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A286452 Compound filter (largest prime factor of n & prime signature of 2n-1): a(n) = P(A061395(n), A046523(2n-1)), where P(n,k) is sequence A000027 used as a pairing function.

%I A286452 #9 Feb 16 2025 08:33:45
%S A286452 0,2,5,2,18,5,14,16,5,9,50,5,42,59,9,2,73,23,44,31,14,20,199,5,18,61,
%T A286452 5,40,115,9,77,67,50,35,40,5,90,179,61,9,391,14,185,50,9,100,205,23,
%U A286452 14,94,35,27,1006,5,20,40,44,115,395,31,228,131,59,2,61,20,295,442,54,14,320,23,346,265,9,44,125,61,275,31,23,104,1349,14,52,314,65,125,430
%N A286452 Compound filter (largest prime factor of n & prime signature of 2n-1): a(n) = P(A061395(n), A046523(2n-1)), where P(n,k) is sequence A000027 used as a pairing function.
%H A286452 Antti Karttunen, <a href="/A286452/b286452.txt">Table of n, a(n) for n = 1..10000</a>
%H A286452 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F A286452 a(n) = (1/2)*(2 + ((A061395(n)+A046523(2n-1))^2) - A061395(n) - 3*A046523(2n-1)).
%o A286452 (Scheme) (define (A286452 n) (* (/ 1 2) (+ (expt (+ (A061395 n) (A046523 (+ n n -1))) 2) (- (A061395 n)) (- (* 3 (A046523 (+ n n -1)))) 2)))
%o A286452 (Python)
%o A286452 from sympy import primepi, primefactors, factorint
%o A286452 def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
%o A286452 def P(n):
%o A286452     f = factorint(n)
%o A286452     return sorted([f[i] for i in f])
%o A286452 def a046523(n):
%o A286452     x=1
%o A286452     while True:
%o A286452         if P(n) == P(x): return x
%o A286452         else: x+=1
%o A286452 def a061395(n): return 0 if n==1 else primepi(primefactors(n)[-1])
%o A286452 def a(n): return T(a061395(n), a046523(2*n - 1)) # _Indranil Ghosh_, May 14 2017
%Y A286452 Cf. A000027, A046523, A061395, A278223, A285334, A286356, A286372, A286453.
%K A286452 nonn
%O A286452 1,2
%A A286452 _Antti Karttunen_, May 14 2017