A383821 3-automorphic numbers: positive integers k such that 3k^2 ends with k.

2, 5, 7, 67, 75, 92, 667, 792, 875, 6667, 6875, 9792, 66667, 69792, 96875, 296875, 369792, 666667, 2369792, 4296875, 6666667, 62369792, 66666667, 262369792, 404296875, 666666667, 6666666667, 7262369792, 9404296875, 27262369792, 39404296875, 66666666667, 639404296875

Offset: 1

Author

Shyam Sunder Gupta, May 11 2025

Keywords

• nonn
• base

Comments

All 3-automorphic numbers end in 2, 5, or 7 only.

From Michael S. Branicky, May 11 2025: (Start)

Terms of successively larger digits can be created by prepending digits on the left of previous terms; for each length, only 3 positive such "seeds" are valid (some may have leading zeros and thus do not contribute terms at that length).

Infinite since 6..67, with i 6's and then a 7 is a term for all i >= 0.

a(2774) has 1001 digits. (End)

Examples

			67 is in the sequence because 3*67^2 = 13467 which ends with 67.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..2773
• Shyam Sunder Gupta, Elegance of Squares, Cubes, and Higher Powers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 2, 29-81.

Crossrefs

Cf. A003226, A033428.

Essentially the union of A030985, A030986, and A067275.

Programs

Mathematica

Select[Range[10^7],IntegerDigits[#]==Take[IntegerDigits[3#^2],-IntegerLength[#]]&] (* James C. McMahon, May 16 2025 *)

Extensions

More terms from Michael S. Branicky, May 11 2025