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A280493 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). 5
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, 42, 52, 56, 44, 72, 32, 63, 68, 123, 56, 140, 38, 60, 88, 135, 42, 87, 72, 84, 112, 120, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This is roughly a GF(2)[X]-analog of A000203. A178908 gives another, maybe a more consistent analog.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..2048

Index entries for sequences operating on polynomials in ring GF(2)[X]

FORMULA

For all n >= 0, a(2^n) = A000203(2^n) = A178908(2^n) = A000225(1+n).

EXAMPLE

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.

PROG

(Scheme)

(define (A280493 n) (let loop ((k n) (s 0)) (if (zero? k) s (loop (- k 1) (+ s (if (= k (A091255bi n k)) k 0))))))

;; A091255bi implements the 2-argument GF(2)[X] GCD-function (A091255) which is used for checking that k is a divisor of n.

;; Another version:

(define (A280493 n) (add A280494 (+ 1 (A000217 (- n 1))) (A000217 n)))

(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (+ 1 i) (+ res (intfun i)))))))

CROSSREFS

Row sums of triangles A280494 and A280499.

Cf. A000225, A000203, A178908.

Cf. A014580 (gives the positions where a(n) = n+1).

Sequence in context: A187477 A111499 A077025 * A032729 A035270 A120451

Adjacent sequences:  A280490 A280491 A280492 * A280494 A280495 A280496

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 09 2017

STATUS

approved

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Last modified December 6 19:30 EST 2021. Contains 349567 sequences. (Running on oeis4.)