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A089237 List of primes and squares.

%I A089237 #42 Oct 13 2024 02:39:49
%S A089237 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,
%T A089237 61,64,67,71,73,79,81,83,89,97,100,101,103,107,109,113,121,127,131,
%U A089237 137,139,144,149,151,157,163,167,169,173,179,181,191,193,196,197,199,211,223,225,227
%N A089237 List of primes and squares.
%C A089237 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
%H A089237 Zak Seidov, <a href="/A089237/b089237.txt">Table of n, a(n) for n = 1..1000</a>
%F A089237 a(A161187(n)+1) = A000290(n); a(A161188(n)+1) = A000040(n). - _Reinhard Zumkeller_, Dec 18 2012
%F A089237 A010051(a(n)) + A010052(a(n)) = 1. - _Reinhard Zumkeller_, Jul 07 2014
%F A089237 a(n) ~ n log n. - _Charles R Greathouse IV_, Oct 14 2016
%t A089237 m=100; Sort[Flatten[{Range[0,m]^2, Prime[Range[PrimePi[m^2]]]}]] (* _Zak Seidov_, Nov 05 2009 *)
%o A089237 (Haskell)
%o A089237 a089237 n = a089237_list !! (n-1)
%o A089237 a089237_list = merge a000040_list a000290_list where
%o A089237    merge xs'@(x:xs) ys'@(y:ys) =
%o A089237          if x < y then x : merge xs ys' else y : merge xs' ys
%o A089237 -- _Reinhard Zumkeller_, Dec 18 2012
%o A089237 (PARI) is(n)=isprime(n) || issquare(n) \\ _Charles R Greathouse IV_, Oct 14 2016
%o A089237 (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
%o A089237 (Python)
%o A089237 from math import isqrt
%o A089237 from sympy import primepi
%o A089237 def A089237(n):
%o A089237     def bisection(f,kmin=0,kmax=1):
%o A089237         while f(kmax) > kmax: kmax <<= 1
%o A089237         while kmax-kmin > 1:
%o A089237             kmid = kmax+kmin>>1
%o A089237             if f(kmid) <= kmid:
%o A089237                 kmax = kmid
%o A089237             else:
%o A089237                 kmin = kmid
%o A089237         return kmax
%o A089237     def f(x): return int(n-1+x-primepi(x)-isqrt(x))
%o A089237     return bisection(f,n-1,n-1) # _Chai Wah Wu_, Oct 12 2024
%Y A089237 Complement of A089229.
%Y A089237 Cf. A000040, A000290, A010051, A010052, A161187, A161188, A268335.
%K A089237 nonn
%O A089237 1,3
%A A089237 _N. J. A. Sloane_, Dec 11 2003