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A167221
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a(n) is the smallest positive number B that yields a solution for k = A167219(n).
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1
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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?
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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