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A286143 Compound filter: a(n) = T(A055881(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

%I A286143 #16 Feb 16 2025 08:33:44
%S A286143 1,5,2,12,2,31,2,38,7,23,2,94,2,23,16,138,2,94,2,80,16,23,2,355,7,23,
%T A286143 29,80,2,499,2,530,16,23,16,706,2,23,16,302,2,499,2,80,67,23,2,1279,7,
%U A286143 80,16,80,2,328,16,302,16,23,2,1894,2,23,67,2082,16,499,2,80,16,467,2,2779,2,23,67,80,16,499,2,1178,121,23,2,1894,16,23,16,302,2,1894,16
%N A286143 Compound filter: a(n) = T(A055881(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.
%H A286143 Antti Karttunen, <a href="/A286143/b286143.txt">Table of n, a(n) for n = 1..10000</a>
%H A286143 MathWorld, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F A286143 a(n) = (1/2)*(2 + ((A055881(n)+A046523(n))^2) - A055881(n) - 3*A046523(n)).
%t A286143 Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 - Boole[n == 1] & @@ {Module[{m = 1}, While[Mod[n, m!] == 0, m++]; m - 1], Times @@ MapIndexed[ Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]}, {n, 92}] (* _Michael De Vlieger_, May 04 2017, after _Robert G. Wilson v_ at A055881 *)
%o A286143 (PARI)
%o A286143 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 A286143 A055881(n) = { my(i); i=2; while((0 == (n%i)), n = n/i; i++); return(i-1); }
%o A286143 A286143(n) = (1/2)*(2 + ((A055881(n)+A046523(n))^2) - A055881(n) - 3*A046523(n));
%o A286143 for(n=1, 10000, write("b286143.txt", n, " ", A286143(n)));
%o A286143 (Scheme) (define (A286143 n) (* (/ 1 2) (+ (expt (+ (A055881 n) (A046523 n)) 2) (- (A055881 n)) (- (* 3 (A046523 n))) 2)))
%o A286143 (Python)
%o A286143 from sympy import factorial, factorint
%o A286143 def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
%o A286143 def P(n):
%o A286143     f = factorint(n)
%o A286143     return sorted([f[i] for i in f])
%o A286143 def a046523(n):
%o A286143     x=1
%o A286143     while True:
%o A286143         if P(n) == P(x): return x
%o A286143         else: x+=1
%o A286143 def a055881(n):
%o A286143     m = 1
%o A286143     while n%factorial(m)==0:
%o A286143         m+=1
%o A286143     return m - 1
%o A286143 def a(n): return T(a055881(n), a046523(n)) # _Indranil Ghosh_, May 05 2017
%Y A286143 Cf. A000027, A046523, A055881, A286144, A286160, A286161, A286162, A286163, A286164.
%Y A286143 Differs from A286142 for the first time at n=24, where a(24) = 355, while A286142(24) = 328.
%K A286143 nonn
%O A286143 1,2
%A A286143 _Antti Karttunen_, May 04 2017