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%I #22 Jan 10 2017 19:12:49
%S 1,3,4,7,9,12,8,15,20,27,12,28,14,24,24,31,41,60,20,63,29,36,40,60,26,
%T 42,52,56,44,72,32,63,68,123,56,140,38,60,88,135,42,87,72,84,112,120,
%U 48,124,68,78,92,98,76,156,56,120,102,132,60,168,62,96,104,127,201,204,68,287,81,168,136,300,74,114,192,140,140,264,112,279,95,126,192,203
%N Sum of GF(2)[X] divisors of n: the sum is ordinary sum of integers, the summands being all the natural numbers whose binary expansions encode such (0,1)-polynomials that divide the (0,1)-polynomial encoded by n when the polynomial factorization is done over the field GF(2).
%C This is roughly a GF(2)[X]-analog of A000203. A178908 gives another, maybe a more consistent analog.
%H Antti Karttunen, <a href="/A280493/b280493.txt">Table of n, a(n) for n = 1..2048</a>
%H <a href="/index/Ge#GF2X">Index entries for sequences operating on polynomials in ring GF(2)[X]</a>
%F For all n >= 0, a(2^n) = A000203(2^n) = A178908(2^n) = A000225(1+n).
%e 9 ("1001" in binary) encodes polynomial X^3 + 1, which is factored over GF(2) as (X+1)(X^2 + X + 1), where polynomial X + 1 is encoded by 3 ("11" in binary), and polynomial X^2 + X + 1 by 7 ("111" in binary), and furthermore (like all polynomials) it is also divisible by 1 and itself, thus a(9) = 1 + 3 + 7 + 9 = 20.
%o (Scheme)
%o (define (A280493 n) (let loop ((k n) (s 0)) (if (zero? k) s (loop (- k 1) (+ s (if (= k (A091255bi n k)) k 0))))))
%o ;; A091255bi implements the 2-argument GF(2)[X] GCD-function (A091255) which is used for checking that k is a divisor of n.
%o ;; Another version:
%o (define (A280493 n) (add A280494 (+ 1 (A000217 (- n 1))) (A000217 n)))
%o (define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (+ 1 i) (+ res (intfun i)))))))
%Y Row sums of triangles A280494 and A280499.
%Y Cf. A000225, A000203, A178908.
%Y Cf. A014580 (gives the positions where a(n) = n+1).
%K nonn
%O 1,2
%A _Antti Karttunen_, Jan 09 2017
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