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Compound filter: a(n) = P(A278222(n), A278222(1+n)), where P(n,k) is sequence A000027 used as a pairing function.
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%I #16 May 13 2017 05:18:22

%S 2,5,12,14,23,42,59,44,23,61,142,117,109,183,261,152,23,61,142,148,

%T 601,850,607,375,109,265,1093,939,473,765,1097,560,23,61,142,148,601,

%U 850,607,430,601,1741,3946,2545,2497,3463,2509,1323,109,265,1093,1117,2497,4525,5707,3153,473,1105,4489,3813,1969,3129,4497,2144,23,61,142,148,601,850,607,430,601,1741

%N Compound filter: a(n) = P(A278222(n), A278222(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

%H Antti Karttunen, <a href="/A286240/b286240.txt">Table of n, a(n) for n = 0..16383</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>

%F a(n) = (1/2)*(2 + ((A278222(n)+A278222(1+n))^2) - A278222(n) - 3*A278222(1+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 A286240(n) = (2 + ((A278222(n)+A278222(1+n))^2) - A278222(n) - 3*A278222(1+n))/2;

%o for(n=0, 16383, write("b286240.txt", n, " ", A286240(n)));

%o (Scheme) (define (A286240 n) (* (/ 1 2) (+ (expt (+ (A278222 n) (A278222 (+ 1 n))) 2) (- (A278222 n)) (- (* 3 (A278222 (+ 1 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(a278222(n), a278222(n + 1)) # _Indranil Ghosh_, May 07 2017

%Y Cf. A000027, A278222, A286241, A286242, A286255.

%Y Cf. A088705 (one of the matches not matched by A278222 alone. Thus also the whole A007814 (A001511) family is included).

%K nonn

%O 0,1

%A _Antti Karttunen_, May 07 2017