OFFSET
1,6
COMMENTS
The alternating sum of digits of n is equal to 1 for base n and equal to n for bases > n.
LINKS
Felix Huber, Table of n, a(n) for n = 1..10000
FORMULA
Trivial bounds: 1 <= a(n) <= n - 2 for n >= 3 because the representation of n in base n-1 is [1,1] and the alternating sum of digits of n is > 0 for bases >= n.
EXAMPLE
a(72) = 10 because the alternating sum of digits of n is equal to 0 in the 10 bases 2 [1, 0, 0, 1, 0, 0, 0], 3 [2, 2, 0, 0], 5 [2, 4, 2], 7 [1, 3, 2], 8 [1, 1, 0], 11 [6, 6], 17 [4, 4], 23 [3, 3], 35 [2, 2] and 71 [1, 1].
MAPLE
A384212:=proc(n)
local a, b, c, i;
a:=0;
for b from 2 to n-1 do
c:=convert(n, 'base', b);
if add(c[i]*(-1)^i, i=1..nops(c))=0 then
a:=a+1
fi
od;
return a
end proc;
seq(A384212(n), n=1..87);
A384212bases:=proc(n)
local L, b, c, i;
L:=[];
for b from 2 to n-1 do
c:=convert(n, 'base', b);
if add(c[i]*(-1)^i, i=1..nops(c))=0 then
L:=[op(L), b, ListTools:-Reverse(c)]
fi
od;
return op(L)
end proc;
A384212bases(72);
MATHEMATICA
q[n_, b_] := Module[{d = IntegerDigits[n, b]}, Sum[(-1)^k*d[[k]], {k, 1, Length[d]}] == 0 ]; a[n_] := Count[Range[2, n-1], _?(q[n, #] &)]; Array[a, 100] (* Amiram Eldar, May 24 2025 *)
PROG
(Python)
from sympy.ntheory import digits
def s(v): return sum(v[::2]) - sum(v[1::2])
def a(n): return sum(1 for b in range(2, n) if s(digits(n, b)[1:]) == 0)
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, May 24 2025
(PARI) a(n) = sum(b=2, n-1, my(d=digits(n, b)); sum(k=1, #d, (-1)^k*d[k]) == 0); \\ Michel Marcus, May 24 2025
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Felix Huber, May 24 2025
STATUS
approved
