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.

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

Offset

1,1

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

Links

Table of n, a(n) for n=1..17.

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

Prog

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

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(n^2) ⊂ {0,1,4}).

Sequence in context: A245425 A000547 A138900 * A202187 A370016 A057373

Adjacent sequences: A384092 A384093 A384094 * A384096 A384097 A384098

Keyword

nonn,base,hard,more

Author

M. F. Hasler, Jun 15 2025

Status

approved