A240752 -id:A240752 - 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

A202089 Numbers n such that n^2 and (n+1)^2 have same digit sum.

Original entry on oeis.org

4, 13, 22, 49, 58, 76, 103, 130, 139, 157, 193, 202, 229, 247, 256, 274, 283, 301, 391, 418, 427, 454, 463, 472, 481, 508, 526, 553, 598, 607, 616, 643, 661, 679, 688, 724, 733, 742, 760, 769, 778, 796, 850, 868, 877, 886, 904, 913, 931, 949, 958, 976, 1003

Offset: 1

Zak Seidov, Dec 11 2011

Or numbers n such that A004159(n)=A004159(n+1), or A007953(n^2)=A007953((n+1)^2).
Corresponding digit sums are of the form 7+9k, with k=1, 2, 3,... .
Numbers n are of the form 4+9m, with  m=0, 1, 2, 5, 6, 8, 11, ... .
A240752(a(n)) = 0. - Reinhard Zumkeller, Apr 12 2014

			4^2=16 and 5^2=25 have same digit sum ds=7.
13^2=169 and 14^2=196 have ds=16.
76^2=5776 and 77^2=5929 have ds=25.
526^2=276676 and 527^2=277729 have ds=34.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A004159, A007953.

Cf. A239878, A240754.

import Data.List (elemIndices)
a202089 n = a202089_list !! (n-1)
a202089_list = elemIndices 0 a240752_list
-- Reinhard Zumkeller, Apr 12 2014

cnt = 0; nn = 10000; n = 4; Reap[While[cnt < nn, While[Total[IntegerDigits[n^2]] != Total[IntegerDigits[(n + 1)^2]], n = n + 9]; cnt++; Sow[n]; n = n + 9]][[2, 1]]

def ok(n): return sum(map(int, str(n*n))) == sum(map(int, str((n+1)**2)))
print(list(filter(ok, range(1004)))) # Michael S. Branicky, Apr 13 2021

A239878 Numbers k with digit_sum(kk) + 1 = digit_sum((k+1)(k+1)).

Original entry on oeis.org

0, 18, 27, 36, 45, 72, 81, 108, 153, 198, 216, 225, 243, 252, 270, 297, 306, 342, 369, 396, 423, 441, 450, 477, 486, 495, 504, 513, 522, 549, 558, 576, 603, 630, 639, 657, 693, 702, 729, 747, 756, 783, 801, 846, 891, 918, 954, 963, 972, 981

Offset: 1

Reiner Moewald, Mar 28 2014

All terms are divisible by 9.

The number of terms is unlimited: n = 3*10^z + 6, i.e., digit_sum(n*n) + 1 = 27 + 1 = 28 = digit_sum((n+1)*(n+1)). - Reiner Moewald, Apr 20 2014

Reiner Moewald, Table of n, a(n) for n = 1..4067

Cf. A202089, A240752, A240754, A004159, A007953, A000290.

import Data.List (elemIndices)
a239878 n = a239878_list !! (n-1)
a239878_list = elemIndices 1 a240752_list
-- Reinhard Zumkeller, Apr 12 2014

isok(n) = (sumdigits(n^2) + 1) == sumdigits((n+1)^2); \\ Michel Marcus, Apr 06 2014

def digit_Sum(n):
   integerString = str(n)
   digit_Sum=0
   for digitLetter in integerString:
      digit_Sum = digit_Sum + int(digitLetter)
   return digit_Sum
count = 0;
for i in range(20000):
   if(digit_Sum(i*i) + 1 == digit_Sum((i+1)*(i+1))):
      count = count +1
      print(count,"   ",i)

A240752(a(n)) = 1. - Reinhard Zumkeller, Apr 12 2014

A240754 Numbers k with digit_sum(kk) - 1 = digit_sum((k+1)(k+1)).

Original entry on oeis.org

8, 26, 53, 98, 107, 143, 161, 170, 179, 188, 224, 233, 242, 260, 269, 278, 287, 296, 350, 368, 386, 404, 413, 431, 449, 476, 494, 503, 539, 548, 557, 584, 593, 629, 638, 647, 674, 683, 737, 746, 773, 791, 818, 827, 863, 872, 908, 926, 944, 998, 1007, 1043

Offset: 1

Reinhard Zumkeller, Apr 12 2014

A240752(a(n)) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A202089, A239878, A004159, A007953, A000290.

import Data.List (elemIndices)
a240754 n = a240754_list !! (n-1)
a240754_list = elemIndices (-1) a240752_list

def ds(n): return sum(map(int, str(n)))
def ok(n): return ds(n*n) - 1 == ds((n+1)*(n+1))
print(list(filter(ok, range(1044)))) # Michael S. Branicky, Aug 26 2021

cp's OEIS Frontend

A004159 Sum of digits of n^2.

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A202089 Numbers n such that n^2 and (n+1)^2 have same digit sum.

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A239878 Numbers k with digit_sum(k*k) + 1 = digit_sum((k+1)*(k+1)).

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Haskell

PARI

Python

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A240754 Numbers k with digit_sum(k*k) - 1 = digit_sum((k+1)*(k+1)).

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Programs

Haskell

Python

A239878 Numbers k with digit_sum(kk) + 1 = digit_sum((k+1)(k+1)).

A240754 Numbers k with digit_sum(kk) - 1 = digit_sum((k+1)(k+1)).