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

Author

Jeffrey Shallit, Aug 21 2020

Keywords

• nonn
• base

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

Examples

			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)
		

Links

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

Crossrefs

Cf. A069656, A235993, A337261.

Programs

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

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

Extensions

56 and 58 added by Robert Israel, Aug 21 2020

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