A004159 Sum of digits of n^2.

0, 1, 4, 9, 7, 7, 9, 13, 10, 9, 1, 4, 9, 16, 16, 9, 13, 19, 9, 10, 4, 9, 16, 16, 18, 13, 19, 18, 19, 13, 9, 16, 7, 18, 13, 10, 18, 19, 13, 9, 7, 16, 18, 22, 19, 9, 10, 13, 9, 7, 7, 9, 13, 19, 18, 10, 13, 18, 16, 16, 9, 13, 19, 27, 19, 13, 18, 25, 16, 18, 13, 10, 18, 19, 22, 18, 25, 25, 18, 13

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• base

Comments

If 3|n then 9|a(n); otherwise, a(n) == 1 (mod 3). - Jon E. Schoenfield, Jun 30 2018

Examples

			Trajectories under the map x -> a(x):
1 ->  1 ->  1 ->  1 ->  1 ->  1 ->  1 ->  1 ->  1 -> ...
2 ->  4 ->  7 -> 13 -> 16 -> 13 -> 16 -> 13 -> 16 -> ...
3 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 -> ...
4 ->  7 -> 13 -> 16 -> 13 -> 16 -> 13 -> 16 -> 13 -> ...
5 ->  7 -> 13 -> 16 -> 13 -> 16 -> 13 -> 16 -> 13 -> ...
6 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 -> ...
7 -> 13 -> 16 -> 13 -> 16 -> 13 -> 16 -> 13 -> 16 -> ...
- _R. J. Mathar_, Jul 08 2012
		

Links

• Zak Seidov, Table of n, a(n) for n = 0..10000
• A. S. Besicovitch, The asymptotic distribution of the numerals in the decimal representation of the squares of the natural numbers, Mathematische Zeitschrift 39 (1934), pp. 146-156.
• H. Davenport and P. Erdős, Note on normal decimals, Canadian Journal of Mathematics 4 (1952), pp. 58-63.
• Michael Drmota, Christian Mauduit and Joël Rivat, The sum-of-digits function of polynomial sequences, J. Lond. Math. Soc. (2) 84(2011), no. 1, 81--102. MR2819691 (2012f:11193)
• Bernt Lindström, On the binary digits of a power, Journal of Number Theory, Volume 65, Issue 2, August 1997, Pages 321-324.
• Christian Mauduit and Joël Rivat, La somme des chiffres des carrés, Acta Mathem. 203 (1) (2009) 107-148. MR2545827 (2010j:11119).
• H. I. Okagbue, M. O. Adamu, S. A. Iyase and A. A. Opanuga, Sequence of Integers Generated by Summing the Digits of their Squares, Indian Journal of Science and Technology, Vol 8(15), DOI: 10.17485/ijst/2015/v8i15/69912, July 2015.
• K. B. Stolarsky, The binary digits of a power, Proc. Amer. Math. Soc. 71 (1978), 1-5.

Crossrefs

Cf. A007953, A159918, A056691, A268226.

Cf. A240752 (first differences), A071317 (partial sums).

Cf. A062685 (smallest square with digit sum n, or 0 if no such square exists).

Programs

Haskell

a004159 = a007953 . a000290  -- Reinhard Zumkeller, Apr 12 2014

Maple

read("transforms"):
A004159 := proc(n)
        digsum(n^2) ;
end proc: # R. J. Mathar, Jul 08 2012

Mathematica

a004159[n_Integer] := Apply[Plus, IntegerDigits[n^2]]; Table[
a004159[n], {n, 0, 100}] (* Michael De Vlieger, Jul 21 2014 *)
Total[IntegerDigits[#]]&/@(Range[0,100]^2) (* Harvey P. Dale, Feb 03 2019 *)

PARI

A004159(n)=sumdigits(n^2) \\ M. F. Hasler, Sep 23 2014

Python

def A004159(n):
    return sum(int(d) for d in str(n*n)) # Chai Wah Wu, Sep 03 2014

Formula

a(n) = A007953(A000290(n)); a(A058369(n)) = A007953(A058369(n)). - Reinhard Zumkeller, Apr 25 2009

a(10n) = a(n). If n > 1 is not a multiple of 10, then a(n)=4 iff n = 10^k+1 = A062397(k), a(n)=7 iff n is in A215614={4, 5, 32, 49, 149, 1049}, and else a(n) >= 9. - M. F. Hasler, Sep 23 2014