A352057 Triangular numbers whose nonzero digits are all the same.

0, 1, 3, 6, 10, 55, 66, 300, 666, 990, 3003, 5050, 10011, 66066, 500500, 600060, 50005000, 5000050000, 500000500000, 50000005000000, 5000000050000000, 500000000500000000, 50000000005000000000, 5000000000050000000000, 500000000000500000000000, 50000000000005000000000000

Offset: 1

Author

Steven Lu, Mar 02 2022

Keywords

• nonn
• base

Comments

This sequence may correspond to "monochromatic step squads" in the British animation "Numberblocks".

Conjecture: the largest term in this sequence whose nonzero digits are not 5 is 600060.

Crossrefs

Supersequence of A037156.

Cf. A000217, A118668, A125289, A343811.

Cf. A352148 (indices of these triangular numbers).

Programs

Mathematica

(* Method1 *)
NonZeroQ[n_Integer] := n != 0; Select[
Table[n (n + 1)/2, {n, 0, 1000000}],
Length[Tally[Select[IntegerDigits[#], NonZeroQ]]] == 1 &]
(* Method2 *)
Sort[Select[
  Flatten[Outer[Times,
    Table[FromDigits[IntegerDigits[n, 2]], {n, 2^16 - 1}], Range[9]]],
   IntegerQ[Sqrt[8 # + 1]] &]]

PARI

isok(k) = my(d=digits(k*(k+1)/2)); d = select(x->(x!=0), d); #Set(d)<=1;
lista(nn) = {for (n=0, nn, if (isok(n), print1(n*(n+1)/2, ", ")););} \\ Michel Marcus, Mar 02 2022

Python

from sympy import integer_nthroot
from sympy.utilities.iterables import multiset_permutations
def istri(n): return integer_nthroot(8*n+1, 2)[1]
def zplus1(digits):
    if digits == 1: yield 0
    for d1 in "123456789":
        digset = "0"*(digits-1) + d1*(digits-1)
        for mp in multiset_permutations(digset, digits-1):
            t = int(d1 + "".join(mp))
            yield t
def afind(maxdigits):
    for digits in range(1, maxdigits+1):
        for t in zplus1(digits):
            if istri(t):
                print(t, end=", ")
afind(22) # Michael S. Branicky, Mar 02 2022

Extensions

a(24)-a(25) from Michael S. Branicky, Mar 02 2022