A325114 Integers k such that no nonzero subsequence of the decimal representation of k is divisible by 7.

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 18, 19, 20, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 36, 38, 39, 40, 41, 43, 44, 45, 46, 48, 50, 51, 52, 53, 54, 55, 58, 59, 60, 61, 62, 64, 65, 66, 68, 69, 80, 81, 82, 83, 85, 86, 88, 89, 90, 92, 93, 94, 95, 96, 99, 100, 101, 102, 103, 104, 106, 108, 109, 110, 111, 113, 115, 116, 118, 120

Offset: 1

Author

Jonathan Kal-El Peréz, Mar 27 2019

Keywords

• nonn
• base

Comments

Does not contain 114 (helps to distinguish this from related sequences).

From David A. Corneth, Sep 10 2024: (Start)

Any term greater than 10^6 must have a digit 0. Proof: Any term between 10^6 and 10^7 has a 0.

Proof via induction and contradiction; any 7 digital number term has a digit 0. Suppose some number with k with q > 7 digits has no digit 0. Then floor(k/10) is a term and has no digit 0 and q - 1 digits. But there is no such number. A contradiction. Therefore any term with at least 7 digits has a digit 0. (End)

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000
• David A. Corneth, PARI program

Crossrefs

Cf. A014261 (for 2), A325112 (for 3), A325113 (for 4), A261189 (for 5).

See A376046 for complement.

Programs

Mathematica

With[{k = 7}, Select[Range@ 100, NoneTrue[DeleteCases[FromDigits /@ Rest@ Subsequences[IntegerDigits@ #], 0], Mod[#, k] == 0 &] &]] (* Michael De Vlieger, Mar 31 2019 *)

PARI

\\ See Corneth link

Extensions

More than the usual number of terms are shown in order to distinguish this from a new sequence arising from the game of "buzz" (cf. A092433). - N. J. A. Sloane, Sep 09 2024