login
Numbers m with the property that shifting the rightmost digit of m to the left end multiplies the number by 5.
8

%I #29 Nov 06 2024 04:19:49

%S 142857,142857142857,142857142857142857,142857142857142857142857,

%T 142857142857142857142857142857,142857142857142857142857142857142857,

%U 102040816326530612244897959183673469387755,122448979591836734693877551020408163265306,142857142857142857142857142857142857142857

%N Numbers m with the property that shifting the rightmost digit of m to the left end multiplies the number by 5.

%C From _Seiichi Manyama_, Aug 22 2017: (Start)

%C For k >= 1, (10^(6*k) - 1)/7 is a term.

%C For 5 <= a <= 9 and k >= 1, a*(10^(42*k) - 1)/49 is a term. (End)

%H Robert Israel, <a href="/A146754/b146754.txt">Table of n, a(n) for n = 1..258</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Parasitic_number">Parasitic number</a>.

%e From _Seiichi Manyama_, Aug 22 2017: (Start)

%e b1 = 14285.

%e a(1) = b1*10 + 7,

%e 5*a(1) = 714285 = 7*10^5 + b1.

%e b7 = 10204081632653061224489795918367346938775.

%e a(7) = b7*10 + 5,

%e 5*a(7) = 510204081632653061224489795918367346938775 = 5*10^41 + b7. (End)

%p f:= proc(d) # solutions with d+1 digits

%p local b,R,a;

%p R:= NULL;

%p for b from ceil(49*10^(d-1)/(10^d - 1)) to 9 do

%p a:= (10^d-5)*b/49;

%p if a::integer then R:= R, 10*a+b fi

%p od;

%p R

%p end proc:

%p map(f, [$1..42]); # _Robert Israel_, Nov 05 2024

%Y Cf. A146088 (k=2), A146561 (k=3), A146569 (k=4), this sequence (k=5), A291215 (k=7).

%Y Cf. A092697, A097717.

%K nonn,base

%O 1,1

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