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A286241 Compound filter: a(n) = P(A278219(n), A278219(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

%I A286241 #19 Feb 16 2025 08:33:44
%S A286241 2,12,14,12,59,86,27,12,109,363,269,86,142,148,27,12,109,1093,1117,
%T A286241 363,1097,1517,489,86,601,1408,619,148,142,148,27,12,109,1093,1117,
%U A286241 1093,5707,8587,2545,363,1969,6153,4529,1517,4489,4537,489,86,601,3946,3976,1408,2509,5719,2545,148,601,1408,619,148,142,148,27,12,109,1093,1117,1093,5707,8587
%N A286241 Compound filter: a(n) = P(A278219(n), A278219(1+n)), where P(n,k) is sequence A000027 used as a pairing function.
%H A286241 Antti Karttunen, <a href="/A286241/b286241.txt">Table of n, a(n) for n = 0..16383</a>
%H A286241 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F A286241 a(n) = (1/2)*(2+((A278219(n)+A278219(1+n))^2) - A278219(n) - 3*A278219(1+n)).
%t A286241 f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; g[n_] := If[n == 1, 1, Times @@ MapIndexed[ Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]]; h[n_] := g@ f[BitXor[n, Floor[n/2]], 1, 1]; Map[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ # & /@ # &, Table[{h[n], h[n + 1]}, {k, 12}, {n, k (k - 1)/2, k (k + 1)/2 - 1}]] // Flatten (* _Michael De Vlieger_, May 09 2017 *)
%o A286241 (PARI)
%o A286241 A003188(n) = bitxor(n, n>>1);
%o A286241 A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of _M. F. Hasler_
%o A286241 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 A286241 A278222(n) = A046523(A005940(1+n));
%o A286241 A278219(n) = A278222(A003188(n));
%o A286241 A286241(n) = (2 + ((A278219(n)+A278219(1+n))^2) - A278219(n) - 3*A278219(1+n))/2;
%o A286241 for(n=0, 16383, write("b286241.txt", n, " ", A286241(n)));
%o A286241 (Scheme) (define (A286241 n) (* (/ 1 2) (+ (expt (+ (A278219 n) (A278219 (+ 1 n))) 2) (- (A278219 n)) (- (* 3 (A278219 (+ 1 n)))) 2)))
%o A286241 (Python)
%o A286241 from sympy import prime, factorint
%o A286241 import math
%o A286241 def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
%o A286241 def A(n): return n - 2**int(math.floor(math.log(n, 2)))
%o A286241 def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
%o A286241 def a005940(n): return b(n - 1)
%o A286241 def P(n):
%o A286241     f = factorint(n)
%o A286241     return sorted([f[i] for i in f])
%o A286241 def a046523(n):
%o A286241     x=1
%o A286241     while True:
%o A286241         if P(n) == P(x): return x
%o A286241         else: x+=1
%o A286241 def a003188(n): return n^int(n/2)
%o A286241 def a243353(n): return a005940(1 + a003188(n))
%o A286241 def a278219(n): return a046523(a243353(n))
%o A286241 def a(n): return T(a278219(n), a278219(n + 1)) # _Indranil Ghosh_, May 07 2017
%Y A286241 Cf. A000027, A003188, A005940, A046523, A243353, A278219, A278222, A286240, A286242, A286255.
%K A286241 nonn
%O A286241 0,1
%A A286241 _Antti Karttunen_, May 07 2017