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A286591 Compound filter: a(n) = P(A009191(n), A009194(n)), where P(n,k) is sequence A000027 used as a pairing function.

%I A286591 #9 Feb 16 2025 08:33:45
%S A286591 1,3,1,1,1,23,1,10,6,5,1,42,1,5,4,1,1,34,1,5,1,5,1,179,1,5,1,408,1,23,
%T A286591 1,3,4,5,1,45,1,5,1,144,1,23,1,12,13,5,1,12,1,3,4,5,1,23,1,113,1,5,1,
%U A286591 265,1,5,6,1,1,23,1,5,4,5,1,103,1,5,6,12,1,23,1,65,1,5,1,753,1,5,4,63,1,259,22,12,1,5,11,265,1,3,13,1,1,23,1,44,4,5,1
%N A286591 Compound filter: a(n) = P(A009191(n), A009194(n)), where P(n,k) is sequence A000027 used as a pairing function.
%H A286591 Antti Karttunen, <a href="/A286591/b286591.txt">Table of n, a(n) for n = 1..10000</a>
%H A286591 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F A286591 a(n) = (1/2)*(2 + ((A009191(n)+A009194(n))^2) - A009191(n) - 3*A009194(n)).
%o A286591 (PARI)
%o A286591 A009191(n) = gcd(n, numdiv(n));
%o A286591 A009194(n) = gcd(n, sigma(n));
%o A286591 A286591(n) = (1/2)*(2 + ((A009191(n)+A009194(n))^2) - A009191(n) - 3*A009194(n));
%o A286591 (Scheme) (define (A286591 n) (* (/ 1 2) (+ (expt (+ (A009191 n) (A009194 n)) 2) (- (A009191 n)) (- (* 3 (A009194 n))) 2)))
%o A286591 (Python)
%o A286591 from sympy import divisor_sigma, divisor_count, gcd
%o A286591 def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
%o A286591 def a(n): return T(gcd(n, divisor_count(n)), gcd(n, divisor_sigma(n))) # _Indranil Ghosh_, May 22 2017
%Y A286591 Cf. A000027, A009191, A009194, A286594.
%K A286591 nonn
%O A286591 1,2
%A A286591 _Antti Karttunen_, May 21 2017