A046404 -id:A046404 - cp's OEIS frontend

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

Patrick De Geest, Jun 15 1998

nonn

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

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

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.

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

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 *)

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

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

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

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

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

A225375 Odd numbers with exactly 2 distinct prime factors.

Original entry on oeis.org

15, 21, 33, 35, 39, 45, 51, 55, 57, 63, 65, 69, 75, 77, 85, 87, 91, 93, 95, 99, 111, 115, 117, 119, 123, 129, 133, 135, 141, 143, 145, 147, 153, 155, 159, 161, 171, 175, 177, 183, 185, 187, 189, 201, 203, 205, 207, 209, 213, 215, 217, 219, 221, 225, 235, 237

Offset: 1

R. J. Mathar, Oct 13 2008

nonn

Numbers of the form p^i*q^j where p, q are distinct odd primes and i>=1, j>=1.

Subset of A098905 (which contains in addition A046390 and numbers like 255255, 285285, 345345, 373065 etc.).

Subsequence of A272592. A046388 is a subsequence.
Cf. A046315, A046390, A046404, A024556, A056913.
Different from A046388, A098905.

isA225375 := proc(n) RETURN( (n mod 2 = 1) and (A001221(n) = 2) ); end proc:
for n from 1 to 840 do if isA225375(n) then printf("%d,",n) ; end if; end do:

Select[2 Range[200] + 1, PrimeNu[#] == 2&] (* Jean-François Alcover, Apr 04 2020 *)

is_A046388(n)={ bittest(n,0) & omega(n)==2 } \\ M. F. Hasler, Feb 13 2012

Entry created by N. J. A. Sloane, May 09 2013 in order to restore the original definition of A046388.

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

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A225375 Odd numbers with exactly 2 distinct prime factors.

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A046372 Odd numbers with exactly 2 palindromic prime factors (counted with multiplicity).

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