A035014 et al.: unique multiple of $2^n$ with $n$ digits in {A, B}

These sequences (mostly in the range A053312-A053338 and A053376-A053380) are of the form

  • a(n) = unique number having $n$ digits A or B which is divisible by $2^n$.

where 0 < A < B < 10 are two nonzero digits with opposite parity.

It is somewhat remarkable that this definition defines a(n) uniquely (i.e., at no length $n$, there will ever be any way to replace a certain number of ‘B’s by ‘A’ in order to get a smaller multiple of $2^n$).
Also note that they are not “the same up to substitution of the digits”.

FORMULA: a(n+1) = a(n) + $10^n \cdot (A \text{ if } a(n)/2^n \text{ is odd, else } B)$, assuming A is the odd digit.

Program (PARI/GP, others to follow)

/* Shortest possibility in the 3rd vector arg is to use d[!(t%2^k)+1]
 * which is = d[2] if !t%2^k <=> t%2^k == 0 <=> need even digit, = d[1] else.
 * So the odd digit must come first.
 */
first(n, d=[1,2], t=0)= d[2]%2 && d=Vecrev(d); vector(n, k, t += d[!(t%2^k)+1]*10^(k-1))

/* function Axxx(n) that only returns the n-th term (less efficient if many terms are needed) */
a(n, d=[1,2], t=0)= d[2]%2 && d=Vecrev(d); for(k=1,n, t += d[!(t%2^k)+1]*10^(k-1)); t

/* explicit exhaustive search (this works only for digits [d,d+1], use e.g. vecextract for other cases) */
find(n,D=[7,8])=forvec(d=vector(n,i,D),fromdigits(d)%2^n||print(d)) \\ not return() to see all solutions

Table of the sequences for given digits A, B:

  • 1,2 : A053312 = 2, 12, 112, 2112, 22112, 122112, 2122112, 12122112, 212122112, 1212122112
  • 1,4 : A053314 = 4, 44, 144, 4144, 14144, 414144, 1414144, 41414144, 441414144, 1441414144
  • 2,3 : A053316 = 2, 32, 232, 3232, 23232, 223232, 2223232, 32223232, 232223232, 3232223232
  • 2,5 : A053317 = 2, 52, 552, 5552, 55552, 255552, 5255552, 55255552, 255255552, 2255255552
  • 2,7 : A053318 = 2, 72, 272, 2272, 22272, 222272, 7222272, 27222272, 727222272, 2727222272
  • 2,9 : A053313 = 2, 92, 992, 2992, 92992, 292992, 2292992, 22292992, 222292992, 2222292992
  • 3,4 : A035014 = 4, 44, 344, 3344, 33344, 433344, 3433344, 33433344, 333433344, 3333433344
  • 4,5 : A053315 = 4, 44, 544, 4544, 44544, 444544, 4444544, 54444544, 454444544, 5454444544
  • 4,7 : A053332 = 4, 44, 744, 7744, 47744, 447744, 4447744, 44447744, 444447744, 4444447744
  • 4,9 : A053333 = 4, 44, 944, 4944, 94944, 994944, 4994944, 94994944, 494994944, 9494994944

  • 1,6 : A053334 = 6, 16, 616, 1616, 11616, 111616, 6111616, 16111616, 616111616, 1616111616
  • 3,6 : A053335 = 6, 36, 336, 6336, 66336, 366336, 6366336, 36366336, 636366336, 3636366336
  • 5,6 : A053336 = 6, 56, 656, 6656, 66656, 566656, 6566656, 66566656, 666566656, 6666566656
  • 6,7 : A053337 = 6, 76, 776, 7776, 67776, 667776, 6667776, 66667776, 766667776, 6766667776
  • 6,9 : A053338 = 6, 96, 696, 9696, 69696, 669696, 6669696, 96669696, 696669696, 9696669696

  • 1,8 : A053376 = 8, 88, 888, 1888, 81888, 181888, 8181888, 18181888, 118181888, 8118181888
  • 3,8 : A053377 = 8, 88, 888, 3888, 33888, 333888, 3333888, 83333888, 383333888, 3383333888
  • 5,8 : A053378 = 8, 88, 888, 5888, 85888, 885888, 8885888, 58885888, 558885888, 8558885888
  • 7,8 : A053379 = 8, 88, 888, 7888, 77888, 877888, 7877888, 87877888, 787877888, 8787877888
  • 8,9 : A053380 = 8, 88, 888, 9888, 89888, 989888, 9989888, 89989888, 989989888, 8989989888