A171671 Square numbers not of form m + sum of digits of m.

1, 9, 64, 121, 400, 5776, 6889, 7396, 8836, 9409, 10816, 12100, 17689, 18769, 27556, 29929, 30976, 33856, 34969, 37636, 49729, 65536, 69169, 69696, 70756, 75076, 75625, 76729, 80656, 110224, 124609, 126736, 132496, 134689, 156816, 162409

Offset

1,2

Comments

We may call these numbers the self or Colombian squares. Subsequence of A003052. There are 446 such self squares < 2*10^7, 218 odd and 228 even.

Kaprekar (1963) introduced these numbers and called them self-square numbers. - N. J. A. Sloane, Oct 30 2014

References

D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately Printed, 311 Devlali Camp, Devlali, India, 1963.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.

D. R. Kaprekar, The Mathematics of the New Self Numbers. [annotated and scanned]

Zak Seidov, Self squares.

Formula

a(n) = A171672(n)^2. - Amiram Eldar, Mar 26 2025

Mathematica

A062028=Table[n+Total[IntegerDigits[n]], {n, 0, 20000000}];
se=Select[Complement[Range[0, 20000000], A062028], IntegerQ[Sqrt[ # ]]&]

Crossrefs

Intersection of A000290 and A003052 (self or Colombian numbers).

Cf. A171672 (m^2 are self numbers), A062028 (a(n) = n + sum of the digits of n), A171673 (n and n^2 are self numbers), A382166.

Sequence in context: A302975 A165447 A050792 * A016886 A099761 A018201

Adjacent sequences: A171668 A171669 A171670 * A171672 A171673 A171674

Keyword

base,nonn

Author

Zak Seidov, Dec 15 2009

Extensions

Changed the word "safe" in this entry to "self". - N. J. A. Sloane, Feb 26 2017

Status

approved