A053057 Squares whose digit sum is also a square.

0, 1, 4, 9, 36, 81, 100, 121, 144, 169, 196, 225, 324, 400, 441, 484, 529, 900, 961, 1521, 1681, 2025, 2304, 2601, 3364, 3481, 3600, 4489, 4624, 5776, 5929, 7225, 7396, 8100, 8836, 9025, 10000, 10201, 10404, 10609, 10816, 11025, 12100, 12321, 12544, 12769

Offset: 1

Author

Felice Russo, Feb 25 2000

Keywords

• nonn
• easy
• base

Comments

The numbers 81, 100, 121, 144, 169, 196, 225 are seven consecutive squares belonging to this sequence. The next set of seven consecutive squares whose digit sum is also a square is {9999^2, 10000^2, 10001^2, 10002^2, 10003^2, 10004^2, 10005^2}. (See Crux Mathematicorum link.) - Bernard Schott, May 24 2017

The first set of 8 consecutive squares begin at 46045846^2. This was already known in 2016, see MathStackExchange link. - Michel Marcus, May 25 2017

The first run of 9 consecutive squares starts at 302260461719025^2. - Giovanni Resta, Jun 08 2017

Examples

			144 is a term: 144 = 12^2 and 1 + 4 + 4 = 9 = 3^2. - _Bernard Schott_, May 24 2017
		

References

• Felice Russo, A set of new Smarandache functions, sequences and conjectures in number theory, American Research Press, 2000.

Links

• Daniel Mondot, Table of n, a(n) for n = 1..1192
• Allan Wm. Johnson Jr., Problem 443, Crux Mathematicorum, Vol. 6, No. 3 (Mar. 1980), page 88.
• MathStackExchange, Numbers n such that the digit sum of n^2 is a square, 2015-2016.

Crossrefs

Subsequence of A000290.

Cf. A028839, A061910.

Programs

Magma

[n^2: n in [0..120] | IsSquare(&+Intseq(n^2))];  // Bruno Berselli, May 26 2011

Mathematica

Select[Range[0,115]^2, IntegerQ[Sqrt[DigitSum[#]]]&] (* Stefano Spezia, Mar 07 2024 *)

PARI

lista(nn) = for (n=1, nn, if (issquare(sumdigits(n^2)), print1(n^2, ", "));); \\ Michel Marcus, May 25 2017

Extensions

More terms from James Sellers, Feb 28 2000