A260191 Numbers m such that there exists no square whose base-m digit sum is binomial(m,2).

3, 5, 13, 17, 21, 29, 37, 45, 49, 53, 61, 65, 69, 77, 81, 85, 93, 101, 109, 113, 117, 125, 133, 141, 145, 149, 157, 165, 173, 177, 181, 189, 193, 197, 205, 209, 213, 221, 229, 237, 241, 245, 253, 257, 261, 269, 273, 277, 285, 293, 301, 305, 309, 317, 321, 325

Offset: 1

Author

Jon E. Schoenfield, Jul 18 2015

Keywords

• nonn
• base

Comments

After the initial term a(1)=3 (see Example), this sequence consists of all numbers of the form (2j-1)*4^k+1 where j and k are positive integers.

For each term m > 3, no square has a base-m digit sum == binomial(m,2) (mod 4).

After the initial term a(1)=3, is this A249034?

Examples

			No square has a base-3 digit sum of exactly binomial(3,2)=3, so 3 is in the sequence.
Binomial(5,2) = 10 == 2 (mod 4), but no square has a base-5 digit sum == 2 (mod 4), so there cannot be a square whose base-5 digit sum is 10; thus, 5 is in the sequence.
		

Crossrefs

Cf. A007953, A096008.

Programs

Python

from itertools import count, islice
def A260191_gen(startvalue=3): # generator of terms >= startvalue
    c = max(startvalue,3)
    if c<=3: yield 3
    for n in count(c+(c&1^1),2):
        if (~(m:=n-1>>1) & m-1).bit_length()&1:
            yield n
A260191_list = list(islice(A260191_gen(),20)) # Chai Wah Wu, Feb 26 2024