A340389 -id:A340389 - 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

A028307 Form a triangle with n numbers in top row; all other numbers are the sum of their parents. E.g.: 4 1 2 7; 5 3 9; 8 12; 20. The numbers must be positive and distinct and the final number is to be minimized. Sequence gives final number.

Original entry on oeis.org

1, 3, 8, 20, 43, 98, 212, 465, 1000, 2144, 4497, 9504, 19872, 41455, 85356, 178630, 363467, 757085, 1541998, 3183600, 6515066, 13357593, 27432649, 55914902, 114683858, 233517515, 478061719, 972479046, 1986013932

Offset: 1

Mauro Fiorentini

Suggested by Problem 401 of the All-Soviet-Union Mathematical Competitions 1961-1986. Two different links are available for this collection.

			Solutions for n = 1, 2, ... are:
  1;
  1, 2;
  2, 1, 4;
  4, 1, 2, 7;
  7, 2, 1, 4, 6;
  8, 6, 1, 3, 2, 10;
  ...

Mauro Fiorentini, Further comments
Vladimir A. Pertsel, Problems of the All-Soviet-Union Mathematical Competitions 1961-1986

Cf. A028308, A062896, A340389.

From A.H.M. Smeets, Feb 25 2022: (Start)
a(n) > 2*a(n-1). Proof: Let x, y be the numbers in the second last row, then x >= a(n-1), y >= a(n-1) and x != y, so a(n) = x + y > 2*a(n-1).
It seems that a(n) > (4/3)*(2*a(n-1)-a(n-2)). (End)

More terms from the author, Jul 03 2001

cp's OEIS Frontend

A089237 List of primes and squares.

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