A007821 Primes p such that pi(p) is not prime.

2, 7, 13, 19, 23, 29, 37, 43, 47, 53, 61, 71, 73, 79, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 151, 163, 167, 173, 181, 193, 197, 199, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 281, 293, 307, 311, 313, 317, 337, 347, 349, 359, 373

Offset: 1

Author

Monte J. Zerger (mzerger(AT)cc4.adams.edu), Clark Kimberling

Keywords

• nonn

Comments

Primes prime(k) such that A049076(k)=1, sorted along increasing k. - R. J. Mathar, Jan 28 2014

The complement of A006450 (primes with prime index) within the primes A000040.

References

• C. Kimberling, Fractal sequences and interspersions, Ars Combinatoria, vol. 45 p 157 1997.

Links

• R. Zumkeller, Table of n, a(n) for n = 1..1000
• Lubomir Alexandrov, On the nonasymptotic prime number distribution, arXiv:math/9811096 [math.NT], 1998.
• N. Fernandez, An order of primeness, F(p)
• N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Crossrefs

Cf. A049076, A049078, A049079, A049080, A049081, A058322, A058324, A058325, A058326, A058327, A058328, A093046, A006450.

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270795, A270796, A102616.

Programs

Haskell

a007821 = a000040 . a018252
a007821_list = map a000040 a018252_list
-- Reinhard Zumkeller, Jan 12 2013

Maple

A007821 := proc(n) ithprime(A018252(n)) ; end proc: # R. J. Mathar, Jul 07 2012

Mathematica

Prime[ Select[ Range[75], !PrimeQ[ # ] &]] (* Robert G. Wilson v, Mar 15 2004 *)
With[{nn=100},Pick[Prime[Range[nn]],Table[If[PrimeQ[n],0,1],{n,nn}],1]] (* Harvey P. Dale, Aug 14 2020 *)

PARI

forprime(p=2, 1e3, if(!isprime(primepi(p)), print1(p, ", "))) \\ Felix Fröhlich, Aug 16 2014

Python

from sympy import primepi
def A007821(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-(p:=primepi(x))+primepi(p)
    return bisection(f,n,n) # Chai Wah Wu, Oct 19 2024

Formula

A137588(a(n)) = n; a(n) = A000040(A018252(n)). - Reinhard Zumkeller, Jan 28 2008

A000040 = A007821 U A006450. - Juri-Stepan Gerasimov, Sep 24 2009

A175247 U { a(n); n > 1 } = A000040. { a(n) } = { 2 } U { primes (A000040) with composite index (A002808) }. - Jaroslav Krizek, Mar 13 2010

G.f. over nonprime powers: Sum_{k >= 1} prime(k)*x^k-prime(prime(k))*x^prime(k). - Benedict W. J. Irwin, Jun 11 2016

Extensions

Edited by M. F. Hasler, Jul 31 2015