A078442 a(p) = a(n) + 1 if p is the n-th prime, prime(n); a(n)=0 if n is not prime.

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

Offset: 1

Author

Henry Bottomley, Dec 31 2002

Keywords

• nonn

Comments

Fernandez calls this the order of primeness of n.

a(A007097(n))=n, for any n >= 0. - Paul Tek, Nov 12 2013

When a nonoriented rooted tree is encoded as a Matula-Goebel number n, a(n) tells how many edges needs to be climbed up from the root of the tree until the first branching vertex (or the top of the tree, if n is one of the terms of A007097) is encountered. Please see illustrations at A061773. - Antti Karttunen, Jan 27 2014

Zero-based column index of n in the Kimberling-style dispersion table of the primes (see A114537). - Allan C. Wechsler, Jan 09 2024

Examples

			a(1) = 0 since 1 is not prime;
a(2) = a(prime(1)) = a(1) + 1 = 1 + 0 = 1;
a(3) = a(prime(2)) = a(2) + 1 = 1 + 1 = 2;
a(4) = 0 since 4 is not prime;
a(5) = a(prime(3)) = a(3) + 1 = 2 + 1 = 3;
a(6) = 0 since 6 is not prime;
a(7) = a(prime(4)) = a(4) + 1 = 0 + 1 = 1.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• N. Fernandez, An order of primeness, F(p)
• N. Fernandez, An order of primeness [cached copy, included with permission of the author]
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

A left inverse of A007097.

One less than A049076.

a(A000040(n)) = A049076(n).

Cf. A373338 (mod 2), A018252 (positions of zeros).

Cf. A000720, A049084, A061773.

Cf. permutations A235489, A250247/A250248, A250249/A250250, A245821/A245822 that all preserve a(n).

Cf. also array A114537 (A138947) and permutations A135141/A227413, A246681.

Programs

Haskell

a078442 n = fst $ until ((== 0) . snd)
                        (\(i, p) -> (i + 1, a049084 p)) (-2, a000040 n)
-- Reinhard Zumkeller, Jul 14 2013

Maple

A078442 := proc(n)
    if not isprime(n) then
        0 ;
    else
        1+procname(numtheory[pi](n)) ;
    end if;
end proc: # R. J. Mathar, Jul 07 2012

Mathematica

a[n_] := a[n] = If[!PrimeQ[n], 0, 1+a[PrimePi[n]]]; Array[a, 105] (* Jean-François Alcover, Jan 26 2018 *)

PARI

A078442(n)=for(i=0,n, isprime(n) || return(i); n=primepi(n)) \\ M. F. Hasler, Mar 09 2010

Formula

a(n) = A049076(n)-1.

a(n) = if A049084(n) = 0 then 0 else a(A049084(n)) + 1. - Reinhard Zumkeller, Jul 14 2013

For all n, a(n) = A007814(A135141(n)) and a(A227413(n)) = A007814(n). Also a(A235489(n)) = a(n). - Antti Karttunen, Jan 27 2014