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The unique function f with f(1)=1 and f(jD!+k)=(-1)^j f(k) for all D, j=1..D, and k=1..D!.
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%I #60 Sep 27 2024 23:16:43

%S 1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,-1,1,1,

%T -1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,1,-1,-1,1,1,-1,

%U -1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1

%N The unique function f with f(1)=1 and f(jD!+k)=(-1)^j f(k) for all D, j=1..D, and k=1..D!.

%C sup_n |Sum_{j=1..n} f(jd)| is finite (but not bounded) for all d, thus giving a counterexample to a strong form of the Erdős discrepancy conjecture (see Remark 1.14 on p. 7 of Tao paper).

%H Antti Karttunen, <a href="/A262725/b262725.txt">Table of n, a(n) for n = 1..40320</a>

%H Terence Tao, <a href="http://arxiv.org/abs/1509.05363">The Erdős discrepancy problem</a>, arXiv:1509.05363 [math.CO], 2016.

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>

%F a(n) = (-1)^A034968(n-1). - _Antti Karttunen_ and _Peter Munn_, Aug 09 2024

%t (* Generates terms recursively, using definition *)

%t Module[{dmax = 4, i = 2, a}, a = Table[1, (dmax+1)!]; For[d = 1, d <= dmax, d++, For[j = 1, j <= d, j++, For[k = 1, k <= d!, k++, a[[i++]] = (-1)^j*a[[k]]]]]; a] (* _Paolo Xausa_, Aug 10 2024 *)

%t (* Generates terms individually, via A034968 (slower) *)

%t A034968[n_] := Module[{a = n, i = 2}, While[i! <= n, a-=(i-1)*Floor[n/i++!]]; a];

%t Array[(-1)^A034968[#] &, 5!, 0] (* _Paolo Xausa_, Aug 10 2024 *)

%o (Sage)

%o A=[1,1]

%o for D in [1..4]:

%o j=1

%o while j<=D:

%o k=1

%o while k<=factorial(D):

%o A.append((-1)^j*A[k])

%o k+=1

%o j+=1

%o A[1:73] # _Tom Edgar_, Sep 29 2015

%o (PARI)

%o A034968(n) = { my(s=0, b=2, d); while(n, d = (n%b); s += d; n = (n-d)/b; b++); (s); };

%o A262725(n) = ((-1)^A034968(n-1)); \\ _Antti Karttunen_, Aug 09 2024

%o (Python)

%o from math import factorial

%o def aupto(n):

%o f = [1, 1]

%o for D in range(1, 5):

%o for j in range(1, D + 1):

%o sign = (-1) ** j

%o f.extend(sign * f[k] for k in range(1, factorial(D) + 1))

%o return f[1:n] # _Paul Muljadi_, Sep 26 2024

%Y Cf. A034968, A237695, A374468.

%Y Cf. also A343785 (another example from Tao paper).

%K easy,sign

%O 1

%A _Terence Tao_, Sep 28 2015