A049076 Number of steps in the prime index chain for the n-th prime.

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

Offset

1,2

Comments

Let p(k) = k-th prime, let S(p) = S(p(k)) = k, the subscript of p; a(n) = order of primeness of p(n) = 1+m where m is largest number such that S(S(..S(p(n))...)) with m S's is a prime.

The record holders correspond to A007097.

Links

N. Fernandez, Table of n, a(n) for n = 1..500

N. Fernandez, An order of primeness, F(p)

N. Fernandez, An order of primeness [cached copy, included with permission of the author]

N. Fernandez, The Exploring Primeness Project

Formula

Let b(n) = 0 if n is nonprime, otherwise b(n) = k where n is the k-th prime. Then a(n) is the number of times you can apply b to the n-th prime before you hit a nonprime.

a(n) = 1 + A078442(n). - R. J. Mathar, Jul 07 2012

a(n) = A078442(A000040(n)). - Alois P. Heinz, Mar 16 2020

Example

11 is 5th prime, so S(11)=5, 5 is 3rd prime, so S(S(11))=3, 3 is 2nd prime, so S(S(S(11)))=2, 2 is first prime, so S(S(S(S(11))))=1, not a prime. Thus a(5)=4.
Alternatively, a(5) = 4: the 5th prime is 11 and its prime index chain is 11->5->3->2->1->0. a(6) = 1: the 6th prime is 13 and its prime index chain is 13->6->0.

Maple

A049076 := proc(n)
    if not isprime(n) then
        1 ;
    else
        1+procname(numtheory[pi](n)) ;
    end if;
end proc:
seq(A049076(n), n=1..30) ; # R. J. Mathar, Jan 28 2014

Mathematica

A049076 f[n_] := Length[ NestWhileList[ PrimePi, n, PrimeQ]]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Mar 11 2004 *)
Table[Length[NestWhileList[PrimePi[#]&, Prime[n], PrimeQ[#]&]]-1, {n, 110}] (* Harvey P. Dale, May 07 2018 *)

Prog

(PARI) apply(p->my(s=1); while(isprime(p=primepi(p)), s++); s, primes(100)) \\ Charles R Greathouse IV, Nov 20 2012
(Haskell)
a049076 = (+ 1) . a078442  -- Reinhard Zumkeller, Jul 14 2013

Crossrefs

Cf. A000040, A049077 - A049081, A006450, A049084, A236542.

Sequence in context: A325542 A365715 A081877 * A097744 A055445 A135560

Adjacent sequences: A049073 A049074 A049075 * A049077 A049078 A049079

Keyword

nice,nonn,easy

Author

Neil Fernandez

Extensions

Additional comments from Gabriel Cunningham (gcasey(AT)mit.edu), Apr 12 2003

Status

approved