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A286251 Compound filter: a(n) = P(A001511(1+n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

%I A286251 #12 Feb 16 2025 08:33:44
%S A286251 3,2,9,7,5,16,14,29,12,16,9,67,5,16,50,121,5,67,9,67,23,16,14,277,12,
%T A286251 16,48,67,5,436,27,497,23,16,31,631,5,16,40,277,5,436,9,67,80,16,20,
%U A286251 1129,12,67,31,67,5,277,40,277,23,16,9,1771,5,16,160,2017,23,436,9,67,23,436,14,2557,5,16,94,67,23,436,20,1129,138,16,9,1771,23,16,40,277,5
%N A286251 Compound filter: a(n) = P(A001511(1+n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.
%H A286251 Antti Karttunen, <a href="/A286251/b286251.txt">Table of n, a(n) for n = 1..10000</a>
%H A286251 MathWorld, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F A286251 a(n) = (1/2)*(2 + ((A001511(1+n)+A046523(n))^2) - A001511(1+n) - 3*A046523(n)).
%o A286251 (PARI)
%o A286251 A001511(n) = (1+valuation(n,2));
%o A286251 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 A286251 A286251(n) = (2 + ((A001511(1+n)+A046523(n))^2) - A001511(1+n) - 3*A046523(n))/2;
%o A286251 for(n=1, 10000, write("b286251.txt", n, " ", A286251(n)));
%o A286251 (Scheme) (define (A286251 n) (* (/ 1 2) (+ (expt (+ (A001511 (+ 1 n)) (A046523 n)) 2) (- (A001511 (+ 1 n))) (- (* 3 (A046523 n))) 2)))
%o A286251 (Python)
%o A286251 from sympy import factorint
%o A286251 def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
%o A286251 def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
%o A286251 def P(n):
%o A286251     f = factorint(n)
%o A286251     return sorted([f[i] for i in f])
%o A286251 def a046523(n):
%o A286251     x=1
%o A286251     while True:
%o A286251         if P(n) == P(x): return x
%o A286251         else: x+=1
%o A286251 def a(n): return T(a001511(n + 1), a046523(n)) # _Indranil Ghosh_, May 07 2017
%Y A286251 Cf. A000027, A001511, A046523, A286161, A286252, A286253, A286254.
%K A286251 nonn
%O A286251 1,1
%A A286251 _Antti Karttunen_, May 07 2017