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A285729 Compound filter: a(n) = T(A032742(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

%I A285729 #16 Feb 16 2025 08:33:44
%S A285729 1,2,2,12,2,31,2,59,18,50,2,142,2,73,50,261,2,199,2,220,73,131,2,607,
%T A285729 33,166,129,314,2,961,2,1097,131,248,73,1396,2,295,166,923,2,1246,2,
%U A285729 550,340,401,2,2509,52,655,248,692,2,1252,131,1303,295,590,2,3946,2,661,517,4497,166,1924,2,1024,401,2051,2,5707,2,898,655,1214,131,2317,2,3781,888
%N A285729 Compound filter: a(n) = T(A032742(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.
%H A285729 Antti Karttunen, <a href="/A285729/b285729.txt">Table of n, a(n) for n = 1..10000</a>
%H A285729 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F A285729 a(n) = (1/2)*(2 + ((A032742(n)+A046523(n))^2) - A032742(n) - 3*A046523(n)).
%t A285729 Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {Sort[Flatten@ Apply[ TensorProduct, # /. {p_, e_} /; p > 1 :> p^Range[0, e]]][[-2]], Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[#[[All, -1]], Greater]] - Boole[n == 1]} &@ FactorInteger@ n, {n, 81}] (* _Michael De Vlieger_, May 04 2017 *)
%o A285729 (PARI)
%o A285729 A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
%o A285729 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 A285729 A285729(n) = (1/2)*(2 + ((A032742(n)+A046523(n))^2) - A032742(n) - 3*A046523(n));
%o A285729 for(n=1, 10000, write("b285729.txt", n, " ", A285729(n)));
%o A285729 (Scheme) (define (A285729 n) (* (/ 1 2) (+ (expt (+ (A032742 n) (A046523 n)) 2) (- (A032742 n)) (- (* 3 (A046523 n))) 2)))
%o A285729 (Python)
%o A285729 from sympy import divisors, factorint
%o A285729 def a032742(n): return 1 if n==1 else max(divisors(n)[:-1])
%o A285729 def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
%o A285729 def P(n):
%o A285729    f = factorint(n)
%o A285729    return sorted([f[i] for i in f])
%o A285729 def a046523(n):
%o A285729     x=1
%o A285729     while True:
%o A285729         if P(n) == P(x): return x
%o A285729         else: x+=1
%o A285729 def a(n): return T(a032742(n), a046523(n)) # _Indranil Ghosh_, May 05 2017
%Y A285729 Cf. A000027, A032742, A046523, A286142, A286143, A286144, A286149, A286152, A286154, A286160, A286161, A286162, A286163, A286164.
%K A285729 nonn
%O A285729 1,2
%A A285729 _Antti Karttunen_, May 04 2017