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

%I A037143 #56 Aug 24 2024 05:56:08
%S A037143 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,
%T A037143 37,38,39,41,43,46,47,49,51,53,55,57,58,59,61,62,65,67,69,71,73,74,77,
%U A037143 79,82,83,85,86,87,89,91,93,94,95,97,101,103,106,107,109,111,113,115,118
%N A037143 Numbers with at most 2 prime factors (counted with multiplicity).
%C A037143 A001222(a(n)) <= 2; A054576(a(n)) = 1. - _Reinhard Zumkeller_, Mar 10 2006
%C A037143 Products of two noncomposite numbers. - _Juri-Stepan Gerasimov_, Apr 15 2010
%C A037143 Also, numbers with permutations of all divisors only with coprime adjacent elements: A109810(a(n)) > 0. - _Reinhard Zumkeller_, May 24 2010
%C A037143 A060278(a(n)) = 0. - _Reinhard Zumkeller_, Apr 05 2013
%C A037143 1 together with numbers k such that sigma(k) + phi(k) - d(k) = 2k - 2. - _Wesley Ivan Hurt_, May 03 2015
%C A037143 Products of two not necessarily distinct terms of A008578 (the same relation between A000040 and A001358). - _Flávio V. Fernandes_, May 28 2021
%H A037143 Felix Fröhlich, <a href="/A037143/b037143.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Reinhard Zumkeller)
%H A037143 Andreas Weingartner, <a href="https://arxiv.org/abs/2303.16819">Uniform distribution of alpha*n modulo one for a family of integer sequences</a>, arXiv:2303.16819 [math.NT], 2023.
%p A037143 with(numtheory): A037143:=n->`if`(bigomega(n)<3,n,NULL): seq(A037143(n), n=1..200); # _Wesley Ivan Hurt_, May 03 2015
%t A037143 Select[Range[120], PrimeOmega[#] <= 2 &] (* _Ivan Neretin_, Aug 16 2015 *)
%o A037143 (Haskell)
%o A037143 a037143 n = a037143_list !! (n-1)
%o A037143 a037143_list = 1 : merge a000040_list a001358_list where
%o A037143    merge xs'@(x:xs) ys'@(y:ys) =
%o A037143          if x < y then x : merge xs ys' else y : merge xs' ys
%o A037143 -- _Reinhard Zumkeller_, Dec 18 2012
%o A037143 (PARI) is(n)=bigomega(n)<3 \\ _Charles R Greathouse IV_, Apr 29 2015
%o A037143 (Python)
%o A037143 from math import isqrt
%o A037143 from sympy import primepi, primerange
%o A037143 def A037143(n):
%o A037143     def f(x): return int(n-2+x-primepi(x)-sum(primepi(x//k)-a for a,k in enumerate(primerange(isqrt(x)+1))))
%o A037143     kmin, kmax = 1,2
%o A037143     while f(kmax) >= kmax:
%o A037143         kmax <<= 1
%o A037143     while True:
%o A037143         kmid = kmax+kmin>>1
%o A037143         if f(kmid) < kmid:
%o A037143             kmax = kmid
%o A037143         else:
%o A037143             kmin = kmid
%o A037143         if kmax-kmin <= 1:
%o A037143             break
%o A037143     return kmax # _Chai Wah Wu_, Aug 23 2024
%Y A037143 Union of A008578 and A001358. Complement of A033942.
%Y A037143 Cf. A063928, A001222, A054576, A109810, A060278.
%Y A037143 A101040(a(n))=1 for n>1.
%Y A037143 Subsequence of A037144. - _Reinhard Zumkeller_, May 24 2010
%Y A037143 A098962 and A139690 are subsequences.
%K A037143 nonn
%O A037143 1,2
%A A037143 _N. J. A. Sloane_
%E A037143 More terms from _Henry Bottomley_, Aug 15 2001