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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

%I #34 Jan 24 2019 17:24:55

%S 3,208050,4,10,208050,5,31,364,3276,6,7,372322860,21,7812,7,

%T 74571415482725702377300,8,1300,24,21328,8,36,208050,63,18,32767,650,

%U 9,10,30,10,182,4869520626260925609644717450,40,83220,10

%N 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.

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

%H Robert G. Wilson v, <a href="/A317168/b317168.txt">Table of n, a(n) for n = 2..1036</a> (rows 1 to 46, flattened; first 231 terms from Charlie Neder; offset changed by Georg Fischer, Jan 24 2019)

%e T(2,3) = 208050 = 110010 110010 110010[2] = 101120 101120[3].

%e Leading zeros are not allowed, so e.g. T(2,4) does not equal 01 01[2] = 11[4] = 5.

%e Triangle begins:

%e 3,

%e 208050, 4,

%e 10, 208050, 5,

%e 31, 364, 3276, 6,

%e 7, 372322860, 21, 7812, 7,

%e 74571415482725702377300, 8, 1300, 24, 21328, 8,

%e 36, 208050, 63, 18, 32767, 650, 9,

%e 10, 30, 10, 182, 4869520626260925609644717450, 40, 83220, 10,

%e ...

%t 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 *)

%K nonn,base,tabl

%O 2,1

%A _Charlie Neder_, Jul 23 2018

%E Offset changed to 2 by _N. J. A. Sloane_, Aug 30 2018