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A069512 Geometric mean of digits = 2 and digits are in nondecreasing order.

%I A069512 #10 Feb 14 2021 19:25:03
%S A069512 2,14,22,118,124,222,1128,1144,1224,2222,11148,11228,11244,12224,
%T A069512 22222,111188,111248,111444,112228,112244,122224,222222,1111288,
%U A069512 1111448,1112248,1112444,1122228,1122244,1222224,2222222,11111488,11112288,11112448,11114444
%N A069512 Geometric mean of digits = 2 and digits are in nondecreasing order.
%C A069512 No number is obtainable by permuting the digits of other members - only one with ascending order of digits is included. Product of the digits = 2^k where k is the number of digits.
%H A069512 Michael S. Branicky, <a href="/A069512/b069512.txt">Table of n, a(n) for n = 1..10000</a>
%e A069512 1128 is a term but 2118 is not.
%t A069512 a = {}; b = 2; Do[c = Apply[ Times, IntegerDigits[n]]/b^Floor[ Log[10, n] + 1]; If[c == 1 && Position[a, FromDigits[ Sort[ IntegerDigits[n]]]] == {}, Print[n]; a = Append[a, n]], {n, 1, 10^7}]
%o A069512 (Python)
%o A069512 from math import prod
%o A069512 from sympy.utilities.iterables import multiset_combinations
%o A069512 def aupton(terms):
%o A069512   n, digits, alst, powsexps2 = 0, 1, [], [(1,0), (2,1), (4,2), (8,3)]
%o A069512   while n < terms:
%o A069512     target = 2**digits
%o A069512     mcstr = "".join(str(d)*(digits//max(1, r)) for d, r in powsexps2)
%o A069512     for mc in multiset_combinations(mcstr, digits):
%o A069512       if prod(map(int, mc)) == target:
%o A069512         n += 1
%o A069512         alst.append(int("".join(mc)))
%o A069512         if n == terms: break
%o A069512     else: digits += 1
%o A069512   return alst
%o A069512 print(aupton(34)) # _Michael S. Branicky_, Feb 14 2021
%Y A069512 Cf. A061426, A069516, A069518.
%K A069512 nonn,base
%O A069512 1,1
%A A069512 _Amarnath Murthy_, Mar 30 2002
%E A069512 Edited and extended by _Robert G. Wilson v_, Apr 01 2002
%E A069512 a(31) corrected by and a(33) and beyond from _Michael S. Branicky_, Feb 14 2021