A242372 n^2 in base b is n in base 10 for some base b < n.

121, 144, 169, 268, 274, 280, 319, 400, 441, 484, 519, 574, 900, 961, 1000, 1191, 1331, 8000, 10000, 14641, 100000, 1000000

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..22.

Example

319 is in the sequence because 319^2 in base 184 is 319 in base 10.

Maple

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:
select(f, [$2..100000]); # Robert Israel, Feb 26 2021

Prog

(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

Crossrefs

Sequence in context: A206288 A287390 A258202 * A182072 A262514 A258693

Adjacent sequences: A242369 A242370 A242371 * A242373 A242374 A242375

Keyword

nonn,base,hard

Author

Michel Lagneau, May 12 2014

Extensions

Definition corrected and more terms by Robert Israel, Feb 26 2021

Status

approved