A089237 List of primes and squares.

0, 1, 2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 36, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 100, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 196, 197, 199, 211, 223, 225, 227

Offset

1,3

Comments

Starting at a(1) = 1, this is the lexicographically earliest sequence of distinct numbers whose partial products are all exponentially odd numbers (A268335). - Amiram Eldar, Sep 24 2024

Links

Zak Seidov, Table of n, a(n) for n = 1..1000

Formula

a(A161187(n)+1) = A000290(n); a(A161188(n)+1) = A000040(n). - Reinhard Zumkeller, Dec 18 2012

A010051(a(n)) + A010052(a(n)) = 1. - Reinhard Zumkeller, Jul 07 2014

a(n) ~ n log n. - Charles R Greathouse IV, Oct 14 2016

Mathematica

m=100; Sort[Flatten[{Range[0, m]^2, Prime[Range[PrimePi[m^2]]]}]] (* Zak Seidov, Nov 05 2009 *)

Prog

(Haskell)
a089237 n = a089237_list !! (n-1)
a089237_list = merge a000040_list a000290_list where
   merge xs'@(x:xs) ys'@(y:ys) =
         if x < y then x : merge xs ys' else y : merge xs' ys
-- Reinhard Zumkeller, Dec 18 2012
(PARI) is(n)=isprime(n) || issquare(n) \\ Charles R Greathouse IV, Oct 14 2016
(PARI) {A89237=List([0..5]); A089237(n)=while(#A89237<n, my(t=A89237[#A89237]); forprime(p=t+1, t=(sqrtint(t)+1)^2, listput(A89237, p)); listput(A89237, t)); A89237[n]} \\ For use in other functions, e.g., A340389. - M. F. Hasler, Jul 24 2021, edited Sep 01 2021
(Python)
from math import isqrt
from sympy import primepi
def A089237(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 int(n-1+x-primepi(x)-isqrt(x))
    return bisection(f, n-1, n-1) # Chai Wah Wu, Oct 12 2024

Crossrefs

Complement of A089229.

Cf. A000040, A000290, A010051, A010052, A161187, A161188, A268335.

Sequence in context: A123193 A345944 A066724 * A352870 A009087 A026477

Adjacent sequences: A089234 A089235 A089236 * A089238 A089239 A089240

Keyword

nonn

Author

N. J. A. Sloane, Dec 11 2003

Status

approved