A286451 - OEIS

%I #8 May 14 2017 15:54:45

%S 0,2,6,4,3,9,10,7,1,5,6,13,3,14,10,11,3,2,6,8,21,9,10,18,1,5,10,19,3,

%T 14,21,16,15,5,15,4,3,9,10,12,3,27,6,13,3,14,15,24,1,2,10,8,3,14,10,

%U 25,15,5,6,19,3,27,10,22,6,20,6,8,21,20,10,7,3,5,6,13,21,14,15,17,1,5,6,34,6,9,10,18,3,5,15,19,36,20,10,31,3,2,6,4,3,14,10

%N Compound filter (2-adic valuation of sigma(n) & 2-adic valuation of n): a(n) = P(A286357(n), A001511(n)), where P(n,k) is sequence A000027 used as a pairing function, with a(1) = 0 by an explicit convention.

%H Antti Karttunen, <a href="/A286451/b286451.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(1) = 0; for n > 1, a(n) = (1/2)*(2 + ((A286357(n)+A001511(n))^2) - A286357(n) - 3*A001511(n)).

%o (PARI)

%o A001511(n) = (1+valuation(n,2));

%o A286357(n) = A001511(sigma(n));

%o A286451(n) = if(1==n,0,(1/2)*(2 + ((A286357(n)+A001511(n))^2) - A286357(n) - 3*A001511(n)));

%o for(n=1, 10000, write("b286451.txt", n, " ", A286451(n)));

%o (Scheme) (define (A286451 n) (if (= 1 n) 0 (* (/ 1 2) (+ (expt (+ (A286357 n) (A001511 n)) 2) (- (A286357 n)) (- (* 3 (A001511 n))) 2))))

%o (Python)

%o from sympy import divisor_sigma as D

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

%o def a001511(n): return bin(n)[2:][::-1].index("1") + 1

%o def a(n): return 0 if n==1 else T(a001511(D(n)), a001511(n)) # _Indranil Ghosh_, May 14 2017

%Y Cf. A000027, A001511, A286357, A286260, A286358.

%K nonn

%O 1,2

%A _Antti Karttunen_, May 13 2017