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

1,1

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?

Links

Table of n, a(n) for n=1..56.

Example

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.

Prog

(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

Crossrefs

Cf. A007953, A096008.

Sequence in context: A049282 A003625 A105900 * A094745 A072742 A076189

Adjacent sequences: A260188 A260189 A260190 * A260192 A260193 A260194

Keyword

nonn,base

Author

Jon E. Schoenfield, Jul 18 2015

Status

approved