cp's OEIS Frontend

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

%I A336615 #15 Jan 31 2025 04:27:33
%S A336615 2,3,5,7,11,12,13,17,18,19,20,23,28,29,31,37,41,43,44,45,47,48,50,52,
%T A336615 53,59,61,63,67,68,71,73,75,76,79,80,83,89,92,97,98,99,101,103,107,
%U A336615 109,112,113,116,117,124,127,131,137,139,147,148,149,151,153,157,162
%N A336615 Numbers of the form p * m^2, where p is prime and m > 0 is not divisible by p.
%C A336615 Numbers k such that A008833(k) is a unitary divisor of k and A007913(k) = k / A008833(k) is a prime number.
%H A336615 Amiram Eldar, <a href="/A336615/b336615.txt">Table of n, a(n) for n = 1..10000</a>
%H A336615 Eckford Cohen, <a href="https://doi.org/10.4064/aa-7-4-417-420">Arithmetical notes, IX. On the set of integers representable as a product of a prime and square</a>, Acta Arithmetica, Vol. 7 (1962), pp. 417-420.
%F A336615 The number of terms not exceeding x is (Pi^2/6) * x/log(x) + O(x/(log(x))^2) (Cohen, 1962).
%t A336615 Select[Range[2, 200], Select[FactorInteger[#][[;;, 2]], OddQ] == {1} &]
%o A336615 (Python)
%o A336615 from math import isqrt
%o A336615 from sympy import primepi, primefactors
%o A336615 def A336615(n):
%o A336615     def bisection(f,kmin=0,kmax=1):
%o A336615         while f(kmax) > kmax: kmax <<= 1
%o A336615         kmin = kmax >> 1
%o A336615         while kmax-kmin > 1:
%o A336615             kmid = kmax+kmin>>1
%o A336615             if f(kmid) <= kmid:
%o A336615                 kmax = kmid
%o A336615             else:
%o A336615                 kmin = kmid
%o A336615         return kmax
%o A336615     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))
%o A336615     return bisection(f,n,n) # _Chai Wah Wu_, Jan 30 2025
%Y A336615 Intersection of A229125 and A335275.
%Y A336615 Cf. A007913, A008833, A013661.
%Y A336615 Subsequences: A000040, A054753, A179643.
%K A336615 nonn
%O A336615 1,1
%A A336615 _Amiram Eldar_, Jul 27 2020