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A167221 Numbers B 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<m) >=0 ; p_0,p_1,..., p_m are prime numbers ; a_0, a_1, ..., a_m, B are integers . For B = (2^r)*3 - r we have n = (2^r)*3 . 0
3, 5, 3, 10, 21, 9, 17, 44, 34, 91, 7 (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

Table of n, a(n) for n=1..11.

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: A076842 A077862 A134061 * A177930 A242034 A205708

Adjacent sequences:  A167218 A167219 A167220 * A167222 A167223 A167224

KEYWORD

easy,nonn

AUTHOR

Ctibor O. Zizka, Oct 30 2009

STATUS

approved

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Last modified August 18 15:00 EDT 2019. Contains 326106 sequences. (Running on oeis4.)