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 A167219 Numbers n such that (p_m ^ a_m)*(p_m-1 ^ a_m-1)*...*(3^a_1)*(2^a_0) = (B^m)*a_m + (B^m-1)*a_m-1 +...+ (B^1)*a_1 + (B^0)*a_0 where n =(p_m ^ a_m)*(p_m-1 ^ a_m-1)*...*(3^a_1)*(2^a_0) ; a_m >= 1 ; a_(i=0 ; p_0,p_1,..., p_m are prime numbers ; a_0, a_1, ..., a_m, B are integers . For n = (2^r)*3 we have B = (2^r)*3 - r. 0
 3, 6, 10, 12, 24, 27, 36, 48, 71, 96, 100 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS B is the base in which we can express n like a sum{i=0 to 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. LINKS EXAMPLE Ex_1 : n = 21 = 2^0 * 3^1 * 5^0 * 7^1 ; n = 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 does exist. Ex_2: n = 10 = 2^1 * 3^0 * 5^1 ; n = 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 . Ex_3: n = 12 = 2^2 * 3^1 ; n = 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 other numbers besides n=12 in base 10 ? CROSSREFS Cf. A054841, A054842 Sequence in context: A082925 A131696 A164114 * A264762 A028357 A310047 Adjacent sequences:  A167216 A167217 A167218 * A167220 A167221 A167222 KEYWORD easy,nonn AUTHOR Ctibor O. Zizka, Oct 30 2009 STATUS approved

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Last modified September 23 20:42 EDT 2021. Contains 347617 sequences. (Running on oeis4.)