A066044 Numbers k that are repdigits in more bases (smaller than k) than any smaller number.

1, 3, 7, 15, 24, 40, 60, 120, 180, 336, 360, 720, 840, 1260, 1440, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, 2882880

Offset: 1

Author

Erich Friedman, Dec 29 2001

Keywords

• nonn
• base

Comments

A repdigit has all digits the same in some base.

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

Examples

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

References

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

Links

• Giovanni Resta, Table of n, a(n) for n = 1..92

Crossrefs

Cf. A066460, A329383.

Programs

Mathematica

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} ]

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;}
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;}
deq2(n, d) = {my(nb=0, nk); for (k=1, #d\2, nk = (n - d[k])/d[k]; if (nk > d[k], nb++);); nb;}
beta23(n) = {if (n<3, return (0)); my(d=divisors(n)); deq2(n, d) + dge3(n, d);}
lista(nn) = {my(m = -1, nm); for (n=1, nn, if ((nm=beta23(n)) > m, print1(n, ", "); m = nm););} \\ Michel Marcus, Jul 13 2019

Extensions

More terms from Robert G. Wilson v, Jan 02 2002

Offset changed to 1 by Giovanni Resta, Apr 05 2017

a(1) changed to 1 and new terms a(32)-a(41) from Giovanni Resta, Apr 05 2017