cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Zak Seidov, Aug 17 2012

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    Select[Range[1500],Mod[#,10]!=0&&Total[IntegerDigits[#^2]]==7&] (* Harvey P. Dale, Aug 21 2022 *)
  • PARI
    for(n=1,9e9, n%10&&sumdigits(n^2)==7&&print1(n",")) \\ M. F. Hasler, Sep 23 2014

Extensions

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

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

Views

Author

M. F. Hasler, Jun 15 2025

Keywords

Comments

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

Crossrefs

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}).

Programs

  • Maple
    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
  • PARI
    select( {is_A384095(n)=n%10 && sumdigits(n^2)==9 && !bittest(36938, fromdigits(Set(digits(n))))}, [1..10^5])
Showing 1-2 of 2 results.