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%I #24 Aug 02 2022 12:37:27
%S 1905,424321,438065,443617,7044945,7899377,7925761,26397649,32286449,
%T 38123521,55759233
%N Numbers k such that k = ivgenpoly(A) for some composite polynomial A in F_2[x] that satisfies the condition sigma(A) = A + 1.
%C Let A be a polynomial in F_2[x]. We let lift(A) in Z[x] denote the same polynomial, but with integer coefficients 0,1.
%C Let ivgenpoly(A) be the positive integer equal to the lift(A) evaluated in x=2. For example, if A = x^2+x+1 in F_2[x], we have lift(A) = x^2+x+1 in Z[x], and ivgenpoly(A) = 2^2+2+1 = 7. Similarly, for every positive integer n, we let genpoly(n) denote the unique polynomial A in F_2[x] such that n = ivgenpoly(A). The coefficients of A, are the digits of the base-2 expansion of n.
%C Over the integers, it is easy to check that sigma(p)=p+1 implies that p is a prime number, where sigma(n) is the sum of all positive divisors of the positive integer n. However, in F_2[x] the analogous result is false.
%C We denote by sigma(A) the sum of all divisors of A. The sequence shows integers k = ivgenpoly(A) such that A is a composite polynomial in F_2[x] for which sigma(A)=A+1.
%e a(1) = 1905, since 1905 = ivgenpoly(A), with A = x^10+x^9+x^8+x^6+x^5+x^4+1, satisfies A = (x^3+x+1)*(x^3+x^2+1)*(x^4+x+1) so that sigma(A) = (x^3+x)*(x^3+x^2)*(x^4+x) = A+1, and for any number m with 0 < m < 1905, with m = ivgenpoly(B), one has that either sigma(B) is unequal to B+1 or B is irreducible.
%e Moreover, a(2) = 424321, since 424321 = ivgenpoly(A), with A = x^18+x^17+x^14+x^13+x^12+x^11+x^8+x^7+1, satisfies A = (x^4+x^3+1)*(x^4+x^3+x^2+x+1)*(x^5+x^2+1)*(x^5+x^4+x^2+1) so that sigma(A) = A+1, and for any number m with 1905 < m < 424321, with m = ivgenpoly(B), one has that either sigma(B) is unequal to B+1 or B is irreducible.
%Y Cf. A000203.
%K nonn,more
%O 1,1
%A _Luis H. Gallardo_, Jul 28 2022
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