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A286163 Compound filter: a(n) = T(A046523(n), A278222(n)), where T(n,k) is sequence A000027 used as a pairing function. 16

%I

%S 2,5,12,14,23,42,38,44,40,61,80,117,80,84,216,152,23,148,80,148,601,

%T 142,302,375,109,142,911,183,302,1020,530,560,61,61,142,856,467,142,

%U 412,430,467,1741,1832,265,2497,412,1178,1323,109,265,826,265,1832,1735,2932,489,412,412,2630,2835,1178,672,2787,2144,61,625,80,148,601,850,302,2998,467,601

%N Compound filter: a(n) = T(A046523(n), A278222(n)), where T(n,k) is sequence A000027 used as a pairing function.

%H Antti Karttunen, <a href="/A286163/b286163.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) = (1/2)*(2 + ((A046523(n)+A278222(n))^2) - A046523(n) - 3*A278222(n)).

%o (PARI)

%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 A286163(n) = (2 + ((A046523(n)+A278222(n))^2) - A046523(n) - 3*A278222(n))/2;

%o for(n=1, 10000, write("b286163.txt", n, " ", A286163(n)));

%o (Scheme) (define (A286163 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A278222 n)) 2) (- (A046523 n)) (- (* 3 (A278222 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 a278222(n): return a046523(a005940(n + 1))

%o def a(n): return T(a046523(n), a278222(n)) # _Indranil Ghosh_, May 05 2017

%Y Cf. A000027, A046523, A278222, A286160, A286161, A286162, A286164.

%K nonn

%O 1,1

%A _Antti Karttunen_, May 04 2017

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Last modified July 30 04:45 EDT 2021. Contains 346348 sequences. (Running on oeis4.)