cp's OEIS Frontend

A004159 Sum of digits of n^2.

%I A004159 #64 Nov 12 2021 12:23:37
%S A004159 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,
%T A004159 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,
%U A004159 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
%N A004159 Sum of digits of n^2.
%C A004159 If 3|n then 9|a(n); otherwise, a(n) == 1 (mod 3). - _Jon E. Schoenfield_, Jun 30 2018
%H A004159 Zak Seidov, <a href="/A004159/b004159.txt">Table of n, a(n) for n = 0..10000</a>
%H A004159 A. S. Besicovitch, <a href="http://dx.doi.org/10.1007/BF01201350">The asymptotic distribution of the numerals in the decimal representation of the squares of the natural numbers</a>, Mathematische Zeitschrift 39 (1934), pp. 146-156.
%H A004159 H. Davenport and P. Erdős, <a href="http://cms.math.ca/10.4153/CJM-1952-005-3">Note on normal decimals</a>, Canadian Journal of Mathematics 4 (1952), pp. 58-63.
%H A004159 Michael Drmota, Christian Mauduit and Joël Rivat, <a href="http://www.dmg.tuwien.ac.at/drmota/dmr6.pdf">The sum-of-digits function of polynomial sequences</a>, J. Lond. Math. Soc. (2) 84(2011), no. 1, 81--102. MR2819691 (2012f:11193)
%H A004159 Bernt Lindström, <a href="http://dx.doi.org/10.1006/jnth.1997.2129">On the binary digits of a power</a>, Journal of Number Theory, Volume 65, Issue 2, August 1997, Pages 321-324.
%H A004159 Christian Mauduit and Joël Rivat, <a href="http://dx.doi.org/10.1007/s11511-009-0040-0">La somme des chiffres des carrés</a>, Acta Mathem. 203 (1) (2009) 107-148.  MR2545827 (2010j:11119).
%H A004159 H. I. Okagbue, M. O. Adamu, S. A. Iyase and A. A. Opanuga, <a href="http://www.indjst.org/index.php/indjst/article/view/69912">Sequence of Integers Generated by Summing the Digits of their Squares</a>, Indian Journal of Science and Technology, Vol 8(15), DOI: 10.17485/ijst/2015/v8i15/69912, July 2015.
%H A004159 K. B. Stolarsky, <a href="http://dx.doi.org/10.1090/S0002-9939-1978-0495823-5">The binary digits of a power</a>, Proc. Amer. Math. Soc. 71 (1978), 1-5.
%F A004159 a(n) = A007953(A000290(n)); a(A058369(n)) = A007953(A058369(n)). - _Reinhard Zumkeller_, Apr 25 2009
%F A004159 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
%e A004159 Trajectories under the map x -> a(x):
%e A004159 1 ->  1 ->  1 ->  1 ->  1 ->  1 ->  1 ->  1 ->  1 -> ...
%e A004159 2 ->  4 ->  7 -> 13 -> 16 -> 13 -> 16 -> 13 -> 16 -> ...
%e A004159 3 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 -> ...
%e A004159 4 ->  7 -> 13 -> 16 -> 13 -> 16 -> 13 -> 16 -> 13 -> ...
%e A004159 5 ->  7 -> 13 -> 16 -> 13 -> 16 -> 13 -> 16 -> 13 -> ...
%e A004159 6 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 ->  9 -> ...
%e A004159 7 -> 13 -> 16 -> 13 -> 16 -> 13 -> 16 -> 13 -> 16 -> ...
%e A004159 - _R. J. Mathar_, Jul 08 2012
%p A004159 read("transforms"):
%p A004159 A004159 := proc(n)
%p A004159         digsum(n^2) ;
%p A004159 end proc: # _R. J. Mathar_, Jul 08 2012
%t A004159 a004159[n_Integer] := Apply[Plus, IntegerDigits[n^2]]; Table[
%t A004159 a004159[n], {n, 0, 100}] (* _Michael De Vlieger_, Jul 21 2014 *)
%t A004159 Total[IntegerDigits[#]]&/@(Range[0,100]^2) (* _Harvey P. Dale_, Feb 03 2019 *)
%o A004159 (Haskell)
%o A004159 a004159 = a007953 . a000290  -- _Reinhard Zumkeller_, Apr 12 2014
%o A004159 (Python)
%o A004159 def A004159(n):
%o A004159     return sum(int(d) for d in str(n*n)) # _Chai Wah Wu_, Sep 03 2014
%o A004159 (PARI) A004159(n)=sumdigits(n^2) \\ _M. F. Hasler_, Sep 23 2014
%Y A004159 Cf. A007953, A159918, A056691, A268226.
%Y A004159 Cf. A240752 (first differences), A071317 (partial sums).
%Y A004159 Cf. A062685 (smallest square with digit sum n, or 0 if no such square exists).
%K A004159 nonn,base
%O A004159 0,3
%A A004159 _N. J. A. Sloane_