A337497 a(n) is the smallest integer k with exactly n bases b such that k in base b contains the digit b-1; or -1 if there is no such integer.

0, 1, 2, 5, 7, 11, 19, 39, 23, 69, 103, 47, 59, 125, 95, 143, 119, 179, 299, 251, 335, 527, 239, 419, 599, 359, 479, 1019, 671, 1619, 1727, 959, 719, 1319, 839, 2039, 1259, 2771, 2339, 2099, 1439, 5471, 1679, 2159, 3695, 3599, 2879, 5939, 3779, 2519, 4619, 3359, 4319

Offset: 0

Author

François Marques, Aug 29 2020

Keywords

• nonn
• base
• hard

Comments

a(n) = Min_({k | A337496(k)=n}) if the set is not empty, else -1.

Conjecture: a(n) > log(n)^(sqrt(2)*log(n)) for n>1. This have been checked for n<3444, and for n<10275 unless if a(n)=-1.

Examples

			a(7) is 39 because 39 has 7 bases b (which are 2,4,5,8,10,20 and 40) where the digits of n contain the digit b-1 and this does not happen for a smaller integer.
		

Links

• François Marques, Table of n, a(n) for n = 0..3443
• François Marques, Table of known a(n) values, for n = 0..10000. Unknown values are replaced by a question mark.

Crossrefs

Cf. A337496, A337536.

Programs

Mathematica

mainBaseQ[n_, b_] := MemberQ[IntegerDigits[n, b], b - 1]; basesCount[n_] := Count[Range[2, n + 1], ?(mainBaseQ[n, #] &)]; m = 50; seq = Table[-1, {m}]; c = 0; n = 0; While[c < m, i = basesCount[n]; If[i <= m - 1 && seq[[i + 1]] < 0, c++; seq[[i + 1]] = n]; n++]; seq (* _Amiram Eldar, Sep 01 2020 *)

PARI

a(n) = for(k=0,+oo, if(sum(b=2, k+1, vecmax(digits(k, b)) == b-1)==n, return(k)) ); \\ François Marques, Nov 19 2020