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A384436 a(n) is the number of distinct ways to represent n in any integer base >= 2 using only square digits.

%I A384436 #9 Jun 03 2025 17:13:05
%S A384436 1,1,1,2,4,3,3,3,3,6,5,4,5,5,4,4,6,5,4,5,7,7,5,5,7,8,6,6,8,7,7,7,7,7,
%T A384436 7,6,11,9,6,7,10,7,7,7,8,8,8,6,8,11,7,7,9,10,7,7,10,10,7,7,11,10,7,7,
%U A384436 13,11,7,7,11,10,7,7,10,11,8,8,11,11,9,8,11,15
%N A384436 a(n) is the number of distinct ways to represent n in any integer base >= 2 using only square digits.
%C A384436 The representations of n remain the same for bases greater than n, as they all consist solely of the digit n.
%H A384436 Felix Huber, <a href="/A384436/b384436.txt">Table of n, a(n) for n = 0..10000</a>
%F A384436 Trivial lower bound for n >= 2: a(n) >= 2 for nonsquares n and a(n) >= 3 for squares n because in base 2 the representations of n consists only of the square digits '0' and '1', in base n the representation of n is [1,0] and in bases > n the representation of n is [n].
%e A384436 The a(36) = 11 distinct ways to represent 36 using only square digits are [1,0,0,1,0,0] in base 2, [1,1,0,0] in base 3, [1,0,0] in base 6, [4,4] in base 8, [4,0] in base 9, [1,16] in base 20, [1,9] in base 27, [1,4] in base 32, [1,1] in base 35, [1,0] in base 36 and [36] in bases >= 37.
%p A384436 A384436:=proc(n)
%p A384436     local a,b,c;
%p A384436     a:=0;
%p A384436     for b from 2 to n+1 do
%p A384436         c:=convert(n,'base',b);
%p A384436         if select(issqr,c)=c then
%p A384436             a:=a+1
%p A384436         fi
%p A384436     od;
%p A384436     return max(1,a)
%p A384436 end proc;
%p A384436 seq(A384436(n),n=0..81);
%t A384436 a[n_] := Sum[Boole[AllTrue[IntegerDigits[n, b], IntegerQ[Sqrt[#]] &]], {b, 2, n+1}]; a[0] = 1; Array[a, 100, 0] (* _Amiram Eldar_, May 29 2025 *)
%o A384436 (PARI) a(n) = sum(b=2, n+1, my(d=digits(n,b)); #select(issquare, d) == #d); \\ _Michel Marcus_, May 29 2025
%Y A384436 Cf. A046030, A055240, A061845, A077268, A126071, A135551, A384211, A384212.
%K A384436 nonn,base
%O A384436 0,4
%A A384436 _Felix Huber_, May 29 2025