A023094 -id:A023094 - cp's OEIS frontend

A279688 Numbers k such that k and 2k are anagrams in some base.

0, 8, 18, 21, 27, 32, 40, 48, 65, 66, 72, 78, 86, 96, 98, 99, 104, 108, 111, 114, 123, 133, 168, 176, 189, 190, 200, 208, 225, 240, 248, 258, 260, 264, 266, 270, 280, 288, 294, 296, 297, 312, 314, 318, 320, 330, 341, 350, 363, 380, 387, 396, 399, 408, 429, 432

Offset: 1

Peter Kagey, Dec 17 2016

			a(2) = 8  because in base 5, 8  = 13_5  and 16 = 31_5.
a(3) = 18 because in base 4, 18 = 102_4 and 36 = 210_4.

Chai Wah Wu, Table of n, a(n) for n = 1..10001

Cf. A023094.

fQ[n_] := Block[{b = 2}, While[ 2b < n +3 && Sort[ IntegerDigits[n, b]] != Sort[ IntegerDigits[ 2n, b]], b++]; 2b < n +3]; fQ[0] = True; Select[Range[0, 435], fQ] (* Robert G. Wilson v, Dec 21 2016 *)

isok(n) = if (n==0, 1, for (b=2, n, if (vecsort(digits(n,b)) == vecsort(digits(2*n,b)), return(1))); 0); \\ Michel Marcus, Dec 17 2016

A087502 Smallest positive integer which when written in base n is doubled when the last digit is put first.

Original entry on oeis.org

32, 18, 8, 10993850, 2129428800, 21, 5064320, 105263157894736842, 40, 64609423538, 5712, 65, 58774271029236501660840264682112, 67650, 96, 833, 586081355679130611935159482937228562988190880, 133

Offset: 3

Pontus von Brömssen, Sep 10 2003

a(n) is the smallest integer of the form x*(n^d-1)/(2n-1) for integer x and d, where 1 < x < n and d > 1. x is the last digit and d is the number of digits of a(n) in base n. - Pontus von Brömssen, Jan 06 2019

			a(10) = 105263157894736842 because 2*105263157894736842 = 210526315789473684 and no smaller number has this property. (Leading zeros are not allowed, otherwise 2*052631578947368421 = 105263157894736842 would be a smaller solution.)

Pontus von Brömssen, Table of n, a(n) for n = 3..221

See A158877 for these numbers written in base n. Cf. A023094, A034089, A081463, A087502.

A087502 := proc(n) local d,a; d := 1; a := n; while a>=n do d := d+1; a := denom((2^d-1)/(2*n-1)); od; return(max(2,a)*(n^d-1)/(2*n-1)); end proc;

A158877 Definition of a(n): in base-n arithmetic a(n) is the smallest positive integer that is doubled when its least significant digit is moved to become the most significant digit.

Original entry on oeis.org

1012, 102, 13, 1031345242, 103524563142, 25, 10467842, 105263157894736842, 37, 10631694842

Offset: 3

Daniel Asimov (asimov(AT)msri.org), Mar 28 2009

The problem has no solution in base 2, so sequence begins with the base-3 solution. The idea was suggested by a NY Times article (Sunday Magazine of Mar 29, 2009) -- in which Freeman Dyson is said to have solved the base-10 question almost instantaneously when it was posed to him -- and by the ensuing math-fun discussion.

			For n = 5, the smallest positive integer whose base-5 representation doubles when the rightmost digit is moved to become the leftmost digit is 8 = 13_5; 31_5 = 16.
For n = 8, the smallest positive integer whose base-8 representation doubles when the rightmost digit is moved to become the leftmost digit is 21 = 25_8; 52_8 = 42. - _Robert Tanniru_, Aug 09 2022
For n = 13, the number can't be represented in this list as it would be 27A5 in base 13.

See A087502 (which is the main entry for this sequence) for these numbers written in base 10. Cf. A023094, A159774.

a(5) corrected by William A. Hoffman III (whoff(AT)robill.com), Apr 19 2009
a(8) corrected by Robert Tanniru, Aug 09 2022
a(11)-a(12) from Robert Tanniru, Aug 11 2022, using A087502

A382946 a(n) is the least positive integer k having a proper divisor d such that the base n expansions of k and d, without leading zeros, have, up to order, the same digits, or a(n) = -1 if no such k exists.

Original entry on oeis.org

-1, 64, 36, 16, 700, 36, 42, 64, 3105, 45, 594, 105, 130, 168, 945, 120, 1666, 96, 266, 275, 2457, 231, 460, 351, 450, 273, 7938, 175, 7714, 280, 682, 1024, 308, 459, 7525, 741, 962, 665, 27300, 288, 17097, 560, 1290, 1265, 18540, 1035, 1974, 540, 952, 715

Offset: 2

Rémy Sigrist, Apr 09 2025

Conjecture: a(n) > 0 for any n > 2.

			The first terms, alongside an appropriate divisor d, in bases 10 and n, are:
  n   a(n)  d     n in base n  d in base n
  --  ----  ----  -----------  -----------
   2    -1  N/A   N/A          N/A
   3    64    32  2,1,0,1      1,0,1,2
   4    36    18  2,1,0        1,0,2
   5    16     8  3,1          1,3
   6   700   350  3,1,2,4      1,3,4,2
   7    36    12  5,1          1,5
   8    42    21  5,2          2,5
   9    64    16  7,1          1,7
  10  3105  1035  3,1,0,5      1,0,3,5
  11    45    15  4,1          1,4
  12   594   198  4,1,6        1,4,6
  13   105    21  8,1          1,8
  14   130    65  9,4          4,9
  15   168    56  11,3         3,11
  16   945   315  3,11,1       1,3,11

Rémy Sigrist, Table of n, a(n) for n = 2..1000

Cf. A023094, A090056, A382945.

a(n) = {
    if (n==2, return (-1));
    for (k = 1, oo,
        my (t = vecsort(digits(k, n)));
        fordiv (k, d,
            if (d < k && vecsort(digits(d, n))==t,
                return (k);););); }

from sympy import divisors
from sympy.ntheory import digits
from itertools import count
def a(n):
    if n == 2:
        return -1
    for k in count(2*n):
        divs, kdigs = divisors(k), sorted(digits(k, n)[1:])
        for d in sorted(divs[:-1], reverse=True):
            ddigs = sorted(digits(d, n)[1:])
            if ddigs == kdigs:
                return k
            if len(ddigs) < len(kdigs):
                break
print([a(n) for n in range(2, 52)]) # Michael S. Branicky, Apr 13 2025

A279916 Least b such that A279688(n) and 2*A279688(n) are anagrams in base b.

Original entry on oeis.org

5, 4, 8, 4, 3, 5, 5, 14, 4, 4, 4, 3, 3, 3, 4, 3, 4, 4, 4, 4, 20, 8, 23, 8, 11, 5, 5, 26, 5, 3, 3, 3, 4, 3, 4, 29, 3, 3, 3, 4, 3, 3, 4, 3, 4, 32, 6, 4, 11, 4, 4, 4, 35, 4, 4, 11, 4, 4, 4, 4, 4, 7, 38, 4, 6, 4, 4, 4, 4, 8, 41, 11, 16, 8, 44, 3, 47, 3, 3, 3, 3, 3

Offset: 2

Peter Kagey, Dec 23 2016

			A279688(2) = 8, and a(2) = 5 because 5 is the least base such that 8 and 16 are anagrams: 8 = 13_5 and 16 = 31_5.

Peter Kagey, Table of n, a(n) for n = 2..10000

Cf. A023094, A279688, A279689.

DeleteCases[#, 0] &@ Table[Module[{b = 2}, While[2 b < n + 3 && Sort[IntegerDigits[n, b]] != Sort[IntegerDigits[2 n, b]], b++]; b Boole[2 b < n + 3]], {n, 780}] (* Michael De Vlieger, Dec 23 2016, after Robert G. Wilson v at A279688 *)

cp's OEIS Frontend

A279688 Numbers k such that k and 2k are anagrams in some base.

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A087502 Smallest positive integer which when written in base n is doubled when the last digit is put first.

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A158877 Definition of a(n): in base-n arithmetic a(n) is the smallest positive integer that is doubled when its least significant digit is moved to become the most significant digit.

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A382946 a(n) is the least positive integer k having a proper divisor d such that the base n expansions of k and d, without leading zeros, have, up to order, the same digits, or a(n) = -1 if no such k exists.

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A279916 Least b such that A279688(n) and 2*A279688(n) are anagrams in base b.

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