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A137110 Numbers k such that k and k^2 use only the digits 2, 5, 6 and 7.

%I A137110 #45 Sep 11 2024 22:52:23
%S A137110 5,25,26,75,76,275,525,526,725,7525,27525,72576,256266,276725,725725,
%T A137110 276726675,756652275
%N A137110 Numbers k such that k and k^2 use only the digits 2, 5, 6 and 7.
%C A137110 Generated with DrScheme.
%C A137110 If it exists, a(18) > 10^34. - _David A. Corneth_ and _Michael S. Branicky_, May 25 2021
%C A137110 From _Pontus von Brömssen_, May 01 2024: (Start)
%C A137110 a(18) > 2*10^43 (if it exists).
%C A137110 If k = x*10^m is a term where 1 < x < 10 and k is not 25 or 76, then 2.5622756725665225652662672277265762225525525 < x < 7.5665565267667667557762552666757226626652626.
%C A137110 (End)
%H A137110 David A. Corneth, <a href="/A137110/a137110.gp.txt">PARI program</a>
%H A137110 Jonathan Wellons, <a href="https://web.archive.org/web/20090206165028/http://jonathanwellons.com/shared-digits/">Tables of Shared Digits</a> [archived].
%e A137110 276726675^2 = 76577652656555625.
%o A137110 (Python)
%o A137110 def auptod(maxdigits, only="2567"):
%o A137110   aset, digset, valid = set(), set(only), set(only)
%o A137110   for e in range(1, maxdigits+1):
%o A137110     newvalid = set()
%o A137110     for tstr in valid:
%o A137110       t = int(tstr)
%o A137110       if set(str(t**2)) <= digset: aset.add(t)
%o A137110       for d in digset:
%o A137110         dtstr = d + tstr
%o A137110         dt = int(dtstr)
%o A137110         remstr = str(dt**2)[-e-1:]
%o A137110         if set(remstr) <= digset: newvalid.add(dtstr)
%o A137110     valid = newvalid
%o A137110   return sorted(aset)
%o A137110 print(auptod(16)) # _Michael S. Branicky_, May 25 2021
%o A137110 (PARI) \\ See PARI link. _David A. Corneth_, May 25 2021
%K A137110 base,nonn,more
%O A137110 1,1
%A A137110 Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008