A023086 -id:A023086 - cp's OEIS frontend

A336670 Numbers that have decimal expansion c(1)c(2)...c(n) with distinct digits that satisfy c(1) <> 0, c(1) is the largest digit, and for each i in 1..n there is j in {0, 1} such that c(i) == 2*c(i-1) + j (mod 10) (with c(0): = c(n)).

0, 9, 63, 512, 874, 5012, 7513, 8624, 9874, 62513, 75013, 86374, 98624, 625013, 875124, 986374, 8750124, 9875124, 86251374, 86375124, 87513624, 98750124, 862501374, 863750124, 875013624, 986251374, 986375124, 987513624, 9862501374, 9863750124, 9875013624

Offset: 1

Petros Hadjicostas, Jul 29 2020

This is one of Schuh's examples of a puzzle tree.
Putting the number on a circle and going clockwise, we observe that a 0 is followed by a 1; a 1 is followed by a 2 or 3; a 2 is followed by a 4 or 5; a 3 is followed by a 6 or 7; a 4 is followed by an 8 or 9; a 5 is followed by a 0 or 1; a 6 is followed by a 2 or 3; a 7 is followed by a 4 or 5; an 8 is followed by a 6 or 7; and a 9 is followed by an 8. (These observations assume the number has at least two digits.)
Schuh (pp. 31-35) uses the solution to this problem to solve the "doubles puzzle": find all numbers (with no initial 0) that are written with the same digits as their double (the double of k is 2*k). These numbers are listed in A023086.
The number 0 has been included here for two reasons: (i) we may assume that it satisfies the conditions of the problem vacuously, and (ii) its inclusion allows Schuh to solve the "doubles puzzle". The numbers in A023086 are all permutations of combinations of numbers in this sequence.

			In all the cases below, the first digit must be the largest and all the digits must be distinct.
9 belongs to this list because c(1) = 9 = c(0) and 9 == 2*9 + 1 (mod 10).
63 belongs to this list because c(1) = 6, c(2) = 3 = c(0), 6 == 2*3 (mod 10), and 3 == 2*6 + 1 (mod 10).
512 belongs to this list because 5 == 2*2 + 1 (mod 10), 1 == 2*5 + 1 (mod 10), and 2 == 2*1 (mod 10).
5012 belongs to this list because 5 == 2*2 + 1 (mod 10), 0 == 2*5 (mod 10), 1 == 2*0 + 1 (mod 10), and 2 == 2*1 (mod 10).
62513 belongs to this list because 6 == 2*3 (mod 10), 2 == 2*6 (mod 10), 5 == 2*2 + 1 (mod 10), 1 = 2*5 + 1 (mod 10), and 3 = 2*1 + 1 (mod 10).

Fred Schuh, The Master Book of Mathematical Recreations, Dover, New York, 1968, pp. 31-35.

David A. Corneth and Petros Hadjicostas, PARI program.

Cf. A023086, A336661.

\\ See the Corneth and Hadjicostas link. David A. Corneth, Jul 30 2020

A306474 Composite numbers that are anagrams of the concatenation of their prime factors.

Original entry on oeis.org

735, 1255, 3792, 7236, 11913, 12955, 13175, 17276, 17482, 19075, 19276, 23535, 25105, 32104, 34112, 37359, 42175, 100255, 101299, 104392, 105295, 107329, 117067, 117873, 121325, 121904, 121932, 123544, 123678, 124483, 127417, 129595, 131832, 132565, 139925

Offset: 1

Michel Lagneau, Feb 18 2019

The sequence contains two subsequences:
Subsequence 1: numbers with distinct digits. This finite subsequence begins with the numbers 735, 3792, 7236, 17482, 19075, 19276, 32104, ...
Subsequence 2: numbers with non-distinct digits. This subsequence begins with the numbers 1255, 11913, 12955, 13175, 17276, 23535, ...

			3792 is in the sequence because the concatenation of the prime distinct divisors {2, 3, 79} is 2379, anagram of 3792.

Robert Israel, Table of n, a(n) for n = 1..500

Cf. A023086, A209799.

A121342 is a subsequence.

with(numtheory):
for n from 1 to 140000 do:
if type(n,prime)=false
  then
  x:=factorset(n):n1:=nops(x): s:=0:s0:=0:
    for i from n1 by -1 to 1 do:
     a:=x[i]:b:=length(a):s:=s+a*10^s0:s0:=s0+b:
    od:
      if sort(convert(n, base, 10)) = sort(convert(s, base, 10))
       then
        printf(`%d, `, n):
        else
      fi:fi:
     od:

Select[Range[2,140000],If [!PrimeQ[#],Sort@IntegerDigits@#==Sort[Join@@IntegerDigits[First/@FactorInteger[#]]]]&]

A345392 a(n) is the least k > 1 such that n and k*n have the same set of decimal digits.

Original entry on oeis.org

2, 11, 11, 11, 11, 11, 11, 11, 11, 11, 10, 101, 101, 87, 101, 77, 101, 101, 66, 101, 10, 101, 101, 14, 101, 9, 87, 101, 101, 101, 10, 43, 101, 101, 101, 101, 101, 101, 101, 87, 10, 101, 101, 8, 101, 99, 14, 101, 101, 101, 10, 101, 101, 101, 101, 101, 101, 101

Offset: 0

Rémy Sigrist, Jun 17 2021

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A023086, A345391.

a(n) = if (n==0, return (2)); { my (d=Set(digits(n))); forstep (m=2*n, oo, max(1, n), if (Set(digits(m))==d, return (m/n))) }

def a(n):
    k, ss = 2, set(str(n))
    while set(str(k*n)) != ss: k += 1
    return k
print([a(n) for n in range(58)]) # Michael S. Branicky, Jun 17 2021

a(n) = A345391(n) / n for any n > 0.

a(n) = 2 for any n in A023086.

A133220 a(n) is the smallest positive number k such that k and n*k are anagrams.

Original entry on oeis.org

1, 125874, 1035, 1782, 142857, 1386, 1359, 113967, 1089

Offset: 1

Tanya Khovanova, Oct 11 2007

			3*1035 = 3105 and this is the smallest such number.

Cf. A023086, A023087, A023088, A023089, A023090, A023091, A023092, A023093.

Table[Select[Range[1000000], Sort[IntegerDigits[ # ]] == Sort[IntegerDigits[n# ]] &][[1]], {n, 9}]

A344436 Numbers k such that k, 2k, 3k, 4k, 5k and 6*k are anagrams and no digit of k is zero.

Original entry on oeis.org

142857, 1429857, 14299857, 142999857, 1429999857, 14299999857, 142857142857, 142999999857, 1428571429857, 1429857142857, 1429999999857, 14285714299857, 14298571429857, 14299857142857, 14299999999857, 137428291864557, 137464282918557, 142829186455737

Offset: 1

Bhupendra Kumar Singh, May 19 2021

All terms are divisible by 9.
a(1) = 143*999 = 1287*111;
a(2) = 143*9999 = 1287*1111;
a(7) = 143*999000999 = 1287*111000111; etc.
a(n) = k is odd. Proof: If k is even then 5*k ends in 0, which is forbidden by definition. - David A. Corneth, May 22 2021

			142857, 1429857, and 14299857 are in the sequence:
.
      k        2*k       3*k       4*k       5*k       6*k
  --------  --------  --------  --------  --------  --------
    142857    285714    428571    571428    714285    857142
   1429857   2859714   4289571   5719428   7149285   8579142
  14299857  28599714  42899571  57199428  71499285  85799142

Cf. A023086, A245682, A323711.

isok(k) = {my(d = vecsort(digits(k))); vecmin(d) && (d==vecsort(digits(2*k))) && (d==vecsort(digits(3*k))) && (d==vecsort(digits(4*k))) && (d==vecsort(digits(5*k))) && (d==vecsort(digits(6*k)));} \\ Michel Marcus, Jun 01 2021

Data corrected by David A. Corneth, May 22 2021

A373407 Smallest positive integer k such that no more than n numbers (formed by multiplying k by a digit) are anagrams of k, or -1 if no such number exists.

Original entry on oeis.org

1, 1035, 123876, 1402857, 1037520684, 142857

Offset: 1

Jean-Marc Rebert, Jun 04 2024

For n = 2..6 all terms are divisible by 9.
For n >= 4, a(n) must be divisible by 9, or a(n) = -1, because all anagrams d*k of k for d = 2, 3, 5, 6, 8 and 9 are divisible by 9. Thus there are only 3 values of d, i.e., 1, 4 and 7, for which k*d must not be divisible by 9.
If a(n) exists for n > 1 then 9|a(n). Holds for n = 2 and n = 3 by inspection. Proof for n >= 4: if k*d is an anagram of k where 2 <= d <= 9 then k*d - k = k*(d-1) is a multiple of 9. For this to be true, k must be a multiple of 9 as d is not of the form 1 (mod 3) for all d. - David A. Corneth, Jun 04 2024
From Michael S. Branicky, Jun 07 2024: (Start)
The following were constructed from multiples of cyclic numbers (cf. A180340, Wikipedia):
a(6)  = 142857 = (10^6 - 1) / 7;
a(7) <= 1304347826086956521739 = 3*(10^22 - 1) / 23;
a(8) <= 1176470588235294 = 2*(10^16 - 1) / 17;
a(9) <= 105263157894736842 = 2*(10^18 - 1) / 19. (End)

			a(2) = 1035, because 1035 * 1 = 1035 and 1035 * 3 = 3105 are anagrams of 1035, and no other number 1035 * i with digit i is an anagram of 1035, and no lesser number verifies this property.
Table n, k, set of multipliers.
  1   1          [1]
  2   1035       [1, 3]
  3   123876     [1, 3, 7]
  4   1402857    [1, 2, 3, 5]
  5   1037520684 [1, 2, 4, 5, 8]
  6   142857     [1, 2, 3, 4, 5, 6]

Wikipedia, Cyclic numbers.

Cf. A023086, A023087, A023088, A023089, A023090, A023091, A023092, A023093.
Cf. A245680, A245682.
Cf. A090061.
Cf. A180340, A001913, A036275.

isok(k, n) = my(d=vecsort(digits(k))); sum(i=1, 9, vecsort(digits(k*i)) == d) == n; \\ Michel Marcus, Jun 04 2024

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A336670 Numbers that have decimal expansion c(1)c(2)...c(n) with distinct digits that satisfy c(1) <> 0, c(1) is the largest digit, and for each i in 1..n there is j in {0, 1} such that c(i) == 2*c(i-1) + j (mod 10) (with c(0): = c(n)).

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A306474 Composite numbers that are anagrams of the concatenation of their prime factors.

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A345392 a(n) is the least k > 1 such that n and k*n have the same set of decimal digits.

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A133220 a(n) is the smallest positive number k such that k and n*k are anagrams.

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Mathematica

A344436 Numbers k such that k, 2*k, 3*k, 4*k, 5*k and 6*k are anagrams and no digit of k is zero.

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A373407 Smallest positive integer k such that no more than n numbers (formed by multiplying k by a digit) are anagrams of k, or -1 if no such number exists.

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PARI

A344436 Numbers k such that k, 2k, 3k, 4k, 5k and 6*k are anagrams and no digit of k is zero.