A118881 Square of sum of decimal digits of n.

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 36, 49, 64, 81, 100, 121, 144

Offset: 0

Author

Giovanni Teofilatto, May 25 2006

Keywords

• base
• easy
• nonn

Comments

a(k) = k iff k = 0, 1, 81; also, the only solution to the double equation a(k) = m and a(m) = k with k < m is (169, 256) (proof in Diophante link, 2ème jonglerie). - Bernard Schott, Mar 08 2021

If the map i->a(i) is iterated starting at i = n, the trajectory will eventually reach one ot the three cycles (1) (if n == +-1 mod 9), (169,256) (if n == 2,4,5,7 mod 9), or (81) (if n == 0 mod 3). - N. J. A. Sloane, Mar 17 2025

Examples

			From _R. J. Mathar_, Jul 08 2012: (Start)
Trajectories of the map x->a(x), A177148:
1 ->1 ->1 ->1 ->1 ->1 ->1 ->1 ->1 ->...
2 ->4 ->16 ->49 ->169 ->256 ->169 ->256 ->169 ->...
3 ->9 ->81 ->81 ->81 ->81 ->81 ->81 ->81 ->...
4 ->16 ->49 ->169 ->256 ->169 ->256 ->169 ->256 ->...
5 ->25 ->49 ->169 ->256 ->169 ->256 ->169 ->256 ->...
6 ->36 ->81 ->81 ->81 ->81 ->81 ->81 ->81 ->...
7 ->49 ->169 ->256 ->169 ->256 ->169 ->256 ->169 ->...
8 ->64 ->100 ->1 ->1 ->1 ->1 ->1 ->1 ->... (End)
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..10000
• Diophante, A1915, Jongleries n°2 avec les chiffres (in French).
• Michael Penn, Squaring the sum of digits, YouTube video, 2021.

Crossrefs

Cf. A007953.

Programs

Maple

read("transforms") :
A118881 := proc(n)
        digsum(n)^2 ;
end proc: # R. J. Mathar, Jul 08 2012

Mathematica

Table[Total[IntegerDigits[n]]^2,{n,0,70}] (* Harvey P. Dale, Jul 31 2012 *)

PARI

a(n) = sumdigits(n)^2; \\ Michel Marcus, Mar 08 2021

Python

def a(n): return sum(map(int, str(n)))**2
print([a(n) for n in range(67)]) # Michael S. Branicky, Nov 19 2021

Formula

a(n) = A007953(n)^2. [R. J. Mathar, Apr 22 2010]