A337252 Digits of 2^n can be rearranged with no leading zeros to form t^2, for t not a power of 2.

8, 10, 12, 14, 20, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150

Offset

1,1

Comments

n has to be even, since odd powers of 2 are congruent to 2,5,8 mod 9, while squares are congruent to 0,1,4,7 mod 9, and two numbers whose digits are rearrangements of each other are congruent modulo 9.

Is it true that all sufficiently large even numbers appear in this list?

22 is a term if leading zeros are allowed. 2^22 = 4194304 and 643^2 = 413449. - Chai Wah Wu, Aug 21 2020

Links

Chai Wah Wu, Table of n, a(n) for n = 1..71

Example

Here are the squares corresponding to the first few powers of 2:
2^8, 25^2
2^10, 49^2
2^12, 98^2
2^14, 178^2
2^20, 1028^2
2^26, 8291^2
2^28, 19112^2
2^30, 33472^2
2^32, 51473^2
2^34, 105583^2
2^36, 129914^2
2^38, 640132^2
2^40, 1081319^2
2^42, 1007243^2
2^44, 3187271^2
2^46, 4058042^2
2^48, 10285408^2
2^50, 32039417^2
2^52, 44795066^2
2^54, 100241288^2
From Robert Israel, Aug 21 2020: (Start)
2^56, 142847044^2
2^58, 318068365^2 (End)
From Chai Wah Wu, Aug 21 2020: (Start)
2^60, 1000562716^2
2^62, 1000709692^2
2^64, 3164169028^2
2^66, 4498215974^2
2^68, 10061077457^2
2^70, 31624545442^2
2^72, 34960642066^2
2^74, 100786105136^2
2^76, 105467328383^2
2^78, 316579648042^2
2^80, 1000556206526^2
2^82, 1001129296612^2
2^84, 3179799285956^2
2^86, 3333501503458^2
2^88, 10000006273742^2
2^90, 31624717039768^2
2^92, 31640399136637^2
2^94, 100001179435324^2
2^96, 100609261981363^2
2^98, 316227945405958^2
2^100, 1000000068136465^2
2^102, 1000000012839623^2
2^104, 3162279442052185^2
2^106, 3162295238497457^2
2^108, 10006109951303125^2
2^110, 31622778376826465^2
2^112, 31626290060004883^2
2^114, 100005555418898327^2
2^116, 100061093137010524^2
2^118, 316229698532373214^2
2^120, 1000000611139735223^2
2^122, 1005540208662183694^2
2^124, 3179814811220058566^2
2^126, 9994442844707576056^2
2^128, 31605185913938432804^2
2^130, 31799720491491676612^2
2^132, 99999944438762188450^2
2^134, 316052017518707374894^2
2^136, 100055595656929586657^2
2^138, 316227783779026656472^2
2^140, 3162277642424057210351^2
2^142, 1000056109592630240914^2
2^144, 3162279417006463372135^2
2^146, 3162279434557126331437^2
2^148, 10005559566228010636663^2
2^150, 99999999444438629490484^2 (End)

Maple

filter:= proc(n) local L, X, S, t, s, x, b;
  b:= 2^(n/2);
  L:= sort(convert(2^n, base, 10));
  S:= map(t -> rhs(op(t)), [msolve(X^2=2^n, 9)]);
  for t from floor(10^((nops(L)-1)/2)/9) to floor(10^(nops(L)/2)/9) do
    for s in S do
       x:= 9*t+s;
       if x = b then next fi;
       if sort(convert(x^2, base, 10))=L then return true fi;
  od od;
  false
end proc:
select(filter, [seq(i, i=2..58, 2)]); # Robert Israel, Aug 21 2020

Prog

(Python)
from math import isqrt
def ok(n, verbose=True):
    s = str(2**n)
    L, target, hi = len(s), sorted(s), int("".join(sorted(s, reverse=True)))
    if '0' not in s: lo = int("".join(target))
    else:
        lownzd, targetcopy = min(set(s) - {'0'}), target[:]
        targetcopy.remove(lownzd)
        rest = "".join(targetcopy)
        lo = int(lownzd + rest)
    for r in range(isqrt(lo), isqrt(hi)+1):
        rr = r*r
        if sorted(str(rr)) == target:
            brr = bin(rr)[2:]
            if brr != '1' + '0'*(len(brr)-1):
                if verbose: print(f"2^{n}, {r}^2")
                return r
    return 0
print(list(filter(ok, range(2, 73, 2)))) # Michael S. Branicky, Aug 10 2021

Crossrefs

Cf. A069656, A235993, A337261.

Sequence in context: A008557 A161425 A096171 * A154786 A335013 A249628

Adjacent sequences: A337249 A337250 A337251 * A337253 A337254 A337255

Keyword

nonn,base

Author

Jeffrey Shallit, Aug 21 2020

Extensions

56 and 58 added by Robert Israel, Aug 21 2020

a(23)-(68) from Chai Wah Wu, Aug 21 2020

Status

approved