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A000469 1 together with products of 2 or more distinct primes.

%I A000469 #54 Aug 02 2024 22:37:54
%S A000469 1,6,10,14,15,21,22,26,30,33,34,35,38,39,42,46,51,55,57,58,62,65,66,
%T A000469 69,70,74,77,78,82,85,86,87,91,93,94,95,102,105,106,110,111,114,115,
%U A000469 118,119,122,123,129,130,133,134,138,141,142,143,145,146,154,155,158
%N A000469 1 together with products of 2 or more distinct primes.
%C A000469 Nonprime squarefree numbers.
%C A000469 Except for 1, composite n such that the squarefree part of n is greater than phi(n). - _Benoit Cloitre_, Apr 06 2002
%H A000469 T. D. Noe, <a href="/A000469/b000469.txt">Table of n, a(n) for n = 1..10000</a>
%F A000469 n such that A007913(n)>A000010(n). - _Benoit Cloitre_, Apr 06 2002
%F A000469 N-floor(N/p1) - floor(N/(p2) - ... - floor(N/p(i) + floor(N/(c2) + floor(N/(c3)+ ... + floor(N/c(j)-1 where N is any number; p1,p2 are the primes with p(i) being the first prime > square root of N and c2, c3 are the numbers other than 1 in this sequence with c(j) <= N will yield the number of primes less than or equal to N other than p1, p2, ..., p(i). - _Ben Paul Thurston_, Aug 15 2007
%F A000469 A005171(a(n))*A008966(a(n)) = 1. - _Reinhard Zumkeller_, Nov 01 2009
%F A000469 Sum(n=1, Infinity, 1/a(n)^s) = Zeta(s)/Zeta(2s) - PrimeZeta(s). - _Enrique Pérez Herrero_, Mar 31 2012
%F A000469 n such that A001221(n) = A001222(n), n nonprime. - _Carlos Eduardo Olivieri_, Aug 06 2015
%F A000469 a(n) = kn + O(n/log n) where k = Pi^2/6. - _Charles R Greathouse IV_, Aug 02 2024
%p A000469 select(numtheory:-issqrfree and not isprime, [$1..1000]); # _Robert Israel_, Aug 06 2015
%t A000469 lst={}; Do[If[SquareFreeQ[n], If[ !PrimeQ[n], AppendTo[lst,n]]], {n,200}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Jan 20 2009 *)
%t A000469 With[{upto=200},Complement[Select[Range[upto],SquareFreeQ],Prime[ Range[ PrimePi[ upto]]]]] (* _Harvey P. Dale_, Oct 01 2011 *)
%t A000469 Select[Range[200], !PrimeQ[#] && PrimeOmega[#] == PrimeNu[#] &] (* _Carlos Eduardo Olivieri_, Aug 06 2015 *)
%o A000469 (PARI) for(n=0,64, if(isprime(n), n+1, if(issquarefree(n),print(n))))
%o A000469 (PARI) for(n=1,160,if(core(n)*(1-isprime(n))>eulerphi(n),print1(n,",")))
%o A000469 (Haskell)
%o A000469 a000469 n = a000469_list !! (n-1)
%o A000469 a000469_list = filter ((== 0) . a010051) a005117_list
%o A000469 -- _Reinhard Zumkeller_, Mar 21 2014
%o A000469 (Python)
%o A000469 from math import isqrt
%o A000469 from sympy import primepi, mobius
%o A000469 def A000469(n):
%o A000469     def f(x): return n+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A000469     m, k = n, f(n)
%o A000469     while m != k:
%o A000469         m, k = k, f(k)
%o A000469     return m # _Chai Wah Wu_, Aug 02 2024
%Y A000469 Cf. A005117, A007913, A000010, A010051, A239508, A239509, A120944 (composite squarefree numbers, same sequence apart from the first term).
%K A000469 nonn,easy,nice
%O A000469 1,2
%A A000469 Dan Bentley (dtb(AT)research.att.com)