OFFSET
1,1
COMMENTS
B is the base in which we can express k as Sum_{i=0..m} B^i * a_i. There is an isomorphism between (Z[B],+) and the positive rationals as the polynomials with integer coefficients considered as a group under addition are isomorphic to the positive rationals considered as a group under multiplication.
EXAMPLE
For k = 21 = 2^0 * 3^1 * 5^0 * 7^1, k = B^0 * 0 + B^1 * 1 + B^2 * 0 + B^3 * 1, so we have to solve the equation 21 = B + B^3 for an integer B. No such B exists.
For k = 10 = 2^1 * 3^0 * 5^1, k = B^0 * 1 + B^1 * 0 + B^2 * 1, so we have to solve the equation 10 = 1 + B^2 for an integer B. B = +-3.
For k = 12 = 2^2 * 3^1, k = B^0 * 2 + B^1 * 1, so we have to solve the equation 12 = 2 + B for an integer B. B = 10. Are there any numbers other than k=12 for which B = 10 yields a solution?
PROG
(PARI) lista(nn) = for (k=2, nn, my(f=factor(k), v=primes(primepi(vecmax(f[, 1])))); my(p=sum(i=1, #v, 'x^(i-1)*valuation(k, v[i]))); p -= k; my(c=-polcoef(p, 0)); my(q=(p+c)/x); my(d=divisors(c)); for (k=1, #d, if(subst(q, x, d[k]) == c/d[k], print1(d[k], ", ")); ); ); \\ Michel Marcus, Aug 09 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Ctibor O. Zizka, Oct 30 2009
EXTENSIONS
Edited by Jon E. Schoenfield, Mar 16 2022
Corrected and extended by Michel Marcus, Aug 09 2022
a(41) and beyond from Michael S. Branicky, Aug 10 2022
STATUS
approved