A004159 -id:A004159 - cp's OEIS frontend

A067552 a(n) = SumOfDigits(n)^2 - SumOfDigits(n^2), where SumOfDigits = A007953.

0, 0, 0, 0, 9, 18, 27, 36, 54, 72, 0, 0, 0, 0, 9, 27, 36, 45, 72, 90, 0, 0, 0, 9, 18, 36, 45, 63, 81, 108, 0, 0, 18, 18, 36, 54, 63, 81, 108, 135, 9, 9, 18, 27, 45, 72, 90, 108, 135, 162, 18, 27, 36, 45, 63, 90, 108, 126, 153, 180, 27, 36, 45, 54, 81, 108, 126, 144, 180, 207

Offset: 0

Reinhard Zumkeller, Jan 28 2002

All terms are divisible by 9; see A069912 for the quotient. - Ivan Neretin, Sep 01 2016

			a(16) = SumOfDigits(16)^2 - SumOfDigits(16^2) = (1+6)^2 - SumOfDigits(256) = 7^2 - (2+5+6) = 49 - 13 = 36.

Ivan Neretin, Table of n, a(n) for n = 0..10000

Cf. A004159, A007953, A061909.

f[n_] := Plus @@ IntegerDigits[n]^2 - Plus @@ (IntegerDigits[n^2]); Table[ f[n], {n, 0, 100}]

a(n) = sumdigits(n)^2 - sumdigits(n^2); \\ Michel Marcus, Sep 01 2016

a(n) = A007953(n)^2 - A004159(n).

Edited by Robert G. Wilson v, May 04 2002

A215614 Numbers k that are not multiples of 10 and such that sum of digits of k^2 is 7.

Original entry on oeis.org

4, 5, 32, 49, 149, 1049

Offset: 1

Zak Seidov, Aug 17 2012

Except for the number 1, the terms of this sequence and numbers 10^k+1 (A062397) are the only numbers (up to trailing 0's) whose square has sum of digits less than 9. - M. F. Hasler, Sep 23 2014
Is this sequence finite? See also A384095 for a similar problem with digit sum 9. - M. F. Hasler, Jun 20 2025
a(7) > 10^15 if it exists. - David A. Corneth, Jun 21 2025
a(7) > 10^65 if it exists. - Michael S. Branicky, Jun 25 2025
a(7) > 10^700 if it exists. - Max Alekseyev, Jun 27 2025

Cf. A004159 (sum of digits of n^2).
Subsequence of A262711.
Cf. A384094 (similar for digit sum 9), A384095 (subset of "sporadic solutions").

Select[Range[1500],Mod[#,10]!=0&&Total[IntegerDigits[#^2]]==7&] (* Harvey P. Dale, Aug 21 2022 *)

for(n=1,9e9, n%10&&sumdigits(n^2)==7&&print1(n",")) \\ M. F. Hasler, Sep 23 2014

Edited and unproven keywords fini,full removed by Max Alekseyev, Jun 20 2025

A056527 Numbers where iterated sum of digits of square settles down to a cyclic pattern (in fact 13, 16, 13, 16, ...).

Original entry on oeis.org

2, 4, 5, 7, 11, 13, 14, 16, 20, 22, 23, 25, 29, 31, 32, 34, 38, 40, 41, 43, 47, 49, 50, 52, 56, 58, 59, 61, 65, 67, 68, 70, 74, 76, 77, 79, 83, 85, 86, 88, 92, 94, 95, 97, 101, 103, 104, 106, 110, 112, 113, 115, 119, 121, 122, 124, 128, 130, 131, 133, 137, 139, 140

Offset: 1

Henry Bottomley, Jun 19 2000

Numbers == 2, 4, 5 or 7 mod 9, i.e. such that n^4 is not congruent to n^2 mod 9.

Numbers congruent to {2, 4, 5, 7} mod 9.

			a(1)=2 because iteration starts 2, 4, 7, 13, 16, 13, 16, ....

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 1, -1).

Cf. A004159 for sum of digits of square, A056020 where iteration settles to 1, A056020 where iteration settles to 9, also A056528, A056529. Unhappy numbers A031177 deal with iteration of square of sum of digits not settling to a single result.

Flatten[Table[9n+{2,4,5,7},{n,0,20}]] (* or *) LinearRecurrence[{1,0,0,1,-1},{2,4,5,7,11},100] (* Harvey P. Dale, Apr 05 2015 *)

Vec(x*(2 + 2*x + x^2 + 2*x^3 + 2*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^80)) \\ Colin Barker, Dec 19 2017

a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - Harvey P. Dale, Apr 05 2015
From Colin Barker, Dec 19 2017: (Start)
G.f.: x*(2 + 2*x + x^2 + 2*x^3 + 2*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = (-9 + (-1)^(1+n) - (3-3*i)*(-i)^n - (3+3*i)*i^n + 18*n) / 8 where i=sqrt(-1).
(End)

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

A371728 a(n) is the largest number that is the digit sum of an n-digit square number.

Original entry on oeis.org

9, 13, 19, 31, 40, 46, 54, 63, 70, 81, 88, 97, 106, 112, 121, 130, 136, 148, 154, 162, 171, 180, 187, 193, 205, 211, 220, 229, 235, 244, 253, 262, 271, 277, 286, 297, 301, 310, 319, 331, 334, 343, 355, 360, 367, 378, 388, 396

Offset: 1

Zhining Yang, Apr 04 2024

a(n) appears to be approximately equal to (33*n-11)/4.

			a(6) = 46 because 46 is the largest digital sum encountered among all 6-digit squares (698896, 779689, 877969).

Shouen Wang, Chinese BBS: How many of these A's are there?

Cf. A004159, A049415, A069665, A069666.

Cf. A056991, A348300, A379298.

Array[Max@Map[Total@IntegerDigits[#^2] &, Range[Floor@Sqrt[10^(#)]], Floor@Sqrt[10^(# + 1) - 1]] &, 15]

a(22)-a(48) from Zhao Hui Du, Apr 05 2024
a(49)-a(62) from Zhining Yang, May 08 2024
a(63)-a(64) from Zhining Yang, May 23 2024
Incorrect a(61) and unverified a(49) onward deleted by Zhining Yang, Mar 03 2025

A062685 Smallest square with digit sum n (or 0 if no such square exists).

Original entry on oeis.org

1, 0, 0, 4, 0, 0, 16, 0, 9, 64, 0, 0, 49, 0, 0, 169, 0, 576, 289, 0, 0, 1849, 0, 0, 4489, 0, 3969, 17956, 0, 0, 6889, 0, 0, 27889, 0, 69696, 98596, 0, 0, 97969, 0, 0, 499849, 0, 1887876, 698896, 0, 0, 2778889, 0, 0, 4999696, 0, 9696996, 19998784, 0, 0

Offset: 1

Erich Friedman, Jul 04 2001

a(n) > 0 iff n mod 9 is 0, 1, 4, or 7. - Jon E. Schoenfield, Jul 06 2018

			16 is the smallest square with digit sum 7, so a(7)=16.

Jon E. Schoenfield, Table of n, a(n) for n = 1..228 (first 213 terms from Giovanni Resta)

Cf. A004159, A056991.

Array[If[FreeQ[{1, 4, 7, 9}, FixedPoint[Total@ IntegerDigits@ # &, #]], 0, Block[{k = 1, s}, While[Total@ IntegerDigits@ Set[s, k^2] != #, k++]; s]] &, 57] (* Michael De Vlieger, Jul 06 2018 *)

A262711 Numbers k such that sum of digits of k^2 is 7.

Original entry on oeis.org

4, 5, 32, 40, 49, 50, 149, 320, 400, 490, 500, 1049, 1490, 3200, 4000, 4900, 5000, 10490, 14900, 32000, 40000, 49000, 50000, 104900, 149000, 320000, 400000, 490000, 500000, 1049000, 1490000, 3200000, 4000000, 4900000, 5000000, 10490000, 14900000

Offset: 1

Vincenzo Librandi, Sep 28 2015

Subsequence of A156638. [Bruno Berselli, Sep 28 2015]

			4 is in sequence because 4^2 = 16 and 1+6 = 7.

Cf. sum of digits of n^2 is k: A052216 (k=4), this sequence (k=7), A262712 (k=9), A262713 (k=10).
Cf. A004159, A061910, A156638.
Cf. A215614.

[n: n in [1..2*10^7] | &+Intseq(n^2) eq 7];

Select[Range[10^7], Total[IntegerDigits[#^2]] == 7 &]

for(n=1, 1e8, if (sumdigits(n^2) == 7, print1(n", "))) \\ Altug Alkan, Sep 28 2015

A336225 Table read by antidiagonals: T(n, k) = digitsum(n*k) with n >= 0 and k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 1, 3, 3, 1, 6, 0, 0, 7, 3, 6, 7, 6, 3, 7, 0, 0, 8, 5, 9, 2, 2, 9, 5, 8, 0, 0, 9, 7, 3, 6, 7, 6, 3, 7, 9, 0, 0, 1, 9, 6, 10, 3, 3, 10, 6, 9, 1, 0, 0, 2, 2, 9, 5, 8, 9, 8, 5, 9, 2, 2, 0

Offset: 0

Stefano Spezia, Jul 12 2020

			The table T(n, k) begins
0   0   0   0   0   0   0   0 ...
0   1   2   3   4   5   6   7 ...
0   2   4   6   8   1   3   5 ...
0   3   6   9   3   6   9   3 ...
0   4   8   3   7   2   6  10 ...
0   5   1   6   2   7   3   8 ...
0   6   3   9   6   3   9   6 ...
0   7   5   3  10   8   6  13 ...
...

Index entries for sequences related to sum of digits.

Cf. A003991, A004092, A004159 (diagonal), A004164 (digitsum of n^3), A004247, A007953, A055565 (digitsum of n^4), A055566 (digitsum of n^5), A055567 (digitsum of n^6).

T[n_,k_]:=Total[IntegerDigits[n*k]]; Table[T[n-k,k],{n,0,12},{k,0,n}]//Flatten

PARI
```
T(n, k) = sumdigits(n*k);
```

T(n, k) = A007953(A004247(n, k)).
T(n, 1) = T(1, n) = A007953(n).
T(n, 2) = T(2, n) = A004092(n).
T(n, k) = A007953(A003991(n, k)) for n*k > 0. - Michel Marcus, Jul 13 2020.

cp's OEIS Frontend

A067552 a(n) = SumOfDigits(n)^2 - SumOfDigits(n^2), where SumOfDigits = A007953.

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A215614 Numbers k that are not multiples of 10 and such that sum of digits of k^2 is 7.

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A056527 Numbers where iterated sum of digits of square settles down to a cyclic pattern (in fact 13, 16, 13, 16, ...).

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

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A371728 a(n) is the largest number that is the digit sum of an n-digit square number.

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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)).