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A007821
Primes p such that pi(p) is not prime.
80
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
Lubomir Alexandrov, On the nonasymptotic prime number distribution, arXiv:math/9811096 [math.NT], 1998.
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
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
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
KEYWORD
nonn
AUTHOR
Monte J. Zerger (mzerger(AT)cc4.adams.edu), Clark Kimberling
EXTENSIONS
Edited by M. F. Hasler, Jul 31 2015
STATUS
approved