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A048639 Binary encoding of A006881, numbers with two distinct prime divisors.

%I A048639 #19 Feb 22 2025 01:22:52
%S A048639 3,5,9,6,10,17,33,18,65,12,129,34,257,66,20,130,513,1025,36,258,2049,
%T A048639 24,4097,68,8193,514,40,1026,16385,132,32769,2050,260,65537,72,131073,
%U A048639 4098,8194,136,262145,16386,524289,48,516,1048577,1028,2097153,32770
%N A048639 Binary encoding of A006881, numbers with two distinct prime divisors.
%H A048639 Michael De Vlieger, <a href="/A048639/b048639.txt">Table of n, a(n) for n = 1..10000</a>
%F A048639 a(n) = 2^(i-1) + 2^(j-1), where A006881(n) = p_i*p_j (p_i and p_j stand for the i-th and j-th primes respectively, where the first prime is 2).
%p A048639 encode_A006881 := proc(upto_n) local b,i; b := [ ]; for i from 1 to upto_n do if((0 <> mobius(i)) and (4 = tau(i))) then b := [ op(b), bef(i) ]; fi; od: RETURN(b); end; # see A048623 for bef
%t A048639 Total[2^PrimePi@ # &@ (Map[First, FactorInteger@ #] - 1)] & /@ Select[Range@ 160, SquareFreeQ@ # && PrimeOmega@ # == 2 &] (* _Michael De Vlieger_, Oct 01 2015 *)
%o A048639 (PARI) lista(nn) = {for (n=1, nn, if (issquarefree(n) && bigomega(n)==2, f = factor(n); x = sum(k=1, #f~, 2^(primepi(f[k,1])-1)); print1(x, ", ");););} \\ _Michel Marcus_, Oct 01 2015
%o A048639 (Python)
%o A048639 from math import isqrt
%o A048639 from sympy import primepi, primerange, primefactors
%o A048639 def A048639(n):
%o A048639     def f(x): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
%o A048639     m, k = n, f(n)
%o A048639     while m != k: m, k = k, f(k)
%o A048639     return sum(1<<primepi(p)-1 for p in primefactors(m)) # _Chai Wah Wu_, Feb 22 2025
%Y A048639 Permutation of A018900. Cf. A048640, A048623.
%K A048639 nonn
%O A048639 1,1
%A A048639 _Antti Karttunen_, Jul 14 1999