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Numbers n with the property that shifting the rightmost digit of n to the left end doubles the number.
10

%I #29 Jun 23 2013 11:20:58

%S 0,105263157894736842,157894736842105263,210526315789473684,

%T 263157894736842105,315789473684210526,368421052631578947,

%U 421052631578947368,473684210526315789,105263157894736842105263157894736842,157894736842105263157894736842105263

%N Numbers n with the property that shifting the rightmost digit of n to the left end doubles the number.

%C The sequence is infinite, since repeating 105263157894736842 any number of times (e.g. 105263157894736842105263157894736842) gives another number with the same property.

%C A number N = 10n+m is in the sequence iff 2N = m*10^d+n, where d is the number of digits of n = [N/10]. This is equivalent to 19n = m(10^d-2), i.e. 10^d=2 (mod 19) and n = m(10^d-2)/19, m=2..9 (to ensure that n has d digits). Thus for each d = 18j-1, j=1,2,3... we have exactly 8 solutions which are the j-fold repetition of one among {a(1),...,a(8)}. - _M. F. Hasler_, May 04 2009

%C Normally lists have offset 1, but there are good reasons to make an exception in this case. - _N. J. A. Sloane_, Dec 24 2012

%H G. P. Michon, <a href="http://www.numericana.com/answer/p-adic.htm#a146088">Deriving A146088 from linear decadic equations</a>.

%F a(n) = ((10^d-2)/19*10+1)m, where m=(n-1)%8+2 is the trailing digit and d=(n+7)\8*18-1 is the number of other digits. - _M. F. Hasler_, May 04 2009

%F a(8k+i) = A217592(9k+i+1)/2 for i=1..8 with any k.

%e The sequence starts with a(0)=0 because rotating a lone 0 does double 0. That initial trivial term was not given an index of 1 when it was added, so that the index of other terms of A146088 would not change and invalidate delicate prior cross-references within OEIS (e.g., A217592) or outside of it.

%e a(4) = 263157894736842105 because 2*a(4) = 526315789473684210.

%t a[n_] := (m = Mod[n - 1, 8] + 2; d = Floor[(n + 7)/8]*18 - 1; ((10/19)*(10^d - 2) + 1)*m); Table[a[n], {n, 0, 10}] (* _Jean-François Alcover_, Jan 16 2013, after _M. F. Hasler_ *)

%o (PARI) A146088(n) = ((10^((n+7)\8*18-1)-2)/19*10+1)*((n-1)%8+2)

%o /* or a more experimental approach: */ for(d=1,99, Mod(10,19)^k-2 & next; for(m=2,9, print1(",",m*(10^k-2)/19,m))) \\\\ _M. F. Hasler_, May 04 2009

%Y Cf. A092697, A146561, A146569, A146754, A217592.

%K nonn,base

%O 0,2

%A _N. J. A. Sloane_, based on correspondence from William A. Hoffman III (whoff(AT)robill.com), Apr 10 2009

%E More terms from _M. F. Hasler_, May 04 2009

%E a(0)=0 added by G. P. Michon, Oct 29 2012