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A286161 Compound filter: a(n) = T(A001511(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

%I A286161 #19 Feb 16 2025 08:33:44
%S A286161 1,5,2,18,2,23,2,59,7,23,2,94,2,23,16,195,2,80,2,94,16,23,2,355,7,23,
%T A286161 29,94,2,467,2,672,16,23,16,706,2,23,16,355,2,467,2,94,67,23,2,1331,7,
%U A286161 80,16,94,2,302,16,355,16,23,2,1894,2,23,67,2422,16,467,2,94,16,467,2,2779,2,23,67,94,16,467,2,1331,121,23,2,1894,16,23,16,355,2,1832
%N A286161 Compound filter: a(n) = T(A001511(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.
%H A286161 Antti Karttunen, <a href="/A286161/b286161.txt">Table of n, a(n) for n = 1..10000</a>
%H A286161 MathWorld, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F A286161 a(n) = (1/2)*(2 + ((A001511(n)+A046523(n))^2) - A001511(n) - 3*A046523(n)).
%o A286161 (PARI)
%o A286161 A001511(n) = (1+valuation(n,2));
%o A286161 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from _Charles R Greathouse IV_, Aug 17 2011
%o A286161 A286161(n) = (2 + ((A001511(n)+A046523(n))^2) - A001511(n) - 3*A046523(n))/2;
%o A286161 for(n=1, 10000, write("b286161.txt", n, " ", A286161(n)));
%o A286161 (Scheme) (define (A286161 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A046523 n)) 2) (- (A001511 n)) (- (* 3 (A046523 n))) 2)))
%o A286161 (Python)
%o A286161 from sympy import factorint
%o A286161 def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
%o A286161 def P(n):
%o A286161     f = factorint(n)
%o A286161     return sorted([f[i] for i in f])
%o A286161 def a046523(n):
%o A286161     x=1
%o A286161     while True:
%o A286161         if P(n) == P(x): return x
%o A286161         else: x+=1
%o A286161 def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
%o A286161 def a(n): return T(a001511(n), a046523(n)) # _Indranil Ghosh_, May 06 2017
%Y A286161 Cf. A000027, A001511, A046523, A286160, A286162, A286163, A286164, A286251.
%K A286161 nonn
%O A286161 1,2
%A A286161 _Antti Karttunen_, May 04 2017