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

Author

Zak Seidov, Dec 15 2009

Keywords

• base
• nonn

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.

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.

Programs

Mathematica

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

Formula

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

Extensions

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