A339996 - OEIS

%I #45 Jan 01 2021 12:27:47

%S 0,1,3,4,9,13,14,31,33,41,83,99,103,104,109,141,247,263,264,283,301,

%T 303,333,401,436,437,446,447,781,813,836,901,947,949,954,959,999,1003,

%U 1004,1009,1053,1054,1291,1349,1367,2467,2486,2609,2849,2949,2986,3001

%N Numbers whose square is rotationally ambigrammatic with no trailing zeros.

%C A rotationally ambigrammatic number (A045574) is one that can be rotated by 180 degrees resulting in a readable, most often new number. Such numbers, by definition, can only contain the digits 0, 1, 6, 8, 9.

%C If the number once rotated happens to be the same number it is a strobogrammatic number (A000787); such numbers form a subset of the ambigrammatic numbers.

%C Numbers (such as 10) whose square has trailing zeros have been excluded because the rotation of such a number by 180 degrees would result in a number with leading zeros. Typically this is not the way we write numbers.

%C The numbers 14 and 31 are interesting numbers in this sequence in that when their square is rotated 180 degrees, the square root results in the other number. I believe this is unique to only these two numbers.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ambigram">Ambigram</a>

%F a(n) = sqrt(A340164(n)).

%e 13^2 = 169. A rotationally ambigrammatic number. Hence, 13 is a term.

%e 15^2 = 225. Not rotationally ambigrammatic and hence 15 is not a term.

%e 10^2 = 100. This number has trailing zeros, so under this definition of rotationally ambigrammatic, 10 is not a term.

%t Select[Range[0, 4001], (# == 0 || !Divisible[#, 10]) && AllTrue[IntegerDigits[#^2], MemberQ[{0, 1, 6, 8, 9}, #1] &] &] (* _Amiram Eldar_, Dec 26 2020 *)

%o (PARI) isra(n) = (n%10) && (!setminus(Set(Vec(Str(n))), Vec("01689"))) || !n; \\ A045574

%o isok(n) = isra(n^2); \\ _Michel Marcus_, Dec 27 2020

%Y Cf. A045574, A340164 (squares).

%K nonn,base

%O 1,3

%A _Philip Mizzi_, Dec 25 2020