A337877 Numbers of the form p^2*q where p and q are primes and p <= q.

8, 12, 20, 27, 28, 44, 45, 52, 63, 68, 76, 92, 99, 116, 117, 124, 125, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 244, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 343, 356, 369, 387, 388, 404, 412, 423, 425, 428, 436, 452, 475, 477, 508, 524, 531, 539, 548, 549, 556, 575, 596, 603

Offset: 1

Author

Robert Israel, Sep 27 2020

Keywords

• nonn

Examples

			a(3) = 20 is a term because 20=2^2*5 with 2 <= 5.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Contained in A337806.

Programs

Maple

N:= 3000: # for terms <= N
P:= select(isprime, [2,seq(i,i=3..N/2,2)]): nP:= nops(P):
R:= NULL:
for i from 1 to nP do
  p2:= P[i]^2;
  for j from i to nP do
    x:= p2*P[j];
    if x > N then break fi;
    R:= R, x
od od:
sort([R]);

Python

from sympy import primepi, primerange, integer_nthroot
def A337877(n):
    def f(x): return int(n+x-sum(primepi(x//k**2)-a for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1))))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f) # Chai Wah Wu, Aug 29 2024