A097717
a(n) = least number m such that the quotient m/n is obtained merely by shifting the leftmost digit of m to the right end.
Original entry on oeis.org
1, 105263157894736842, 1034482758620689655172413793, 102564, 714285, 1016949152542372881355932203389830508474576271186440677966, 1014492753623188405797, 1012658227848, 10112359550561797752808988764044943820224719
Offset: 1
We have a(5)=714285 since 714285/5=142857.
Likewise, a(4)=102564 since this is the smallest number followed by 205128, 307692, 410256, 512820, 615384, 717948, 820512, 923076, ... which all get divided by 4 when the first digit is made last.
- R. Sprague, Recreation in Mathematics, Problem 21 pp. 17; 47-8 Dover NY 1963.
A097717: when move L digit to R, divides by n (infinite)
A094676: when move L digit to R, divides by n, no. of digits is unchanged (finite)
A092697: when move R digit to L, multiplies by n (finite)
A128857 is the same sequence as
A097717 except that m must begin with 1.
-
Min[Table[Block[{d=Ceiling[Log[10,n]],m=(10n-1)/GCD[10n-1,a]}, If[m!=1, While[PowerMod[10,d,m]!=n,d++ ],d=1]; ((10^(d+1)-1) a n)/(10n-1)], {a,9}]] (* Anton V. Chupin (chupin(X)icmm.ru), Apr 12 2007 *)
a(9) from Anton V. Chupin (chupin(X)icmm.ru), Apr 12 2007
A094676
a(n) = least number m such that the quotient m/n is obtained merely by shifting the leftmost digit of m to the right end and the second digit of m is not zero.
Original entry on oeis.org
1, 210526315789473684, 3103448275862068965517241379, 410256, 714285, 6101694915254237288135593220338983050847457627118644067796, 7101449275362318840579, 8101265822784, 91011235955056179775280898876404494382022471
Offset: 1
a(4) = 410256 = 4*102564.
- H. Camous, Jouer Avec Les Maths, "Chassez le naturel", Section I, Problem 3 pp. 20; 31-2, Les Editions D'Organisation, Paris 1984.
- L. A. Graham, Ingenious Mathematical Problems and Methods, "End At The Beginning", Problem 72 pp. 44; 212-3, Dover NY 1959.
Original entry on oeis.org
1, 16, 3348, 411184, 5, 1262796336, 31415153952, 128, 639, 46402790906782052954848931760, 9548, 37884308119951668432, 507, 483747841655344, 2949712546290578068913640, 368402917173844349535205696, 3162, 1149642179207353109724066230688
Offset: 1
a(2)=16, because in base 5, 16 is written 31 and 16/2 is 8 and written 13.
-
A249599 := proc(n)
local m,b,mbas,msf ;
b := 5;
for m from 1 to 1999999 do
mbas := convert(m,base,b) ;
msf := [op(-1,mbas),op(1..nops(mbas)-1,mbas)] ;
msf := add(op(i,msf)*b^(i-1),i=1..nops(msf)) ;
if m/n = msf then
return m;
end if;
end do:
-1 ;
end proc:
for n from 1 do
print(n,A249599(n)) ;
end do: # R. J. Mathar, Nov 11 2014
Original entry on oeis.org
1, 18, 279, 4, 68985, 1094166, 49, 264, 1053, 1050, 4191, 17966487875148, 65, 266910, 73278909191113155, 16, 18722068612123127013, 299304917928357795234, 265639, 76514292380672732576340, 1223491190935287357533961, 67880230, 1035, 17360709912, 775
Offset: 1
279 is (10113)_4 which shift-rotates into (01131)_4 = 93 = 279/3, so 279 qualifies as a(3).
Original entry on oeis.org
1, 32, 3, 88, 260, 15192960, 28, 61616, 9, 7888549122400, 55, 182208, 132538588, 2240, 165, 32048741728, 1185506696, 2194329698227926780769440, 247, 23264534699960, 69737318935284, 179872, 14559920, 16912071582760464, 130885300, 69521680967024, 27
Offset: 1
Showing 1-5 of 5 results.
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