A071317 -id:A071317 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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

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.

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).

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

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

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 *)

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

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

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

A353974 a(n) is the n-th partial sum of A056992.

Original entry on oeis.org

0, 1, 5, 14, 21, 28, 37, 41, 42, 51, 52, 56, 65, 72, 79, 88, 92, 93, 102, 103, 107, 116, 123, 130, 139, 143, 144, 153, 154, 158, 167, 174, 181, 190, 194, 195, 204, 205, 209, 218, 225, 232, 241, 245, 246, 255, 256, 260, 269, 276, 283, 292, 296, 297, 306, 307, 311

Offset: 0

Stefano Spezia, May 12 2022

Also the n-th partial sum of the main diagonal of A353109, or equivalently, the trace of the matrix M(n) whose permanent is A353933(n) for n > 0.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).

Cf. A003991, A010888, A056992, A071317, A353109, A353933.

CoefficientList[Series[x(1+4x+9x^2+7x^3+7x^4+9x^5+4x^6+x^7+9x^8)/((1-x)^2(1+x+x^2)(1+x^3+x^6)),{x,0,56}],x]

G.f.: x*(1 + 4*x + 9*x^2 + 7*x^3 + 7*x^4 + 9*x^5 + 4*x^6 + x^7 + 9*x^8)/((1 - x)^2*(1 + x + x^2)*(1 + x^3 + x^6)).
a(n) = a(n-1) + a(n-9) - a(n-10) for n > 9.
a(n) ~ 51*n/9.

A071121 a(n) = a(n-1) + sum of decimal digits of n^3.

Original entry on oeis.org

1, 9, 18, 28, 36, 45, 55, 63, 81, 82, 90, 108, 127, 144, 162, 181, 198, 216, 244, 252, 270, 289, 306, 324, 343, 369, 396, 415, 441, 450, 478, 504, 531, 550, 576, 603, 622, 648, 675, 685, 711, 738, 766, 792, 810, 838, 855, 873, 901, 909, 927, 946, 981, 1008

Offset: 1

Labos Elemer, May 27 2002

N. Agronomof, Question 4420, L'Intermédiaire des Math. 21 (1914), 147.

Cf. A037123, A000788, A051351, A071317.

Partial sums of A004164.

s=0; Do[s=s+Apply[Plus, IntegerDigits[n^3]]; Print[s], {n, 1, 128}]
nxt[{n_,a_}]:={n+1,a+Total[IntegerDigits[(n+1)^3]]}; NestList[nxt,{1,1},60][[;;,2]] (* Harvey P. Dale, Aug 30 2025 *)

A333034 a(n) is the sum of the digits of the squares of all n-digit numbers.

Original entry on oeis.org

69, 1410, 21921, 298725, 3792660, 46016727, 541129686, 6221175405, 70311424443, 784112741880, 8651123311875, 94611219470547

Offset: 1

Robert Israel, Mar 05 2020

a(n) == 6 (mod 9). - Robert Israel, Mar 06 2020

			a(1) = 1+4+9+(1+6)+(2+5)+(3+6)+(4+9)+(6+4)+(8+1) = 69.

Cf. A071317.

ds:= proc(x) local t,s,r;
  s:= x; t:= 0;
  while s > 0 do
    r:= s mod 10;
    t:= t + r;
    s:= (s-r)/10;
  od;
t
end proc:
seq(add(ds(x^2),x=10^(n-1)..10^n-1), n=1..5);

for(d=0,6,print1(sum(k=10^d,10^(d+1)-1,vecsum(digits(k^2))),", ")) \\ Hugo Pfoertner, Mar 05 2020

def A333034(n):
    return sum(int(d) for i in range(10**(n-1),10**n) for d in str(i**2)) # Chai Wah Wu, Mar 05 2020

a(n) = A071317(10^n) - A071317(10^(n-1)).

a(8)-a(9) from Hugo Pfoertner, Mar 05 2020

a(10)-a(12) from Giovanni Resta, Mar 07 2020

A071122 a(n) = a(n-1) + sum of decimal digits of 2^n.

Original entry on oeis.org

2, 6, 14, 21, 26, 36, 47, 60, 68, 75, 89, 108, 128, 150, 176, 201, 215, 234, 263, 294, 320, 345, 386, 423, 452, 492, 527, 570, 611, 648, 695, 753, 815, 876, 935, 999, 1055, 1122, 1193, 1254, 1304, 1350, 1406, 1464, 1526, 1596, 1664, 1737, 1802, 1878, 1958

Offset: 1

Labos Elemer, May 27 2002

Cf. A037123, A000788, A051351, A071317, A071121.

s=0; Do[s=s+Apply[Plus, IntegerDigits[2^n]]; Print[s], {n, 1, 128}]

A099358 a(n) = sum of digits of k^4 as k runs from 1 to n.

Original entry on oeis.org

1, 8, 17, 30, 43, 61, 68, 87, 105, 106, 122, 140, 162, 184, 202, 227, 246, 273, 283, 290, 317, 339, 370, 397, 422, 459, 477, 505, 530, 539, 561, 592, 619, 644, 663, 699, 727, 752, 770, 783, 814, 841, 866, 903, 921, 958, 1001, 1028, 1059, 1072, 1099, 1124, 1161

Offset: 1

Yalcin Aktar, Nov 16 2004

Partial sums of A055565.

			a(3) = sum_digits(1^4) + sum_digits(2^4) + sum_digits(3^4) = 1 + 7 + 9 = 17.

Cf. k^1 in A037123, k^2 in A071317 & k^3 in A071121.

f[n_] := Block[{s = 0, k = 1}, While[k <= n, s = s + Plus @@ IntegerDigits[k^4]; k++ ]; s]; Table[ f[n], {n, 50}] (* Robert G. Wilson v, Nov 18 2004 *)
Accumulate[Table[Total[IntegerDigits[n^4]],{n,60}]] (* Harvey P. Dale, Jun 08 2021 *)

a(n) = a(n-1) + sum of decimal digits of n^4.
a(n) = sum(k=1, n, sum(m=0, floor(log(k^4)), floor(10((k^4)/(10^(((floor(log(k^4))+1))-m)) - floor((k^4)/(10^(((floor(log(k^4))+1))-m))))))).
General formula: a(n)_p = sum(k=1, n, sum(m=0, floor(log(k^p)), floor(10((k^p)/(10^(((floor(log(k^p))+1))-m)) - floor ((k^p)/(10^(((floor(log(k^p))+1))-m))))))). Here a(n)_p is a sum of digits of k^p from k=1 to n.

Edited and extended by Robert G. Wilson v, Nov 18 2004

Existing example replaced with a simpler one by Jon E. Schoenfield, Oct 20 2013

cp's OEIS Frontend

A004159 Sum of digits of n^2.

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A353974 a(n) is the n-th partial sum of A056992.

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A071121 a(n) = a(n-1) + sum of decimal digits of n^3.

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A333034 a(n) is the sum of the digits of the squares of all n-digit numbers.

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A071122 a(n) = a(n-1) + sum of decimal digits of 2^n.

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A099358 a(n) = sum of digits of k^4 as k runs from 1 to n.

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