A046388 Odd numbers of the form p*q where p and q are distinct primes.

15, 21, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, 95, 111, 115, 119, 123, 129, 133, 141, 143, 145, 155, 159, 161, 177, 183, 185, 187, 201, 203, 205, 209, 213, 215, 217, 219, 221, 235, 237, 247, 249, 253, 259, 265, 267, 287, 291, 295, 299, 301, 303

Offset: 1

Author

Patrick De Geest, Jun 15 1998

Keywords

• nonn

Comments

These are the odd squarefree semiprimes.

These numbers k have the property that k is a Fermat pseudoprime for at least two bases 1 < b < k - 1. That is, b^(k - 1) == 1 (mod k). See sequence A175101 for the number of bases. - Karsten Meyer, Dec 02 2010

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Crossrefs

Intersection of A005117 and A046315, or equally, of A005408 and A006881, or of A001358 and A056911.

Union of A080774 and A190299, which the latter is the union of A131574 and A016105.

Subsequence of A024556 and of A225375.

Cf. A353481 (characteristic function).

Cf. A001221, A046404, A056911, A175101.

Different from A056913, A098905, A225375.

Programs

Haskell

a046388 n = a046388_list !! (n-1)
a046388_list = filter ((== 2) . a001221) a056911_list
-- Reinhard Zumkeller, Jan 02 2014

Mathematica

max = 300; A046388 = Sort@Flatten@Table[Prime[m] Prime[n], {n, 3, Ceiling[PrimePi[max/3]]}, {m, 2, n - 1}]; Select[A046388, # < max &] (* Alonso del Arte based on Robert G. Wilson v's program for A006881, Oct 24 2011 *)

PARI

isok(n) = (n % 2) && (bigomega(n) == 2) && (omega(n)==2); \\ Michel Marcus, Feb 05 2015

Python

from sympy import factorint
def ok(n):
    if n < 2 or n%2 == 0: return False
    f = factorint(n)
    return len(f) == 2 and sum(f.values()) == 2
print([k for k in range(304) if ok(k)]) # Michael S. Branicky, May 03 2022

Python

from math import isqrt
from sympy import primepi, primerange
def A046388(n):
    if n == 1: return 15
    def f(x): return int(n-1+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(3, s+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,n,n) # Chai Wah Wu, Sep 10 2024

Formula

Sum_{n>=1} 1/a(n)^s = (1/2)*(P(s)^2 - P(2*s)) + 1/4^s - P(s)/2^s, for s>1, where P is the prime zeta function. - Amiram Eldar, Nov 21 2020

Extensions

I removed some ambiguity in the definition and edited the entry, merging in some material from A146166. - N. J. A. Sloane, May 09 2013