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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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified August 10 12:58 EDT 2022. Contains 356039 sequences. (Running on oeis4.)