A229125 Numbers of the form p * m^2, where p is prime and m > 0: union of A228056 and A000040.

2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, 27, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 48, 50, 52, 53, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 79, 80, 83, 89, 92, 97, 98, 99, 101, 103, 107, 108, 109, 112, 113, 116, 117, 124, 125, 127, 128, 131, 137, 139, 147, 148, 149

Offset: 1

Author

Chris Boyd, Sep 14 2013

Keywords

• nonn

Comments

No term is the product of two other terms.

Squares of terms and pairwise products of distinct terms form a subsequence of A028260.

Numbers n such that A162642(n) = 1. - Jason Kimberley, Oct 10 2016

Numbers k such that A007913(k) is a prime number. - Amiram Eldar, Jul 27 2020

Links

• Chris Boyd, Table of n, a(n) for n = 1..10000
• Eckford Cohen, Arithmetical notes, IX. On the set of integers representable as a product of a prime and square, Acta Arithmetica, Vol. 7 (1962), pp. 417-420.

Crossrefs

Subsequence of A026424.

Cf. A028260, A162642, A229153.

Cf. A007913, A013661.

Subsequences: A000040, A030078, A050997, A054753, A092759, A179643, A179665, A246551.

Programs

Mathematica

With[{nn=70},Take[Union[Flatten[Table[p*m^2,{p,Prime[Range[nn]]},{m,nn}]]], nn]] (* Harvey P. Dale, Dec 02 2014 *)

PARI

test(n)=isprime(core(n))
for(n=1,200,if(test(n), print1(n",")))

Python

from math import isqrt
from sympy import primepi
def A229125(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = 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-sum(primepi(x//y**2) for y in range(1,isqrt(x)+1))
    return bisection(f,n,n) # Chai Wah Wu, Jan 30 2025

Formula

The number of terms not exceeding x is (Pi^2/6) * x/log(x) + O(x/(log(x))^2) (Cohen, 1962). - Amiram Eldar, Jul 27 2020