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

1,1

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.

Formula

The number of terms not exceeding x is (Pi^2/6) * x/log(x) + O(x/(log(x))^2) (Cohen, 1962).

Mathematica

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

Prog

(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

Crossrefs

Intersection of A229125 and A335275.

Cf. A007913, A008833, A013661.

Subsequences: A000040, A054753, A179643.

Sequence in context: A336418 A212166 A293511 * A325337 A342119 A372742

Adjacent sequences: A336612 A336613 A336614 * A336616 A336617 A336618

Keyword

nonn

Author

Amiram Eldar, Jul 27 2020

Status

approved