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

Author

Antti Karttunen, Jan 09 2017

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..2048
• Index entries for sequences operating on polynomials in ring GF(2)[X]

Crossrefs

Row sums of triangles A280494 and A280499.

Cf. A000225, A000203, A178908.

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

Programs

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)))))))

Formula

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