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

%I A260191 #18 Feb 26 2024 17:09:10
%S A260191 3,5,13,17,21,29,37,45,49,53,61,65,69,77,81,85,93,101,109,113,117,125,
%T A260191 133,141,145,149,157,165,173,177,181,189,193,197,205,209,213,221,229,
%U A260191 237,241,245,253,257,261,269,273,277,285,293,301,305,309,317,321,325
%N A260191 Numbers m such that there exists no square whose base-m digit sum is binomial(m,2).
%C A260191 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.
%C A260191 For each term m > 3, no square has a base-m digit sum == binomial(m,2) (mod 4).
%C A260191 After the initial term a(1)=3, is this A249034?
%e A260191 No square has a base-3 digit sum of exactly binomial(3,2)=3, so 3 is in the sequence.
%e A260191 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.
%o A260191 (Python)
%o A260191 from itertools import count, islice
%o A260191 def A260191_gen(startvalue=3): # generator of terms >= startvalue
%o A260191     c = max(startvalue,3)
%o A260191     if c<=3: yield 3
%o A260191     for n in count(c+(c&1^1),2):
%o A260191         if (~(m:=n-1>>1) & m-1).bit_length()&1:
%o A260191             yield n
%o A260191 A260191_list = list(islice(A260191_gen(),20)) # _Chai Wah Wu_, Feb 26 2024
%Y A260191 Cf. A007953, A096008.
%K A260191 nonn,base
%O A260191 1,1
%A A260191 _Jon E. Schoenfield_, Jul 18 2015