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A286571 Compound filter (prime signature of n & n/gcd(n, sigma(n))): a(n) = P(A046523(n), A017666(n)), where P(n,k) is sequence A000027 used as a pairing function.

%I A286571 #8 May 26 2017 16:38:20
%S A286571 1,5,8,25,17,21,30,113,70,51,68,103,93,72,51,481,155,148,192,222,331,
%T A286571 126,278,324,382,159,569,78,437,591,498,1985,126,237,786,2521,705,282,
%U A286571 952,375,863,660,948,243,337,384,1130,1759,1330,1842,237,678,1433,520,1776,459,1897,567,1772,2076,1893,636,2713,8065,2421,810,2280,1002,384,2046
%N A286571 Compound filter (prime signature of n & n/gcd(n, sigma(n))): a(n) = P(A046523(n), A017666(n)), where P(n,k) is sequence A000027 used as a pairing function.
%H A286571 Antti Karttunen, <a href="/A286571/b286571.txt">Table of n, a(n) for n = 1..10000</a>
%F A286571 a(n) = (1/2)*(2 + ((A046523(n)+A017666(n))^2) - A046523(n) - 3*A017666(n)).
%o A286571 (PARI)
%o A286571 A017666(n) = (n/gcd(n, sigma(n)));
%o A286571 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 A286571 A286571(n) = (1/2)*(2 + ((A046523(n)+A017666(n))^2) - A046523(n) - 3*A017666(n));
%o A286571 (Scheme) (define (A286571 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A017666 n)) 2) (- (A046523 n)) (- (* 3 (A017666 n))) 2)))
%o A286571 (Python)
%o A286571 from sympy import factorint, gcd, divisor_sigma
%o A286571 def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
%o A286571 def P(n):
%o A286571     f = factorint(n)
%o A286571     return sorted([f[i] for i in f])
%o A286571 def a046523(n):
%o A286571     x=1
%o A286571     while True:
%o A286571         if P(n) == P(x): return x
%o A286571         else: x+=1
%o A286571 def a(n): return T(a046523(n), n/gcd(n, divisor_sigma(n))) # _Indranil Ghosh_, May 26 2017
%Y A286571 Cf. A000027, A017666, A046523, A286360, A286570.
%K A286571 nonn
%O A286571 1,2
%A A286571 _Antti Karttunen_, May 26 2017