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%I #8 May 26 2017 16:38:13
%S 1,3,3,10,3,61,3,36,10,27,3,117,3,27,34,136,3,103,3,90,21,27,3,619,10,
%T 27,36,753,3,625,3,528,34,27,21,666,3,27,21,552,3,625,3,117,103,27,3,
%U 1323,10,78,34,90,3,430,21,489,21,27,3,2545,3,27,78,2080,21,625,3,90,34,495,3,2773,3,27,78,117,21,625,3,1224,136,27,3,3801,21,27,34,375,3
%N Compound filter (prime signature of n & gcd(n, sigma(n))): a(n) = P(A046523(n), A009194(n)), where P(n,k) is sequence A000027 used as a pairing function.
%H Antti Karttunen, <a href="/A286570/b286570.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = (1/2)*(2 + ((A046523(n)+A009194(n))^2) - A046523(n) - 3*A009194(n)).
%o (PARI)
%o A009194(n) = gcd(n, sigma(n));
%o 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 A286570(n) = (1/2)*(2 + ((A046523(n)+A009194(n))^2) - A046523(n) - 3*A009194(n));
%o (Scheme) (define (A286570 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A009194 n)) 2) (- (A046523 n)) (- (* 3 (A009194 n))) 2)))
%o (Python)
%o from sympy import factorint, gcd, divisor_sigma
%o def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
%o def P(n):
%o f = factorint(n)
%o return sorted([f[i] for i in f])
%o def a046523(n):
%o x=1
%o while True:
%o if P(n) == P(x): return x
%o else: x+=1
%o def a(n): return T(a046523(n), gcd(n, divisor_sigma(n))) # _Indranil Ghosh_, May 26 2017
%Y Cf. A000027, A009194, A046523, A286360, A286571, A286591.
%K nonn
%O 1,2
%A _Antti Karttunen_, May 26 2017
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