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A029783 Exclusionary squares: numbers n such that no digit of n is present in n^2.

%I A029783 #62 Dec 10 2024 03:23:39
%S A029783 2,3,4,7,8,9,17,18,22,24,29,33,34,38,39,44,47,53,54,57,58,59,62,67,72,
%T A029783 77,79,84,88,92,94,144,157,158,173,187,188,192,194,209,212,224,237,
%U A029783 238,244,247,253,257,259,307,313,314,333,334,338,349,353,359
%N A029783 Exclusionary squares: numbers n such that no digit of n is present in n^2.
%C A029783 Complement of A189056; A076493(a(n)) = 0. - _Reinhard Zumkeller_, Apr 16 2011
%C A029783 A258682(a(n)) = a(n)^2. - _Reinhard Zumkeller_, Jun 07 2015
%C A029783 a(78) = 567 and a(112) = 854 are the only two numbers k such that the equation k^2 = m uses only once each of the digits 1 to 9 (reference David Wells). Exactly: 567^2 = 321489, and, 854^2 = 729316 (see A059930). - _Bernard Schott_, Jan 28 2021
%D A029783 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
%D A029783 David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, 1997, page 144, entry 567.
%H A029783 Giovanni Resta, <a href="/A029783/b029783.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Reinhard Zumkeller)
%H A029783 Michael S. Branicky, <a href="/A029783/a029783.py.txt">Python program</a>
%H A029783 Cliff Pickover et al, <a href="https://groups.google.com/forum/#!topic/rec.puzzles/ubSItPD_DGY">Exclusionary Squares and Cubes</a>, rec.puzzles topic on google groups, January 2002
%e A029783 From _M. F. Hasler_, Oct 16 2018: (Start)
%e A029783 It is easy to construct infinite subsequences of the form S(a,b)(n) = a*R(n) + b, where R(n) = (10^n-1)/9 is the repunit of length n. Among these are:
%e A029783 S(3,0) = (3, 33, 333, ...), S(3,1) = (4, 34, 334, 3334, ...), S(3,5) = (8, 38, 338, ...), also b = 26, 44, 434, ... (with a = 3); S(6,1) = (7, 67, 667, ...), S(6,6) = (72, 672, 6672, ...) (excluding n=1), S(6,7) = (673, 6673, ...) (excluding also n=2 here), S(6,-7) = (59, 659, 6659, ...), and others. (End)
%t A029783 Select[Range[1000], Intersection[IntegerDigits[ # ], IntegerDigits[ #^2]] == {} &] (* _Tanya Khovanova_, Dec 25 2006 *)
%o A029783 (Haskell)
%o A029783 a029783 n = a029783_list !! (n-1)
%o A029783 a029783_list = filter (\x -> a258682 x == x ^ 2) [1..]
%o A029783 -- _Reinhard Zumkeller_, Jun 07 2015, Apr 16 2011
%o A029783 (PARI) is_A029783(n)=!#setintersect(Set(digits(n)),Set(digits(n^2))) \\ _M. F. Hasler_, Oct 16 2018
%o A029783 (Python) # see linked program
%o A029783 (Python)
%o A029783 from itertools import count, islice
%o A029783 def A029783_gen(startvalue=0): # generator of terms >= startvalue
%o A029783     return filter(lambda n:not set(str(n))&set(str(n**2)),count(max(startvalue,0)))
%o A029783 A029783_list = list(islice(A029783_gen(),30)) # _Chai Wah Wu_, Feb 12 2023
%Y A029783 Cf. A059930 (n and n^2 use different digits), A112736 (numbers whose squares are exclusionary).
%Y A029783 Cf. A029784, A029785, A029786, A111116, A113316, A189056, A076493, A258682.
%K A029783 nonn,base
%O A029783 1,1
%A A029783 _Patrick De Geest_
%E A029783 Definition slightly reworded at the suggestion of _Franklin T. Adams-Watters_ by _M. F. Hasler_, Oct 16 2018