A004159 -id:A004159 - cp's OEIS frontend

A153752 Numbers k such that there are 16 digits in k^2 and for each factor f of 16 (1,2,4,8) the sum of digit groupings of size f is a square.

31883334, 31886667, 31956690, 31970049, 32469999, 33338100, 33341067, 33870000, 34140000, 34149999, 34713042, 34763334, 34856667, 35780000, 36356249, 36356480, 36359065, 37523635, 37737452, 37949451, 38362409

Offset: 1

Doug Bell, Dec 31 2008

This sequence is a subsequence of both A153745 and A061910.

This sequence contains 124 terms, with a(124) = 9998956. - Giovanni Resta, Jun 06 2015

			31883334^2 = 1016546986955556;
1+0+1+6+5+4+6+9+8+6+9+5+5+5+5+6 = 81 = 9^2;
10+16+54+69+86+95+55+56 = 441 = 21^2;
1016+5469+8695+5556 = 20736 = 144^2;
10165469+86955556 = 97121025 = 9855^2.

Giovanni Resta, Table of n, a(n) for n = 1..124 (all terms)

Cf. A004159, A061910, A153745.

okQ[n_]:=Module[{n2=IntegerDigits[n^2]},And@@(IntegerQ[Sqrt[ #]]&/@ (Total/@(Table[ FromDigits/@Partition[n2,2^i],{i,0,3}])))]; Select[ Range[31622777,38400000],okQ] (* Harvey P. Dale, Aug 12 2012 *)

A164818 Integer average digit of squares.

Original entry on oeis.org

1, 4, 9, 5, 3, 3, 3, 3, 6, 6, 3, 4, 4, 4, 4, 4, 4, 5, 5, 2, 2, 5, 5, 5, 5, 2, 5, 2, 5, 5, 5, 5, 5, 2, 5, 5, 5, 5, 5, 5, 5, 5, 5, 8, 3, 3, 3, 3, 3, 3, 3, 3, 6, 3, 3, 3, 6, 3, 6, 3, 6, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 6, 6, 3, 6, 3, 3, 3, 3, 3, 3, 3, 6, 3, 3, 3, 3, 3, 6, 6, 6, 6, 3, 3, 3, 6, 3, 3, 3, 3, 6, 6, 3, 6

Offset: 1

Zak Seidov, Aug 27 2009

a(n)= average digit of squares of A164817(n).

The only n for which a(n) = 9 is 3. a(44) = a(26165) = 8. - Robert Israel, Feb 24 2016

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A004159, A164817.

f:= proc(n) local L,r;
L:= convert(n^2,base,10);
r:= convert(L,`+`)/nops(L);
if r::integer then r else NULL fi
end proc:
map(f, [$1..1000]); # Robert Israel, Feb 24 2016

A262712 Numbers k such that sum of digits of k^2 is 9.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 30, 39, 45, 48, 51, 60, 90, 102, 105, 111, 120, 150, 180, 201, 210, 249, 300, 318, 321, 348, 351, 390, 450, 480, 501, 510, 549, 600, 900, 1002, 1005, 1011, 1020, 1050, 1101, 1110, 1149, 1200, 1500, 1761, 1800, 2001, 2010, 2100, 2490

Offset: 1

Vincenzo Librandi, Sep 28 2015

Subsequence of A008585.

			6 is in sequence because 6^2 = 36 and 3+6 = 9.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. similar sequences listed in A262711.

Cf. A004159, A008585, A061910.

[n: n in [1..2*10^4] | &+Intseq(n^2) eq 9 ];

filter:= proc(n) convert(convert(n^2,base,10),`+`) = 9 end proc:select(filter, [$1..10^5]); # Robert Israel, Jan 04 2024

Select[Range[10^5], Total[IntegerDigits[#^2]] == 9 &]

for(n=1, 1e8, if (sumdigits(n^2) == 9, print1(n", "))) \\ Altug Alkan, Sep 28 2015

A262713 Numbers k such that the sum of digits of k^2 is 10.

Original entry on oeis.org

8, 19, 35, 46, 55, 71, 80, 145, 152, 179, 190, 251, 332, 350, 361, 449, 451, 460, 548, 550, 649, 710, 800, 1450, 1520, 1790, 1900, 2510, 3320, 3500, 3610, 4490, 4499, 4510, 4600, 5480, 5500, 6490, 7100, 8000, 14500, 15200, 17900, 19000, 20249, 20251, 24499

Offset: 1

Vincenzo Librandi, Sep 28 2015

From Altug Alkan, Sep 29 2015: (Start)
Subsequence of A001651.
If a(n)+1 mod 9 != 0 then a(n)-1 mod 9 = 0;
if a(n)-1 mod 9 != 0 then a(n)+1 mod 9 = 0;
a(n)^2 - 1 mod 9 = 0. (End)
A135027(n)*10^k is a term for all n > 0, k >= 0. - Michael S. Branicky, Aug 19 2021

			19 is in sequence because 19^2 = 361 and 3+6+1 = 10.

Michael S. Branicky, Table of n, a(n) for n = 1..244
Michael S. Branicky, Python program

Cf. similar sequences listed in A262711.

Cf. A004159, A061910, A135027.

[n: n in [1..3*10^4] | &+Intseq(n^2) eq 10 ];

Select[Range[10^5], Total[IntegerDigits[#^2]] == 10 &]

for(n=1, 1e6, if (sumdigits(n^2) == 10, print1(n", "))) \\ Altug Alkan, Sep 28 2015

# See linked program to go to larger numbers
def ok(n): return sum(map(int, str(n*n))) == 10
print(list(filter(ok, range(25000)))) # Michael S. Branicky, Aug 19 2021

A268227 a(n) = sum of digits of (2n)^2.

Original entry on oeis.org

0, 4, 7, 9, 10, 1, 9, 16, 13, 9, 4, 16, 18, 19, 19, 9, 7, 13, 18, 13, 7, 18, 19, 10, 9, 7, 13, 18, 13, 16, 9, 19, 19, 18, 16, 13, 18, 22, 25, 18, 10, 19, 18, 25, 22, 9, 22, 25, 18, 19, 1, 9, 16, 13, 18, 4, 16, 27, 19, 19, 9, 25, 22, 27, 22, 16, 18, 28, 28, 18, 16, 13, 18, 13, 16, 9, 10, 19, 18

Offset: 0

N. J. A. Sloane, Jan 31 2016

Harvey P. Dale, Table of n, a(n) for n = 0..1000
H. I. Okagbue, M. O. Adamu, S. A. Iyase, 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.

Bisection of A004159. Cf. A268228.

Total[IntegerDigits[#]]&/@((2*Range[0,80])^2) (* Harvey P. Dale, Sep 14 2016 *)

a(n) = sumdigits((2*n)^2); \\ Michel Marcus, Jul 23 2016

A268228 a(n) = sum of digits of (2n + 1)^2.

Original entry on oeis.org

1, 9, 7, 13, 9, 4, 16, 9, 19, 10, 9, 16, 13, 18, 13, 16, 18, 10, 19, 9, 16, 22, 9, 13, 7, 9, 19, 10, 18, 16, 13, 27, 13, 25, 18, 10, 19, 18, 25, 13, 18, 31, 16, 27, 19, 19, 27, 16, 22, 18, 4, 16, 9, 19, 19, 9, 25, 13, 27, 13, 16, 18, 19, 19, 18, 16, 31, 18, 31, 16, 27, 19, 10, 18, 7, 13, 18

Offset: 0

N. J. A. Sloane, Jan 31 2016

Robert Israel, Table of n, a(n) for n = 0..10000
H. I. Okagbue, M. O. Adamu, S. A. Iyase, 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.

Bisection of A004159. Cf. A007953, A016754, A268227.

[&+Intseq((2*n+1)^2): n in [0..87] ]; // Vincenzo Librandi, Jul 23 2016

f:= n -> convert(convert((2*n+1)^2,base,10),`+`):
map(f, [$0..100]); # Robert Israel, Apr 26 2020

Table[Sum[DigitCount[(2 n + 1)^2] [[i]] i, {i, 9}], {n, 0, 70}] (* Vincenzo Librandi, Jul 23 2016, after G. C. Greubel *)

a(n) = sumdigits((2*n+1)^2); \\ Michel Marcus, Jul 23 2016

a(n) = A007953(A016754(n)). - Michel Marcus, Oct 13 2017

A360803 Numbers whose squares have a digit average of 8 or more.

Original entry on oeis.org

3, 313, 94863, 298327, 987917, 3162083, 9893887, 29983327, 99477133, 99483667, 197483417, 282753937, 314623583, 315432874, 706399164, 773303937, 894303633, 947047833, 948675387, 989938887, 994927133, 994987437, 998398167, 2428989417, 2754991833, 2983284917, 2999833327

Offset: 1

Dmitry Kamenetsky, Feb 21 2023

This sequence is infinite. For example, numbers floor(30*100^k - (5/3)*10^k) beginning with 2 followed by k 9s, followed by 8 and k 3s, have a square whose digit average converges to (but never equals) 8.25. [Corrected and formula added by M. F. Hasler, Apr 11 2023]
Only a few examples are known whose square has a digit average of 8.25 and above: 3^2 = 9, 707106074079263583^2 = 499998999999788997978888999589997889 (digit average 8.25), 94180040294109027313^2 = 8869879989799999999898984986998979999969 (digit average 8.275).
This is the union of A164772 (digit average = 8) and A164841 (digit average > 8). - M. F. Hasler, Apr 11 2023

			94863 is in the sequence, because 94863^2 = 8998988769, which has a digit average of 8.1 >= 8.

Michael S. Branicky, Python program for OEIS A360822 and searching for squares with < given number of digits < 8
Matthieu Dufour and Silvia Heubach, Squares with Large Digit Average, 2020.
Dmitry Kamenetsky, The first 68 terms and their digit average

Cf. A004159, A185679.

Cf. A164772 (digit average = 8), A164841 (digit average > 8).

isok(k) = my(d=digits(k^2)); vecsum(d)/#d >= 8; \\ Michel Marcus, Feb 22 2023

def ok(n): d = list(map(int, str(n**2))); return sum(d) >= 8*len(d)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Feb 22 2023

A369955 a(n) is the least integer m such that the sum of the digits of m^2 is 9*(k+n) where k is the number of digits of m.

Original entry on oeis.org

3, 63, 3114, 8937, 94863, 5477133, 82395381, 706399164, 9380293167, 99497231067, 4472135831667, 62441868958167, 836594274358167, 9983486364492063, 435866837461509417, 707106074079263583, 77453069648658793167, 754718284918279954614, 8882505274864168010583

Offset: 0

Zhining Yang, Feb 06 2024

3|a(n).

			a(2)=3114 because 3114 is the least 4-digit integer whose square has digit sum 9*(4+2) = 9*6 = 54: 3114^2 = 9696996 and 9+6+9+6+9+9+6 = 54.

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

Cf. A004159, A067179, A369953.

n=0;For[k=0,k<10^8/3,k++,If[Total[IntegerDigits[9k^2]]==9*(n+Ceiling@Log10@(3k)),Print[{n,3k}];n++]]

a(n) = my(m=1); while (sumdigits(m^2) != 9*(#Str(m)+n), m++); m; \\ Michel Marcus, Feb 10 2024

def sd(n):
    return sum(int(d) for d in str(n*n))
n=0
for k in range(0,10**8,3):
    if sd(k)==9*(len(str(k))+n):
        print([n,k])
        n+=1

a(9)-a(18) from Zhao Hui Du, Feb 19 2024

A384094 Numbers whose square has digit sum 9 and no trailing zero.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 39, 45, 48, 51, 102, 105, 111, 201, 249, 318, 321, 348, 351, 501, 549, 1002, 1005, 1011, 1101, 1149, 1761, 2001, 4899, 5001, 10002, 10005, 10011, 10101, 10149, 11001, 14499, 20001, 50001, 100002, 100005, 100011, 100101, 101001, 110001, 200001, 375501, 500001, 1000002

Offset: 1

M. F. Hasler, Jun 15 2025

All numbers of the form 10^a + 10^b + 1 (i.e., A052216+1 = 3*A237424) and of the form 10^a + 5*10^b with min(a, b) = 0 (i.e., A133472 U A199685), are in this sequence. Terms not of this form are (9, 18, 39, 45, 48, 249, 318, 321, 348, 351, 549, 1149, 1761, 4899, 10149, 14499, 375501, ...), see subsequence A384095. (Is this sequence finite? What is the next term?)

Is it true that no number > 1049 = A215614(6) has a square with digit sum less than 9, other than the trivial 1 and 4?

Cf. A004159 (sum of digits of n^2), A215614 (sumdigits(n^2) = 7), A133472 (10^n + 5), A199685 (5*10^n + 1), A052216 (10^a + 10^b), A237424 ((10^a + 10^b + 1)/3).

See also: A058414 (digits(n^2) in {0,1,4}).

select( {is_A384094(n)=n%10 && sumdigits(n^2)==9}, [1..10^5])

A384095 Numbers other than {10^a + 10^b + 1} and {10^a + 5*10^b, min(a, b) = 0} whose square has digit sum 9 and no trailing zero.

Original entry on oeis.org

9, 18, 39, 45, 48, 249, 318, 321, 348, 351, 549, 1149, 1761, 4899, 10149, 14499, 375501

Offset: 1

M. F. Hasler, Jun 15 2025

The definition excludes the two "regular" subsequences of A384094, namely A052216+1 = 3*A237424 and A133472 U A199685, which provide most of its terms.
Is it true that no number > 1049 = A215614(6) has a square with digit sum less than 9, other than the trivial 1 and 4?
The next term, if it exists, is a(18) > 10^8.
a(18) > 10^14 if it exists. - Robert Israel, Jun 15 2025
a(18) > 10^40 if it exists. - Chai Wah Wu, Jun 19 2025

Cf. A004159 (sum of digits of n^2), A384094 (sumdigits(n^2) = 9), A133472 (10^n+5), A199685 (5*10^n + 1), A052216 (10^a+10^b), A237424 ((10^a+10^b+1)/3).

See also: A215614 (sumdigits(n^2) = 7), A058414 (digits(n²) ⊂ {0,1,4}).

extend:= proc(a,d) local i,s;
    s:= convert(convert(a,base,10),`+`);
    op(select(t -> numtheory:-quadres(t,10^d)=1, [seq(i*10^(d-1)+a, i=0 .. 9 - s)]))
end proc:
istriv:= proc(n) local L;
   L:= subs(0=NULL,convert(n,base,10));
   member(L, [[4],[5],[6],[1,1],[1,1,1],[1,2],[2,1],[1,5],[5,1]])
end proc:
R:= NULL:
A:= [1,4,5,6,9]:
for d from 2 to 20 do
  A:= map(extend,A,d);
  V:= select(t -> t > 10^(d-1) and issqr(t) and convert(convert(t,base,10),`+`)=9, A);
  if V <> [] then V:= sort(remove(istriv,map(sqrt,V))); R:= R,op(V); fi
od:
R;# Robert Israel, Jun 15 2025

select( {is_A384095(n)=n%10 && sumdigits(n^2)==9 && !bittest(36938, fromdigits(Set(digits(n))))}, [1..10^5])

cp's OEIS Frontend

A153752 Numbers k such that there are 16 digits in k^2 and for each factor f of 16 (1,2,4,8) the sum of digit groupings of size f is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A164818 Integer average digit of squares.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A262712 Numbers k such that sum of digits of k^2 is 9.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

A262713 Numbers k such that the sum of digits of k^2 is 10.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

A268227 a(n) = sum of digits of (2n)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A268228 a(n) = sum of digits of (2n + 1)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A360803 Numbers whose squares have a digit average of 8 or more.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

A369955 a(n) is the least integer m such that the sum of the digits of m^2 is 9*(k+n) where k is the number of digits of m.

Views