A167221 a(n) is the smallest positive number B that yields a solution for k = A167219(n).

3, 5, 3, 10, 21, 9, 17, 44, 91, 7, 70, 5, 186, 71, 3, 377, 97, 285, 760, 194, 323, 1527, 574, 1148, 3062, 25, 6133, 4603, 12276, 4605, 2499, 2187, 5182, 24563, 18426, 7775, 49138, 12440, 9997, 98289, 36860, 73721, 196592, 82941, 393199, 294904, 786414, 49, 294907

Offset: 1

Author

Ctibor O. Zizka, Oct 30 2009

Keywords

• nonn

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.

Examples

			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?
		

Crossrefs

Cf. A054841, A054842, A167219.

Programs

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

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