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).

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

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