A366106 Primes that are the concatenation of three squares in base 10.

101, 109, 149, 191, 199, 401, 409, 419, 449, 491, 499, 911, 919, 941, 991, 1049, 1181, 1259, 1361, 1481, 1499, 1601, 1609, 1619, 1699, 1811, 1949, 2549, 2591, 3691, 4049, 4259, 4481, 4649, 4909, 4919, 4999, 6449, 6491, 8101, 8111, 8191, 9049, 9161, 9181, 9491, 9649, 9811, 9949, 10009, 10091

Offset

1,1

Comments

The three squares need not be distinct.

At least one of the squares must be divisible by 9.

The first term that is a concatenation of three squares in two different ways is 14411, the concatenation of 1 = 1^2, 441 = 21^2 and 1 = 1^2 and also 144 = 12^2, 1 = 1^2 and 1 = 1^2.

The first term that is a concatenation of three squares in three different ways is 1961441, the concatenation of 196 = 14^2, 144 = 12^2 and 1 = 1^2, of 196, 1 and 441 = 21^2, and of 1, 961 = 31^2 and 441.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(16) = 1049 is a term because it is the concatenation of 1 = 1^2, 0 = 0^2 and 49 = 7^2.

Maple

M:= 5: # for terms < 10^M
S:= {}:
for a from 1 while a^2 < 10^(M-2) do
  x:= a^2; mx:= length(x);
  for b from 0 while b^2 < 10^(M-1-mx) do
    y:= b^2; my:= max(1, length(y));
    for c from 0 while c^2 < 10^(M-mx-my) do
      v:= parse(cat(x, y, c^2));
      if isprime(v) then S:= S union {v} fi;
od od od:
sort(convert(S, list));

Mathematica

a[maxSquareIndex_Integer?Positive]:=Select[Flatten[Table[ToExpression[IntegerString[a^2]<>IntegerString[b^2]<>IntegerString[c^2]], {a, 1, maxSquareIndex}, {b, 0, maxSquareIndex}, {c, 0, maxSquareIndex}]], PrimeQ]//Sort; a[10][[1;; 51]] (* Robert P. P. McKone, Oct 02 2023 *)

Crossrefs

Cf. A167535.

Sequence in context: A118773 A092628 A107219 * A140799 A086918 A259300

Adjacent sequences: A366103 A366104 A366105 * A366107 A366108 A366109

Keyword

nonn,base

Author

Robert Israel, Sep 29 2023

Status

approved