cp's OEIS Frontend

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

%I A337252 #49 Aug 11 2021 01:30:21
%S A337252 8,10,12,14,20,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,
%T A337252 62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,
%U A337252 106,108,110,112,114,116,118,120,122,124,126,128,130,132,134,136,138,140,142,144,146,148,150
%N A337252 Digits of 2^n can be rearranged with no leading zeros to form t^2, for t not a power of 2.
%C A337252 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.
%C A337252 Is it true that all sufficiently large even numbers appear in this list?
%C A337252 22 is a term if leading zeros are allowed. 2^22 = 4194304 and 643^2 = 413449. - _Chai Wah Wu_, Aug 21 2020
%H A337252 Chai Wah Wu, <a href="/A337252/b337252.txt">Table of n, a(n) for n = 1..71</a>
%e A337252 Here are the squares corresponding to the first few powers of 2:
%e A337252 2^8, 25^2
%e A337252 2^10, 49^2
%e A337252 2^12, 98^2
%e A337252 2^14, 178^2
%e A337252 2^20, 1028^2
%e A337252 2^26, 8291^2
%e A337252 2^28, 19112^2
%e A337252 2^30, 33472^2
%e A337252 2^32, 51473^2
%e A337252 2^34, 105583^2
%e A337252 2^36, 129914^2
%e A337252 2^38, 640132^2
%e A337252 2^40, 1081319^2
%e A337252 2^42, 1007243^2
%e A337252 2^44, 3187271^2
%e A337252 2^46, 4058042^2
%e A337252 2^48, 10285408^2
%e A337252 2^50, 32039417^2
%e A337252 2^52, 44795066^2
%e A337252 2^54, 100241288^2
%e A337252 From _Robert Israel_, Aug 21 2020: (Start)
%e A337252 2^56, 142847044^2
%e A337252 2^58, 318068365^2 (End)
%e A337252 From _Chai Wah Wu_, Aug 21 2020: (Start)
%e A337252 2^60, 1000562716^2
%e A337252 2^62, 1000709692^2
%e A337252 2^64, 3164169028^2
%e A337252 2^66, 4498215974^2
%e A337252 2^68, 10061077457^2
%e A337252 2^70, 31624545442^2
%e A337252 2^72, 34960642066^2
%e A337252 2^74, 100786105136^2
%e A337252 2^76, 105467328383^2
%e A337252 2^78, 316579648042^2
%e A337252 2^80, 1000556206526^2
%e A337252 2^82, 1001129296612^2
%e A337252 2^84, 3179799285956^2
%e A337252 2^86, 3333501503458^2
%e A337252 2^88, 10000006273742^2
%e A337252 2^90, 31624717039768^2
%e A337252 2^92, 31640399136637^2
%e A337252 2^94, 100001179435324^2
%e A337252 2^96, 100609261981363^2
%e A337252 2^98, 316227945405958^2
%e A337252 2^100, 1000000068136465^2
%e A337252 2^102, 1000000012839623^2
%e A337252 2^104, 3162279442052185^2
%e A337252 2^106, 3162295238497457^2
%e A337252 2^108, 10006109951303125^2
%e A337252 2^110, 31622778376826465^2
%e A337252 2^112, 31626290060004883^2
%e A337252 2^114, 100005555418898327^2
%e A337252 2^116, 100061093137010524^2
%e A337252 2^118, 316229698532373214^2
%e A337252 2^120, 1000000611139735223^2
%e A337252 2^122, 1005540208662183694^2
%e A337252 2^124, 3179814811220058566^2
%e A337252 2^126, 9994442844707576056^2
%e A337252 2^128, 31605185913938432804^2
%e A337252 2^130, 31799720491491676612^2
%e A337252 2^132, 99999944438762188450^2
%e A337252 2^134, 316052017518707374894^2
%e A337252 2^136, 100055595656929586657^2
%e A337252 2^138, 316227783779026656472^2
%e A337252 2^140, 3162277642424057210351^2
%e A337252 2^142, 1000056109592630240914^2
%e A337252 2^144, 3162279417006463372135^2
%e A337252 2^146, 3162279434557126331437^2
%e A337252 2^148, 10005559566228010636663^2
%e A337252 2^150, 99999999444438629490484^2 (End)
%p A337252 filter:= proc(n) local L,X,S,t,s,x,b;
%p A337252   b:= 2^(n/2);
%p A337252   L:= sort(convert(2^n,base,10));
%p A337252   S:= map(t -> rhs(op(t)), [msolve(X^2=2^n,9)]);
%p A337252   for t from floor(10^((nops(L)-1)/2)/9) to floor(10^(nops(L)/2)/9) do
%p A337252     for s in S do
%p A337252        x:= 9*t+s;
%p A337252        if x = b then next fi;
%p A337252        if sort(convert(x^2,base,10))=L then return true fi;
%p A337252   od od;
%p A337252   false
%p A337252 end proc:
%p A337252 select(filter, [seq(i,i=2..58,2)]); # _Robert Israel_, Aug 21 2020
%o A337252 (Python)
%o A337252 from math import isqrt
%o A337252 def ok(n, verbose=True):
%o A337252     s = str(2**n)
%o A337252     L, target, hi = len(s), sorted(s), int("".join(sorted(s, reverse=True)))
%o A337252     if '0' not in s: lo = int("".join(target))
%o A337252     else:
%o A337252         lownzd, targetcopy = min(set(s) - {'0'}), target[:]
%o A337252         targetcopy.remove(lownzd)
%o A337252         rest = "".join(targetcopy)
%o A337252         lo = int(lownzd + rest)
%o A337252     for r in range(isqrt(lo), isqrt(hi)+1):
%o A337252         rr = r*r
%o A337252         if sorted(str(rr)) == target:
%o A337252             brr = bin(rr)[2:]
%o A337252             if brr != '1' + '0'*(len(brr)-1):
%o A337252                 if verbose: print(f"2^{n}, {r}^2")
%o A337252                 return r
%o A337252     return 0
%o A337252 print(list(filter(ok, range(2, 73, 2)))) # _Michael S. Branicky_, Aug 10 2021
%Y A337252 Cf. A069656, A235993, A337261.
%K A337252 nonn,base
%O A337252 1,1
%A A337252 _Jeffrey Shallit_, Aug 21 2020
%E A337252 56 and 58 added by _Robert Israel_, Aug 21 2020
%E A337252 a(23)-(68) from _Chai Wah Wu_, Aug 21 2020