A045914 Triangular numbers with all digits the same.

0, 1, 3, 6, 55, 66, 666

Offset: 1

Author

Felice Russo

Keywords

• fini
• full
• nonn
• base

Comments

Escott (1905) proved that there are no more terms with fewer than 30 digits. The complete proof that there are no more terms was given by Ballew and Weger (1972). - Amiram Eldar, Jan 22 2022

Hercher and Fegert pointed out that the proof by Ballew and Weger was flawed, and provided an alternative proof (2025). - Seiichi Azuma, Mar 25 2025

References

• L. E. Dickson, History of the Theory of Numbers, Vol. II, p. 33, Chelsea NY, 1952.
• E. B. Escott, Math. Quest. Educational Times, New Series, Vol. 8 (1905), pp. 33-34. - N. J. A. Sloane, Mar 31 2014

Links

• David W. Ballew and Ronald C. Weger, Triangular Numbers with Repeated Digits, Proc. S. D. Acad. Sci., Vol. 51 (1972), pp. 52-55.
• David W. Ballew and Ronald C. Weger, Repdigit triangular numbers, J. Rec. Math., Vol. 8, No. 2 (1975-76), pp. 96-98.
• Christian Hercher and Karl Fegert, Triangular Numbers With a Single Repeated Digit, Journal of Integer Sequences, Vol. 28 (2025), Article 25.2.1.
• Bir Kafle, Florian Luca and Alain Togbé, Triangular Repblocks, Fibonacci Quart., Vol. 56, No. 4 (2018), pp. 325-328.
• C. E. Youngman, Problem 15648, Educational Times, Vol. 58, 1905, p. 87; with a solution by E. B. Escott.

Crossrefs

Cf. A213516 (triangular numbers having only 1 or 2 different digits).

Cf. A118668.

Programs

Mathematica

Select[Union[Flatten[Table[FromDigits[PadRight[{},n,k]],{n,3},{k,0,9}]]],OddQ[ Sqrt[8#+1]]&] (* Harvey P. Dale, Feb 11 2020 *)

Formula

A118668(a(n)) = 1. - Reinhard Zumkeller, Jul 11 2015

Extensions

0 inserted by T. D. Noe, Jun 22 2012