A037143 Numbers with at most 2 prime factors (counted with multiplicity).

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

A001222(a(n)) <= 2; A054576(a(n)) = 1. - Reinhard Zumkeller, Mar 10 2006

Products of two noncomposite numbers. - Juri-Stepan Gerasimov, Apr 15 2010

Also, numbers with permutations of all divisors only with coprime adjacent elements: A109810(a(n)) > 0. - Reinhard Zumkeller, May 24 2010

A060278(a(n)) = 0. - Reinhard Zumkeller, Apr 05 2013

1 together with numbers k such that sigma(k) + phi(k) - d(k) = 2k - 2. - Wesley Ivan Hurt, May 03 2015

Products of two not necessarily distinct terms of A008578 (the same relation between A000040 and A001358). - Flávio V. Fernandes, May 28 2021

Links

• Felix Fröhlich, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)
• Andreas Weingartner, Uniform distribution of alpha*n modulo one for a family of integer sequences, arXiv:2303.16819 [math.NT], 2023.

Crossrefs

Union of A008578 and A001358. Complement of A033942.

Cf. A063928, A001222, A054576, A109810, A060278.

A101040(a(n))=1 for n>1.

Subsequence of A037144. - Reinhard Zumkeller, May 24 2010

A098962 and A139690 are subsequences.

Programs

Haskell

a037143 n = a037143_list !! (n-1)
a037143_list = 1 : merge a000040_list a001358_list where
   merge xs'@(x:xs) ys'@(y:ys) =
         if x < y then x : merge xs ys' else y : merge xs' ys
-- Reinhard Zumkeller, Dec 18 2012

Maple

with(numtheory): A037143:=n->`if`(bigomega(n)<3,n,NULL): seq(A037143(n), n=1..200); # Wesley Ivan Hurt, May 03 2015

Mathematica

Select[Range[120], PrimeOmega[#] <= 2 &] (* Ivan Neretin, Aug 16 2015 *)

PARI

is(n)=bigomega(n)<3 \\ Charles R Greathouse IV, Apr 29 2015

Python

from math import isqrt
from sympy import primepi, primerange
def A037143(n):
    def f(x): return int(n-2+x-primepi(x)-sum(primepi(x//k)-a for a,k in enumerate(primerange(isqrt(x)+1))))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

Extensions

More terms from Henry Bottomley, Aug 15 2001