A352148 Integers m such that nonzero digits of A000217(m) are all the same.

0, 1, 2, 3, 4, 10, 11, 24, 36, 44, 77, 100, 141, 363, 1000, 1095, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 10000000000000000

Offset: 1

Author

Steven Lu, Mar 06 2022

Keywords

• nonn
• base
• more

Crossrefs

Cf. A000217, A352057 (resulting triangular numbers).

Programs

Mathematica

(Sqrt[8 # + 1] - 1)/2 & /@
Sort[Select[
   Flatten[Outer[Times,
     Table[FromDigits[IntegerDigits[n, 2]], {n, 2^22 - 1}],
     Range[9]]], IntegerQ[Sqrt[8 # + 1]] &]]

PARI

isok(m) = #Set(select(x->(x>0), digits(m*(m+1)/2))) <= 1; \\ Michel Marcus, Mar 06 2022

Python

from itertools import count, islice
from sympy import integer_nthroot
def A352148_gen(): # generator of terms
    yield 0
    for l in count(0):
        for d in range(1,10):
            for m in range(2**l,2**(l+1)):
                a, b = integer_nthroot(8*d*int(bin(m)[2:])+1,2)
                if b:
                    yield (a-1)//2
A352148_list = list(islice(A352148_gen(),10)) # Chai Wah Wu, Apr 08 2022

Formula

Conjecture: a(n) = 10^(n-13) for n >= 17.

Extensions

a(25)-a(27) from Chai Wah Wu, Apr 08 2022

a(28)-a(29) from Chai Wah Wu, Apr 09 2022