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

1,3

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

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.

Example

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

Mathematica

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

Prog

(Magma) [n^2: n in [0..120] | IsSquare(&+Intseq(n^2))];  // Bruno Berselli, May 26 2011
(PARI) lista(nn) = for (n=1, nn, if (issquare(sumdigits(n^2)), print1(n^2, ", ")); ); \\ Michel Marcus, May 25 2017

Crossrefs

Subsequence of A000290.

Cf. A028839, A061910.

Sequence in context: A267430 A117756 A326182 * A118547 A363408 A115700

Adjacent sequences: A053054 A053055 A053056 * A053058 A053059 A053060

Keyword

nonn,easy,base

Author

Felice Russo, Feb 25 2000

Extensions

More terms from James A. Sellers, Feb 28 2000

Status

approved