

A317168


Triangular array read by rows: T(n,k) (2 <= k <= n) is the smallest number whose representations in bases n and k both consist of repeated identical blocks of digits.


1



3, 208050, 4, 10, 208050, 5, 31, 364, 3276, 6, 7, 372322860, 21, 7812, 7, 74571415482725702377300, 8, 1300, 24, 21328, 8, 36, 208050, 63, 18, 32767, 650, 9, 10, 30, 10, 182, 4869520626260925609644717450, 40, 83220, 10
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OFFSET

2,1


COMMENTS

Trivially, T(n,n) = n + 1.


LINKS

Robert G. Wilson v, Table of n, a(n) for n = 2..1036 (rows 1 to 46, flattened; first 231 terms from Charlie Neder; offset changed by Georg Fischer, Jan 24 2019)


EXAMPLE

T(2,3) = 208050 = 110010 110010 110010[2] = 101120 101120[3].
Leading zeros are not allowed, so e.g. T(2,4) does not equal 01 01[2] = 11[4] = 5.
Triangle begins:
3,
208050, 4,
10, 208050, 5,
31, 364, 3276, 6,
7, 372322860, 21, 7812, 7,
74571415482725702377300, 8, 1300, 24, 21328, 8,
36, 208050, 63, 18, 32767, 650, 9,
10, 30, 10, 182, 4869520626260925609644717450, 40, 83220, 10,
...


MATHEMATICA

sol[b1_, t1_, n1_, b2_, t2_, n2_] := Block[{k = (b1^t1  1)/(b1^n1  1), r, s, t, i1, i2}, r = (b2^t2  1)/(b2^n2  1)/k; s=Numerator@ r; t=Denominator@ r; i1 = {Ceiling[b1^(n1  1)/s], Floor[(b1^n1  1)/s]}; i2 = {Ceiling[ b2^(n2  1)/t], Floor[(b2^n2  1)/t]}; If[i1[[2]] < i1[[1]]  i2[[2]] < i2[[1]], 0, k * s * Min[ IntervalIntersection @@ Interval /@ {i1, i2}]]]; T[b1_, b2_] := Block[{m = Infinity, t1=1, v}, While[m == Infinity, t1++; Do[ v = sol[b1, t1, n1, b2, t2, n2]; If[v > 0, m = Min[m, v]], {t2, IntegerLength[b1^(t1  1), b2], IntegerLength[b1^t1  1, b2]}, {n1, Most@ Divisors@ t1}, {n2, Most@ Divisors@ t2}]]; m]; Flatten@ Table[ T[b1, b2], {b1, 2, 10}, {b2, 2, b1}] (* Giovanni Resta, Jul 24 2018 *)


CROSSREFS



KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



