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A286463 Compound filter (3-adic valuation & prime-signature): a(n) = P(A051064(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

%I A286463 #11 Feb 16 2025 08:33:45
%S A286463 1,2,5,7,2,23,2,29,18,16,2,80,2,16,23,121,2,94,2,67,23,16,2,302,7,16,
%T A286463 59,67,2,467,2,497,23,16,16,706,2,16,23,277,2,467,2,67,94,16,2,1178,7,
%U A286463 67,23,67,2,355,16,277,23,16,2,1832,2,16,94,2017,16,467,2,67,23,436,2,2704,2,16,80,67,16,467,2,1129,195,16,2,1832,16,16,23,277,2,1894,16
%N A286463 Compound filter (3-adic valuation & prime-signature): a(n) = P(A051064(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.
%H A286463 Antti Karttunen, <a href="/A286463/b286463.txt">Table of n, a(n) for n = 1..10000</a>
%H A286463 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F A286463 a(n) = (1/2)*(2 + ((A051064(n)+A046523(n))^2) - A051064(n) - 3*A046523(n)).
%o A286463 (PARI)
%o A286463 A051064(n) = if(n<1, 0, 1+valuation(n, 3));
%o A286463 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 A286463 A286463(n) = (1/2)*(2 + ((A051064(n)+A046523(n))^2) - A051064(n) - 3*A046523(n));
%o A286463 for(n=1, 10000, write("b286463.txt", n, " ", A286463(n)));
%o A286463 (Scheme) (define (A286463 n) (* (/ 1 2) (+ (expt (+ (A051064 n) (A046523 n)) 2) (- (A051064 n)) (- (* 3 (A046523 n))) 2)))
%o A286463 (Python)
%o A286463 from sympy import factorint, divisors, divisor_count, mobius
%o A286463 def a051064(n): return -sum([mobius(3*d)*divisor_count(n/d) for d in divisors(n)])
%o A286463 def P(n):
%o A286463     f = factorint(n)
%o A286463     return sorted([f[i] for i in f])
%o A286463 def a046523(n):
%o A286463     x=1
%o A286463     while True:
%o A286463         if P(n) == P(x): return x
%o A286463         else: x+=1
%o A286463 def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
%o A286463 def a(n): return T(a051064(n), a046523(n)) # _Indranil Ghosh_, May 11 2017
%Y A286463 Cf. A000027, A046523, A051064, A286161, A286461, A286462, A286464.
%K A286463 nonn
%O A286463 1,2
%A A286463 _Antti Karttunen_, May 10 2017