A178909 -id:A178909 - cp's OEIS frontend

A178908 GF(2) sum of divisors of n.

1, 3, 2, 7, 7, 6, 6, 15, 12, 9, 10, 14, 12, 10, 8, 31, 25, 20, 18, 21, 19, 30, 24, 30, 24, 20, 18, 18, 20, 24, 30, 63, 60, 43, 40, 36, 36, 54, 54, 45, 40, 53, 48, 54, 48, 40, 46, 62, 60, 40, 42, 36, 36, 54, 54, 34, 36, 60, 58, 56, 60, 34, 38, 127, 121, 68, 66, 79, 79, 120, 120

Offset: 1

Franklin T. Adams-Watters, Jun 22 2010

nonn

Take the n-th GF(2) polynomial, compute its sum of divisors, and find the index of that polynomial in the list of GF(2) polynomials.
If 2^k <= n < 2^(k+1), then also 2^k <= a(n) < 2^(k+1), since any proper divisor of a GF(2) polynomial has lower degree.
Numbers whose binary representations correspond to the divisors occur as the nonzero terms on row n of A280499, and they are XORed together to obtain a(n). A280493 gives another GF(2)[X]-analog of A000203. - Antti Karttunen, Jan 11 2017

			5 => x^2 + 1 = (x+1)^2. sigma((x+1)^2) = (x+1)^2 + x+1 + 1 = x^2 + x + 1 => 7, so a(5) = 7. (All polynomials here are over GF(2).)

Antti Karttunen, Table of n, a(n) for n = 1..2047
Index entries for sequences operating on polynomials in ring GF(2)[X]

Cf. A000203, A000225, A003987, A091220, A091255, A178909, A178910, A280493, A280499.

a(n)={local(p,fm,r,k);
while(n>0,p+=Mod(n,2)*x^k;n\=2;k++);
r=Mod(1,2);fm=factor(p);for(k=1,matsize(fm)[1],r*=(fm[k,1]^(fm[k,2]+1)-1)/(fm[k,1]-1));
subst(lift(r),x,2)}

a(n) = {my(s = vecsum(divisors(Mod(1,2)*Pol(binary(n))))); subst(lift(s), x, 2);} \\ Michel Marcus, Jan 13 2019

;; A003987bi implements the 2-argument bitwise-XOR function (A003987).
;; A091255bi implements the 2-argument GF(2)[X] GCD-function (A091255) which is used for checking that k is a divisor of n.
(define (A178908 n) (let loop ((k n) (s 0)) (if (zero? k) s (loop (- k 1) (A003987bi s (if (= k (A091255bi n k)) k 0))))))
;; Antti Karttunen, Jan 11 2017

For all n >= 0, a(2^n) = A000203(2^n) = A280493(2^n) = A000225(1+n). - Antti Karttunen, Jan 11 2017

A178911 Perfex numbers: n = binary XOR of divisors of n.

Original entry on oeis.org

1, 6, 120, 198, 3696, 6240, 32640, 56160, 1941408, 3592200, 8119800, 15628032, 27125280, 59032080, 61788240, 125859840, 1635834720, 2147450880, 3709081680, 16328199552, 26198072160, 52344970080, 52396088160, 209584184160, 210197601120, 236223190200, 237385437360

Offset: 1

Franklin T. Adams-Watters, Jun 22 2010

a(17) > 1e9.
10^11 < a(24) <= 209584184160. a(25) <= 210197601120. - Donovan Johnson, Mar 12 2011
a(28) > 3*10^11. - Giovanni Resta, Aug 14 2019

Cf. A178910, A178909, A000396, A003987.

lst = {}; k = 1; While[k < 10^9, If[ BitXor @@ Divisors@k == k, AppendTo[lst, k]; Print@k]; k++ ]; lst (* Robert G. Wilson v, Jun 27 2010 *)

xigma(n)=local(ds,r);ds=divisors(n);for(k=1,#ds,r=bitxor(r,ds[k]));r
for(n=1,1000000000,if(xigma(n)==n,print1(n",")))

a(17) from Robert G. Wilson v, Jul 30 2010
a(18)-a(23) from Donovan Johnson, Mar 12 2011
a(24)-a(27) from Giovanni Resta, Aug 14 2019

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A178908 GF(2) sum of divisors of n.

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A178911 Perfex numbers: n = binary XOR of divisors of n.

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