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A286163 Compound filter: a(n) = T(A046523(n), A278222(n)), where T(n,k) is sequence A000027 used as a pairing function.

%I A286163 #13 Feb 16 2025 08:33:44
%S A286163 2,5,12,14,23,42,38,44,40,61,80,117,80,84,216,152,23,148,80,148,601,
%T A286163 142,302,375,109,142,911,183,302,1020,530,560,61,61,142,856,467,142,
%U A286163 412,430,467,1741,1832,265,2497,412,1178,1323,109,265,826,265,1832,1735,2932,489,412,412,2630,2835,1178,672,2787,2144,61,625,80,148,601,850,302,2998,467,601
%N A286163 Compound filter: a(n) = T(A046523(n), A278222(n)), where T(n,k) is sequence A000027 used as a pairing function.
%H A286163 Antti Karttunen, <a href="/A286163/b286163.txt">Table of n, a(n) for n = 1..10000</a>
%H A286163 MathWorld, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F A286163 a(n) = (1/2)*(2 + ((A046523(n)+A278222(n))^2) - A046523(n) - 3*A278222(n)).
%o A286163 (PARI)
%o A286163 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 A286163 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 A286163 A278222(n) = A046523(A005940(1+n));
%o A286163 A286163(n) = (2 + ((A046523(n)+A278222(n))^2) - A046523(n) - 3*A278222(n))/2;
%o A286163 for(n=1, 10000, write("b286163.txt", n, " ", A286163(n)));
%o A286163 (Scheme) (define (A286163 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A278222 n)) 2) (- (A046523 n)) (- (* 3 (A278222 n))) 2)))
%o A286163 (Python)
%o A286163 from sympy import prime, factorint
%o A286163 import math
%o A286163 def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
%o A286163 def A(n): return n - 2**int(math.floor(math.log(n, 2)))
%o A286163 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 A286163 def a005940(n): return b(n - 1)
%o A286163 def P(n):
%o A286163     f = factorint(n)
%o A286163     return sorted([f[i] for i in f])
%o A286163 def a046523(n):
%o A286163     x=1
%o A286163     while True:
%o A286163         if P(n) == P(x): return x
%o A286163         else: x+=1
%o A286163 def a278222(n): return a046523(a005940(n + 1))
%o A286163 def a(n): return T(a046523(n), a278222(n)) # _Indranil Ghosh_, May 05 2017
%Y A286163 Cf. A000027, A046523, A278222, A286160, A286161, A286162, A286164.
%K A286163 nonn
%O A286163 1,1
%A A286163 _Antti Karttunen_, May 04 2017