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.

Showing 1-4 of 4 results.

A058413 Numbers k such that k^2 contains only digits {0,1,4}, not ending with zero.

Original entry on oeis.org

1, 2, 12, 21, 38, 102, 201, 1002, 1049, 2001, 10002, 10679, 20001, 20348, 100002, 100549, 200001, 202512, 375501, 643771, 1000002, 1020348, 2000001, 2002848, 10000002, 10200201, 20000001, 20024988, 20100102, 100000002, 100202001
Offset: 1

Views

Author

Patrick De Geest, Nov 15 2000

Keywords

Links

Crossrefs

Cf. A058414.

Programs

  • Mathematica
    Select[Range[1, 10000], (ContainsOnly[#, {0, 1, 4}] && #[[-1]] != 0) &@ IntegerDigits[#^2] &] (* Julien Kluge, Jul 08 2016 *)

A119037 Triangular numbers composed of digits {0,1,4}.

Original entry on oeis.org

1, 10, 10011, 10440, 41041, 41001040, 141044410, 1010004040, 4104044101, 14041444410, 114011410040100, 440404440401100, 1041401040044401, 40114114410110140, 14001400401400441011, 1040410404040110411004140, 1014411414040144040404100100
Offset: 1

Views

Author

Giovanni Resta, May 10 2006

Keywords

Links

Crossrefs

Cf. A000217, A058414, A119038. See A119033 for a table of cross-references.

Programs

  • Magma
    [t: n in [1..2*10^7] | Set(Intseq(t)) subset {0,1,4} where t is n*(n+1) div 2]; // Vincenzo Librandi, Dec 18 2015
  • Mathematica
    Rest[Select[FromDigits/@Tuples[{0, 1, 4}, 12], IntegerQ[(Sqrt[8 # + 1] - 1)/2] &]] (* Vincenzo Librandi, Dec 18 2015 *)

Formula

a(n) = A000217(A119038(n)). - Tyler Busby, Mar 27 2023

Extensions

a(17) from Tyler Busby, Mar 27 2023

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

Views

Author

M. F. Hasler, Jun 15 2025

Keywords

Comments

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?

Crossrefs

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}).

Programs

  • PARI
    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

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 1-4 of 4 results.

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