A161188 -id:A161188 - cp's OEIS frontend

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

N. J. A. Sloane, Dec 11 2003

nonn

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

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

Complement of A089229.

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

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

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

is(n)=isprime(n) || issquare(n) \\ Charles R Greathouse IV, Oct 14 2016

{A89237=List([0..5]); A089237(n)=while(#A89237A340389. - M. F. Hasler, Jul 24 2021, edited Sep 01 2021

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

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

A161187 Let S = A089237\{0} = union of primes and nonzero squares; sequence gives indices of squares.

Original entry on oeis.org

1, 4, 7, 10, 14, 17, 22, 26, 31, 35, 41, 46, 52, 58, 63, 70, 78, 84, 91, 98, 106, 114, 122, 129, 139, 148, 156, 165, 175, 184, 193, 204, 214, 225, 235, 246, 256, 266, 279, 291, 304, 316, 326, 339, 351, 365, 376, 390, 406, 417, 429, 445, 462, 475, 489, 501, 514

Offset: 1

Daniel Tisdale, Jun 06 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A089237. Complement of A161188.

Cf. A050052.

a161187 n = a161187_list !! (n-1)
a161187_list = tail $ findIndices ((== 1) . a010052) a089237_list
-- Reinhard Zumkeller, Dec 18 2012

[1] cat [#PrimesUpTo(n^2-1)+n: n in [2..100]]; // Vincenzo Librandi, Feb 18 2016

Table[PrimePi[n^2 - 1] + n, {n, 60}] (* Vincenzo Librandi, Feb 18 2016 *)

from sympy import primepi
def A161187(n): return n+primepi(n**2) # Chai Wah Wu, Oct 12 2024

A089237(a(n)+1) = A000290(n). - Reinhard Zumkeller, Dec 18 2012

Edited by N. J. A. Sloane, Jun 07 2009

Extended by Ray Chandler, May 06 2010

cp's OEIS Frontend

A089237 List of primes and squares.

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