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A286162 Compound filter: a(n) = T(A001511(n), A278222(n)), where T(n,k) is sequence A000027 used as a pairing function.

%I A286162 #14 Feb 16 2025 08:33:44
%S A286162 2,5,7,9,16,12,29,14,16,23,67,18,67,38,121,20,16,23,67,31,436,80,277,
%T A286162 25,67,80,631,48,277,138,497,27,16,23,67,31,436,80,277,40,436,467,
%U A286162 1771,94,1771,302,1129,33,67,80,631,94,1771,668,2557,59,277,302,2557,156,1129,530,2017,35,16,23,67,31,436,80,277,40,436,467,1771,94,1771,302,1129,50
%N A286162 Compound filter: a(n) = T(A001511(n), A278222(n)), where T(n,k) is sequence A000027 used as a pairing function.
%H A286162 Antti Karttunen, <a href="/A286162/b286162.txt">Table of n, a(n) for n = 1..10000</a>
%H A286162 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F A286162 a(n) = (1/2)*(2 + ((A001511(n)+A278222(n))^2) - A001511(n) - 3*A278222(n)).
%o A286162 (PARI)
%o A286162 A001511(n) = (1+valuation(n,2));
%o A286162 A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of _M. F. Hasler_
%o A286162 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 A286162 A278222(n) = A046523(A005940(1+n));
%o A286162 A286162(n) = (2 + ((A001511(n)+A278222(n))^2) - A001511(n) - 3*A278222(n))/2;
%o A286162 for(n=1, 10000, write("b286162.txt", n, " ", A286162(n)));
%o A286162 (Scheme) (define (A286162 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A278222 n)) 2) (- (A001511 n)) (- (* 3 (A278222 n))) 2)))
%o A286162 (Python)
%o A286162 from sympy import prime, factorint
%o A286162 import math
%o A286162 def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
%o A286162 def A(n): return n - 2**int(math.floor(math.log(n, 2)))
%o A286162 def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
%o A286162 def a005940(n): return b(n - 1)
%o A286162 def P(n):
%o A286162     f = factorint(n)
%o A286162     return sorted([f[i] for i in f])
%o A286162 def a046523(n):
%o A286162     x=1
%o A286162     while True:
%o A286162         if P(n) == P(x): return x
%o A286162         else: x+=1
%o A286162 def a278222(n): return a046523(a005940(n + 1))
%o A286162 def a001511(n): return bin(n)[2:][::-1].index("1") + 1
%o A286162 def a(n): return T(a001511(n), a278222(n)) # _Indranil Ghosh_, May 05 2017
%Y A286162 Cf. A000027, A001511, A278222, A286160, A286161, A286163, A286164.
%K A286162 nonn
%O A286162 1,1
%A A286162 _Antti Karttunen_, May 04 2017