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A286364 Compound filter: a(n) = P(A286361(n), A286363(n)), where P(n,k) is sequence A000027 used as a pairing function.

%I A286364 #16 Feb 16 2025 08:33:45
%S A286364 1,1,2,1,3,2,2,1,7,3,2,2,3,2,5,1,3,7,2,3,16,2,2,2,10,3,29,2,3,5,2,1,
%T A286364 16,3,5,7,3,2,5,3,3,16,2,2,12,2,2,2,7,10,5,3,3,29,5,2,16,3,2,5,3,2,67,
%U A286364 1,21,16,2,3,16,5,2,7,3,3,14,2,16,5,2,3,121,3,2,16,21,2,5,2,3,12,5,2,16,2,5,2,3,7,67,10,3,5,2,3,23,3,2,29,3,5,5,2,3
%N A286364 Compound filter: a(n) = P(A286361(n), A286363(n)), where P(n,k) is sequence A000027 used as a pairing function.
%C A286364 This sequence packs the values of A286361(n) and A286363(n) to a single value with the pairing function A000027. These two components essentially give the prime signature of 4k+1 part and the prime signature of 4k+3 part, and they can be accessed from a(n) with functions A002260 and A004736. For example, A004431 lists all such numbers that the first component is larger than one and the second component is a perfect square.
%H A286364 Antti Karttunen, <a href="/A286364/b286364.txt">Table of n, a(n) for n = 1..10000</a>
%H A286364 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F A286364 a(n) = (1/2)*(2+((A286361(n)+A286363(n))^2) - A286361(n) - 3*A286363(n)).
%F A286364 Other identities. For all n >= 1:
%F A286364 a(A267099(n)) = A038722(a(n)).
%o A286364 (Scheme) (define (A286364 n) (* (/ 1 2) (+ (expt (+ (A286361 n) (A286363 n)) 2) (- (A286361 n)) (- (* 3 (A286363 n))) 2)))
%o A286364 (Python)
%o A286364 from sympy import factorint
%o A286364 from operator import mul
%o A286364 def P(n):
%o A286364     f = factorint(n)
%o A286364     return sorted([f[i] for i in f])
%o A286364 def a046523(n):
%o A286364     x=1
%o A286364     while True:
%o A286364         if P(n) == P(x): return x
%o A286364         else: x+=1
%o A286364 def A(n, k):
%o A286364     f = factorint(n)
%o A286364     return 1 if n == 1 else reduce(mul, [1 if i%4==k else i**f[i] for i in f])
%o A286364 def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
%o A286364 def a(n): return T(a046523(n/A(n, 1)), a046523(n/A(n, 3))) # _Indranil Ghosh_, May 09 2017
%Y A286364 Cf. A000027, A002260, A004431, A004736, A038722, A267099, A286361, A286363, A286365.
%K A286364 nonn
%O A286364 1,3
%A A286364 _Antti Karttunen_, May 08 2017