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1 together with products of 2 or more distinct primes.
43

%I #54 Aug 02 2024 22:37:54

%S 1,6,10,14,15,21,22,26,30,33,34,35,38,39,42,46,51,55,57,58,62,65,66,

%T 69,70,74,77,78,82,85,86,87,91,93,94,95,102,105,106,110,111,114,115,

%U 118,119,122,123,129,130,133,134,138,141,142,143,145,146,154,155,158

%N 1 together with products of 2 or more distinct primes.

%C Nonprime squarefree numbers.

%C Except for 1, composite n such that the squarefree part of n is greater than phi(n). - _Benoit Cloitre_, Apr 06 2002

%H T. D. Noe, <a href="/A000469/b000469.txt">Table of n, a(n) for n = 1..10000</a>

%F n such that A007913(n)>A000010(n). - _Benoit Cloitre_, Apr 06 2002

%F 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 A005171(a(n))*A008966(a(n)) = 1. - _Reinhard Zumkeller_, Nov 01 2009

%F Sum(n=1, Infinity, 1/a(n)^s) = Zeta(s)/Zeta(2s) - PrimeZeta(s). - _Enrique PĂ©rez Herrero_, Mar 31 2012

%F n such that A001221(n) = A001222(n), n nonprime. - _Carlos Eduardo Olivieri_, Aug 06 2015

%F a(n) = kn + O(n/log n) where k = Pi^2/6. - _Charles R Greathouse IV_, Aug 02 2024

%p select(numtheory:-issqrfree and not isprime, [$1..1000]); # _Robert Israel_, Aug 06 2015

%t lst={}; Do[If[SquareFreeQ[n], If[ !PrimeQ[n], AppendTo[lst,n]]], {n,200}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Jan 20 2009 *)

%t With[{upto=200},Complement[Select[Range[upto],SquareFreeQ],Prime[ Range[ PrimePi[ upto]]]]] (* _Harvey P. Dale_, Oct 01 2011 *)

%t Select[Range[200], !PrimeQ[#] && PrimeOmega[#] == PrimeNu[#] &] (* _Carlos Eduardo Olivieri_, Aug 06 2015 *)

%o (PARI) for(n=0,64, if(isprime(n), n+1, if(issquarefree(n),print(n))))

%o (PARI) for(n=1,160,if(core(n)*(1-isprime(n))>eulerphi(n),print1(n,",")))

%o (Haskell)

%o a000469 n = a000469_list !! (n-1)

%o a000469_list = filter ((== 0) . a010051) a005117_list

%o -- _Reinhard Zumkeller_, Mar 21 2014

%o (Python)

%o from math import isqrt

%o from sympy import primepi, mobius

%o def A000469(n):

%o def f(x): return n+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))

%o m, k = n, f(n)

%o while m != k:

%o m, k = k, f(k)

%o return m # _Chai Wah Wu_, Aug 02 2024

%Y Cf. A005117, A007913, A000010, A010051, A239508, A239509, A120944 (composite squarefree numbers, same sequence apart from the first term).

%K nonn,easy,nice

%O 1,2

%A Dan Bentley (dtb(AT)research.att.com)