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A242372
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n^2 in base b is n in base 10 for some base b < n.
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0
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121, 144, 169, 268, 274, 280, 319, 400, 441, 484, 519, 574, 900, 961, 1000, 1191, 1331, 8000, 10000, 14641, 100000, 1000000
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OFFSET
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1,1
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COMMENTS
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The corresponding bases b are 120, 142, 166, 188, 192, 196, 184, 200, 220, 241, 232, 256, 300, 320, 100, 112, 120, ...
The squares of the sequence are 121, 144, 169, 400, 441, 484, 900, ...
Includes 10^k for all k > 2, as 10^(2k) in base 100 is 10^k in base 10. - Robert Israel, Feb 26 2021
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LINKS
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EXAMPLE
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319 is in the sequence because 319^2 in base 184 is 319 in base 10.
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MAPLE
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for n from 1 to 2000 do: x:=convert(n, base, 10): for b from 2 to n-1 do: y:=convert(n^2, base, b): if x=y then printf ( "%d %d \n", n, b) fi:od:od:
# Alternative:
f:= proc(n) local L, k, x, P, S;
L:= convert(n, base, 10);
P:= add(L[k]*x^(k-1), k=1..nops(L)) - n^2;
S:= map(t -> rhs(op(t)), [isolve(P)]);
ormap(t -> t > max(L) and t < n , S);
end proc:
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PROG
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(PARI) isok(m) = {for (b=2, m-1, my(d=digits(m^2, b)); if ((vecmax(d) < 10) && (fromdigits(d) == m), return (1)); ); } \\ Michel Marcus, Feb 27 2021
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CROSSREFS
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KEYWORD
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nonn,base,hard
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AUTHOR
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EXTENSIONS
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Definition corrected and more terms by Robert Israel, Feb 26 2021
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STATUS
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approved
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