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Numbers k that are repdigits in more bases (smaller than k) than any smaller number.
5

%I #49 Jul 23 2020 13:53:59

%S 1,3,7,15,24,40,60,120,180,336,360,720,840,1260,1440,1680,2520,5040,

%T 7560,10080,15120,20160,25200,27720,45360,50400,55440,83160,110880,

%U 166320,221760,277200,332640,498960,554400,665280,720720,1081080,1441440,2162160,2882880

%N Numbers k that are repdigits in more bases (smaller than k) than any smaller number.

%C A repdigit has all digits the same in some base.

%C The number 3 isn't Brazilian (A125134) because 3 = 11_2 is an expansion of the type n = 11_(n-1), which is forbidden for Brazilian numbers. So, except for 3, all the terms here are highly Brazilian numbers (A329383). - _Daniel Lignon_, Dec 30 2019

%D D. Lignon, Dictionnaire de (presque) tous les nombres entiers, Editions Ellipses, 2012, see p. 420. [In French.]

%H Giovanni Resta, <a href="/A066044/b066044.txt">Table of n, a(n) for n = 1..92</a>

%e 15 is in the sequence since 15 = 1111_2 = 33_4 = 11_14 and no smaller number is a repdigit in 3 different bases.

%t a = 0 Range[100]; Do[ c = 1; k = 2; While[ k < n-1, If[ Length[ Union[ IntegerDigits[n, k]]] == 1, c++ ]; k++ ]; If[a[[c]] == 0, a[[c]] = n; Print[c, " = ", n]], {n, 1, 200000} ]

%o (PARI) okrepu3(b, target, lim) = {my(k = 3, nb = 0, x); while ((x=(b^k-1)/(b-1)) <= target, if (x==target, nb++); k++); nb;}

%o dge3(n, d) = {my(nb=0, ndi, limi); for (i=1, #d, ndi = n/d[i]; limi = sqrtint(ndi); for (k=d[i]+1, limi, nb += okrepu3(k, ndi, limi););); nb;}

%o deq2(n, d) = {my(nb=0, nk); for (k=1, #d\2, nk = (n - d[k])/d[k]; if (nk > d[k], nb++);); nb;}

%o beta23(n) = {if (n<3, return (0)); my(d=divisors(n)); deq2(n, d) + dge3(n, d);}

%o lista(nn) = {my(m = -1, nm); for (n=1, nn, if ((nm=beta23(n)) > m, print1(n, ", "); m = nm););} \\ _Michel Marcus_, Jul 13 2019

%Y Cf. A066460, A329383.

%K nonn,base

%O 1,2

%A _Erich Friedman_, Dec 29 2001

%E More terms from _Robert G. Wilson v_, Jan 02 2002

%E Offset changed to 1 by _Giovanni Resta_, Apr 05 2017

%E a(1) changed to 1 and new terms a(32)-a(41) from _Giovanni Resta_, Apr 05 2017