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