login
Compound filter: a(n) = P(A278222(n), A278219(n)), where P(n,k) is sequence A000027 used as a pairing function.
4

%I #11 May 13 2017 17:23:38

%S 1,5,12,14,12,84,40,44,12,142,216,183,40,265,86,152,12,142,826,265,

%T 216,1860,607,489,40,832,607,1117,86,619,226,560,12,142,826,265,826,

%U 5080,2497,619,216,2956,4308,4155,607,8575,1105,1533,40,832,2497,2116,607,5731,4501,3475,86,1402,1105,3475,226,1759,698,2144,12,142,826,265,826,5080,2497,619,826

%N Compound filter: a(n) = P(A278222(n), A278219(n)), where P(n,k) is sequence A000027 used as a pairing function.

%H Antti Karttunen, <a href="/A286242/b286242.txt">Table of n, a(n) for n = 0..16383</a>

%H MathWorld, <a href="http://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>

%F a(n) = (1/2)*(2 + ((A278222(n)+A278219(n))^2) - A278222(n) - 3*A278219(n)).

%F a(n) = (1/2)*(2 + ((A278222(n)+A278222(A003188(n)))^2) - A278222(n) - 3*A278222(A003188(n))).

%o (PARI)

%o A003188(n) = bitxor(n, n>>1);

%o 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 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 A278222(n) = A046523(A005940(1+n));

%o A278219(n) = A278222(A003188(n));

%o A286242(n) = (2 + ((A278222(n)+A278219(n))^2) - A278222(n) - 3*A278219(n))/2;

%o for(n=0, 16383, write("b286242.txt", n, " ", A286242(n)));

%o (Scheme) (define (A286242 n) (* (/ 1 2) (+ (expt (+ (A278222 n) (A278219 n)) 2) (- (A278222 n)) (- (* 3 (A278219 n))) 2)))

%o (Python)

%o from sympy import prime, factorint

%o import math

%o def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2

%o def A(n): return n - 2**int(math.floor(math.log(n, 2)))

%o 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 def a005940(n): return b(n - 1)

%o def P(n):

%o f=factorint(n)

%o return sorted([f[i] for i in f])

%o def a046523(n):

%o x=1

%o while True:

%o if P(n) == P(x): return x

%o else: x+=1

%o def a003188(n): return n^(n>>1)

%o def a243353(n): return a005940(1 + a003188(n))

%o def a278219(n): return a046523(a243353(n))

%o def a278222(n): return a046523(a005940(n + 1))

%o def a(n): return T(a278222(n), a278219(n)) # _Indranil Ghosh_, May 07 2017

%Y Cf. A000027, A003188, A278219, A278222, A286240, A286241, A286242.

%K nonn

%O 0,2

%A _Antti Karttunen_, May 07 2017