A215614 -id:A215614 - 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

A226803 Primes p where the digital sum of p^2 is equal to 7.

Original entry on oeis.org

5, 149, 1049

Offset: 1

Vincenzo Librandi, Jun 24 2013

No more terms below 10^9. - Michel Marcus, Nov 02 2013

No more terms below 10^20. - Hiroaki Yamanouchi, Sep 23 2014

			5 is in the sequence because 5^2 = 25 and 2 + 5 = 7.
149 is in the sequence because 149^2 = 22201 and 2 + 2 + 2 + 0 + 1 = 7.

Subsequence of A215614.

Cf. A226802, A165492, A165459, A165493.

[p: p in PrimesUpTo(6*10^6) | &+Intseq(p^2) eq 7];

Select[Prime[Range[10000]], Total[IntegerDigits[#^2]] == 7 &]

lista(nn) = {forprime(p=2, nn, if (sumdigits(p^2)==7, print1(p, ", ")););} \\ Michel Marcus, Nov 02 2013

Keywords fini,full, since unproven, removed by Max Alekseyev, Jun 20 2025

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

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])

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A004159 Sum of digits of n^2.

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A226803 Primes p where the digital sum of p^2 is equal to 7.

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

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

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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.

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