A309631 -id:A309631 - cp's OEIS frontend

A328567 a(n) is the smallest positive integer divisible by n such that it is possible to strike out a digit from its binary expansion (apart from trailing zeros) so that the resulting number is nonzero and divisible by n.

3, 6, 21, 12, 75, 42, 105, 24, 279, 150, 341, 84, 403, 210, 465, 48, 1071, 558, 1197, 300, 1323, 682, 1449, 168, 1575, 806, 1701, 420, 1827, 930, 1953, 96, 4191, 2142, 4445, 1116, 4699, 2394, 4953, 600, 5207, 2646, 5461, 1364, 5715, 2898, 5969, 336, 6223, 3150

Offset: 1

Rémy Sigrist, Oct 20 2019

This sequence is a binary variant of A309631.

This kind of sequence is well defined for any fixed base b > 1: for any n > 0: consider the concatenation in base b, say m, of n, "0", and n; m is a multiple of n, and removing the central "0" (which is not a trailing zero), gives another multiple of n.

			For n = 3:
- the first multiples of 3 are (in decimal and in binary), alongside the possible values resulting from striking out a non-trailing zero:
    3*k  bin(3*k)  striked (binary)
    ---  --------  ----------------
      3        11  1
      6       110  10
      9      1001  1, 100, 101
     12      1100  100
     15      1111  111
     18     10010  10, 1000, 1010
     21     10101  101, 1001, 1010, 1011, 1101
- 21 is the least appropriate multiple,
- so a(3) = 21.

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, PARI program for A328567

Cf. A070940, A309631.

PARI
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See Links section.
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Apparently, a(n)/n = 2^(1+A070940(n)) - 1.

A332876 a(n) is the smallest positive multiple of n whose decimal expansion includes a digit (other than a trailing zero) whose removal yields a proper multiple of n.

Original entry on oeis.org

12, 14, 36, 28, 105, 102, 147, 136, 108, 120, 242, 204, 286, 238, 330, 352, 374, 306, 2109, 140, 462, 484, 2047, 408, 150, 572, 594, 756, 3219, 360, 682, 864, 2937, 1326, 770, 792, 4107, 2128, 4329, 280, 3649, 1638, 3827, 1232, 990, 2530, 5217, 1344, 5439, 1050

Offset: 1

Bernard Schott, Feb 28 2020

This sequence is a variant of A309631; but here, when we strike out the right digit, it is forbidden that the obtained number is equal to n.
About the origin of this sequence, see comments in A309631.
The first quotients a(n)/n are 12, 7, 12, 7, 21 ,17, 21, 17, 12, 12, 22, 17, 22, 17, 22, ...

			a(7) = 147 because 147 = 7*21 and if we strike out "7", 14 is also divisible by 7, and there is no integer < 147 with that property.

Roman Fedorov, Alexei Belov, Alexander Kovaldzhi, Ivan Yashchenko, Moscow Mathematical Olympiads, 2000-2005,Problem 3, Level D, 2004, MSRI, 2011, p. 21 and 130/131

Index to sequences related to Olympiads.

Cf. A309631 (original version), A328567 (binary variant).

del[n_] := Block[{m = 10^IntegerExponent[n, 10], d}, d = IntegerDigits[n/m]; Table[ FromDigits[Delete[d, k]] m, {k, Length@ d}]]; a[n_] := Block[{k = n, v},  While[! AnyTrue[del[k], # > n && Mod[#, n] == 0 &], k += n]; k]; Array[a, 50] (* Giovanni Resta, Feb 28 2020 *)

More terms from Giovanni Resta, Feb 28 2020

Name improved by Rémy Sigrist and Jon E. Schoenfield, Feb 28 2020

cp's OEIS Frontend

A328567 a(n) is the smallest positive integer divisible by n such that it is possible to strike out a digit from its binary expansion (apart from trailing zeros) so that the resulting number is nonzero and divisible by n.

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