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

1,1

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]

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

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 *)

Prog

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

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.

Sequence in context: A168465 A140562 A155547 * A156007 A067774 A063637

Adjacent sequences: A007818 A007819 A007820 * A007822 A007823 A007824

Keyword

nonn

Author

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

Extensions

Edited by M. F. Hasler, Jul 31 2015

Status

approved