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%I #26 Mar 24 2017 21:46:30
%S 2,2,2,3,2,5,3,2,2,3,5,11,4,3,2,5,2,17,3,19,5,2,11,23,4,5,3,3,2,29,6,
%T 2,2,11,3,7,6,37,19,3,8,41,2,6,5,9,23,3,4,7,5,17,13,53,3,11,2,7,29,59,
%U 6,61,2,4,2,13,11,67,17,23,10,71,6,2,37,5,19,11
%N Least b>1 such that n, when expressed in base b and then interpreted in base b^2, yields a multiple of n.
%C a(n) <= max(2, n) for any n >= 0.
%C a(n*a(n)) <= a(n) for any n >= 0 (see also A283927).
%C a(n) = 2 iff n belongs to A062846.
%C Conjecture: if n is composite, then a(n) < n (see also A283937).
%C Theorem: If n is composite, then a(n) < n. Proof: If n=ab with 1<a<b, then n_b=a0, which interpreted as ab^2 is a multiple of n; if n=b^2 then n_b=100, which interpreted as b^4 is again a multiple of n. - _Michael R Peake_, Mar 25 2017
%C First occurrence of b > 1: 1, 4, 13, 6, 31, 36, 41, 46, 71, 12, 133, 53, 155, 106, 161, 18, 199, 20, 261, ..., . - _Robert G. Wilson v_, Mar 19 2017
%H Rémy Sigrist, <a href="/A283751/b283751.txt">Table of n, a(n) for n = 0..10000</a>
%e The number 5, when expressed in base 2 and then interpreted in base 4, yields 17, not a multiple of 5.
%e The number 5, when expressed in base 3 and then interpreted in base 9, yields 11, not a multiple of 5.
%e The number 5, when expressed in base 4 and then interpreted in base 16, yields 17, not a multiple of 5.
%e The number 5, when expressed in base 5 and then interpreted in base 25, yields 25, a multiple of 5.
%e Hence, a(5)=5.
%t f[n_] := Block[{b = 2}, While[ Mod[ FromDigits[ IntegerDigits[n, b], b^2], n] > 0, b++]; b]; Array[f, 80, 0] (* _Robert G. Wilson v_, Mar 19 2017 *)
%o (PARI) a(n) = my (b=2); if (n>0, while (fromdigits(digits(n,b), b^2)%n, b++)); return (b)
%Y Cf. A062846, A155078, A283927, A283937.
%K nonn,base
%O 0,1
%A _Rémy Sigrist_, Mar 15 2017
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