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A286460 Compound filter (2-adic valuation & sum of the divisors): a(n) = P(A001511(n), A000203(n)), where P(n,k) is sequence A000027 used as a pairing function.

%I A286460 #18 Feb 16 2025 08:33:45
%S A286460 1,8,7,39,16,80,29,157,79,173,67,438,92,302,277,600,154,782,191,949,
%T A286460 497,668,277,1957,466,905,781,1656,436,2630,497,2284,1129,1487,1129,
%U A286460 4281,704,1832,1541,4282,862,4658,947,3658,3004,2630,1129,8133,1597,4373,2557,4953,1432,7262,2557,7507,3161,4097,1771,14368,1892,4658,5357,8785,3487,10442,2279
%N A286460 Compound filter (2-adic valuation & sum of the divisors): a(n) = P(A001511(n), A000203(n)), where P(n,k) is sequence A000027 used as a pairing function.
%H A286460 Antti Karttunen, <a href="/A286460/b286460.txt">Table of n, a(n) for n = 1..10000</a>
%H A286460 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%H A286460 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A286460 a(n) = (1/2)*(2 + ((A001511(n)+A000203(n))^2) - A001511(n) - 3*A000203(n)).
%o A286460 (PARI)
%o A286460 A000203(n) = sigma(n);
%o A286460 A001511(n) = (1+valuation(n,2));
%o A286460 A286460(n) = (1/2)*(2 + ((A001511(n)+A000203(n))^2) - A001511(n) - 3*A000203(n));
%o A286460 for(n=1, 10000, write("b286460.txt", n, " ", A286460(n)));
%o A286460 (Scheme) (define (A286460 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A000203 n)) 2) (- (A001511 n)) (- (* 3 (A000203 n))) 2)))
%o A286460 (Python)
%o A286460 from sympy import divisor_sigma as D
%o A286460 def a001511(n): return bin(n)[2:][::-1].index("1") + 1
%o A286460 def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
%o A286460 def a(n): return T(a001511(n), D(n))
%o A286460 print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, May 12 2017
%Y A286460 Cf. A000027, A286360.
%Y A286460 Cf. A000593, A146076 (sequences matching to this filter), also A000203, A161942, A286260, A286357.
%K A286460 nonn
%O A286460 1,2
%A A286460 _Antti Karttunen_, May 10 2017