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%I #28 Aug 11 2022 03:26:59
%S 3,5,3,10,21,9,17,44,91,7,70,5,186,71,3,377,97,285,760,194,323,1527,
%T 574,1148,3062,25,6133,4603,12276,4605,2499,2187,5182,24563,18426,
%U 7775,49138,12440,9997,98289,36860,73721,196592,82941,393199,294904,786414,49,294907
%N a(n) is the smallest positive number B that yields a solution for k = A167219(n).
%C 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.
%e 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.
%e 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.
%e 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?
%o (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
%Y Cf. A054841, A054842, A167219.
%K nonn
%O 1,1
%A _Ctibor O. Zizka_, Oct 30 2009
%E Edited by _Jon E. Schoenfield_, Mar 16 2022
%E Corrected and extended by _Michel Marcus_, Aug 09 2022
%E a(41) and beyond from _Michael S. Branicky_, Aug 10 2022
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