A143541 -id:A143541 - cp's OEIS frontend

A143542 Triangle read by rows, A051731 * A143541 as infinite lower triangular matrices; 1<=k<=n.

0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 2, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Aug 23 2008

Row sums = A066328: (0, 1, 2, 1, 3, 3, 4,...)

Prime rows = first n terms of A010051, the characteristic function of the primes.

			First few rows of the triangle =
0;
0, 1;
0, 1, 1;
0, 1, 0, 0;
0, 1, 1, 0, 1;
0, 2, 1, 0, 0, 0;
0, 1, 1, 0, 1, 0, 1;
...

Cf. A143541, A051731, A010051.

Triangle read by rows, A051731 * A143541 as infinite lower triangular matrices; 1<=k<=n. A051731 = the inverse Mobius transform.

A049084 a(n) = pi(n) if n is prime, otherwise 0.

Original entry on oeis.org

0, 1, 2, 0, 3, 0, 4, 0, 0, 0, 5, 0, 6, 0, 0, 0, 7, 0, 8, 0, 0, 0, 9, 0, 0, 0, 0, 0, 10, 0, 11, 0, 0, 0, 0, 0, 12, 0, 0, 0, 13, 0, 14, 0, 0, 0, 15, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 17, 0, 18, 0, 0, 0, 0, 0, 19, 0, 0, 0, 20, 0, 21, 0, 0, 0, 0, 0, 22, 0, 0, 0, 23, 0, 0, 0, 0, 0, 24, 0, 0, 0

Offset: 1

N. J. A. Sloane

pi(n) is the prime counting function, A000720.

Equals row sums of triangle A143541. - Gary W. Adamson, Aug 23 2008

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

a(n) = A091227(A091202(n)).

Cf. A143541.

import Data.List (unfoldr)
a049084 n = a049084_list !! (fromInteger n - 1)
a049084_list = unfoldr x (1, 1, a000040_list) where
   x (i, z, ps'@(p:ps)) | i == p = Just (z, (i + 1, z + 1, ps))
                        | i /= p = Just (0, (i + 1, z, ps'))
-- Reinhard Zumkeller, Apr 17 2012, Mar 31 2012, Sep 15 2011

A049084 := proc(n)
    local i;
    if isprime(n) then
        for i from 1 do
            if ithprime(i) = n then
                return i;
            end if;
        end do;
    else
        return 0 ;
    fi;
end proc:
seq(A049084(n),n=1..120) ;

Table[PrimePi[n] * Boole[PrimeQ[n]], {n, 92}] (* Jean-François Alcover, Nov 07 2011, after R. J. Mathar *)
Table[If[PrimeQ[n],PrimePi[n],0],{n,100}] (* Harvey P. Dale, Jan 09 2022 *)

a(n)=if(isprime(n),primepi(n),0) \\ Charles R Greathouse IV, Jan 08 2013

a(n) = pi(n)*(pi(n) - pi(n-1)), pi = A000720. - Reinhard Zumkeller, Nov 30 2003
a(n) = A000720(n*A010051(n)). - Labos Elemer, Jan 09 2004
a(n) = A000720(n)*A010051(n). - R. J. Mathar, Mar 01 2011

Name clarified by Alonso del Arte, Feb 07 2020 at the suggestion of David A. Corneth

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A143542 Triangle read by rows, A051731 * A143541 as infinite lower triangular matrices; 1<=k<=n.

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A049084 a(n) = pi(n) if n is prime, otherwise 0.

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