A336615 Numbers of the form p * m^2, where p is prime and m > 0 is not divisible by p.

2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 37, 41, 43, 44, 45, 47, 48, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 80, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 112, 113, 116, 117, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153, 157, 162

Offset: 1

Author

Amiram Eldar, Jul 27 2020

Keywords

• nonn

Comments

Numbers k such that A008833(k) is a unitary divisor of k and A007913(k) = k / A008833(k) is a prime number.

Links

• Amiram Eldar, 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

Intersection of A229125 and A335275.

Cf. A007913, A008833, A013661.

Subsequences: A000040, A054753, A179643.

Programs

Mathematica

Select[Range[2, 200], Select[FactorInteger[#][[;;, 2]], OddQ] == {1} &]

Python

from math import isqrt
from sympy import primepi, primefactors
def A336615(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(m:=x//y**2)-sum(1 for p in primefactors(y) if p<=m) 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).