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

0,3

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.

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.

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A337496, A337536.

Sequence in context: A038884 A338345 A252282 * A040122 A318207 A038955

Adjacent sequences: A337494 A337495 A337496 * A337498 A337499 A337500

Keyword

nonn,base,hard

Author

François Marques, Aug 29 2020

Status

approved