georg Fischer, Oct 31 2021

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

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A143537 Triangle read by rows: T(n,k) = number of primes in the interval [k..n], n >= 1, 1 <= k <= n.

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A143544 Triangle read by rows, T(n,k) = 2 if n is prime, 1 otherwise; 1<=k<=n.

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