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A065081 Alternating bit sum (A065359) for n-th prime p: replace 2^k with (-1)^k in binary expansion of p.

%I A065081 #22 Jul 20 2024 12:50:07
%S A065081 -1,0,2,1,-1,1,2,1,2,2,1,1,-1,-2,-1,2,-1,1,1,2,1,1,2,2,1,2,1,-1,1,2,1,
%T A065081 -1,-1,-2,2,1,1,-2,-1,-1,-1,1,-1,1,2,1,1,1,-1,1,-1,-1,1,-1,2,2,2,1,4,
%U A065081 2,1,2,1,2,1,2,1,4,2,4,2,2,1,4,1,2,2,1,2,1,-1,1,-1,1,1,-1,2,1,2,1,2,2,1,-1,1,2,2,-1,-2,1
%N A065081 Alternating bit sum (A065359) for n-th prime p: replace 2^k with (-1)^k in binary expansion of p.
%C A065081 Only 3d = 11b has an alternating sum of 0.
%H A065081 Harry J. Smith, <a href="/A065081/b065081.txt">Table of n, a(n) for n=1..1000</a>
%H A065081 William Paulsen, wpaulsen(AT)csm.astate.edu, <a href="http://www.csm.astate.edu/~wpaulsen/primemaze/mazepart.html">Partitioning the [prime] maze</a>
%e A065081 The sixth prime is 13d = 1101b -> -(1)+(1)-(0)+(1) = 1 = a(6)
%t A065081 f[n_] := (d = Reverse[ IntegerDigits[n, 2]]; l = Length[d]; s = 0; k = 1; While[k < l + 1, s = s - (-1)^k*d[[k]]; k++ ]; s); Table[ Prime[ f[n]], {n, 1, 100} ]
%o A065081 (PARI)
%o A065081 baseE(x, b)=
%o A065081 {
%o A065081   local(d, e=0, f=1);
%o A065081   while (x>0, d=x-b*(x\b); x\=b; e+=d*f; f*=10);
%o A065081   return(e)
%o A065081 }
%o A065081 SumAD(x)=
%o A065081 {
%o A065081   local(a=1, s=0);
%o A065081   while (x>9, s+=a*(x-10*(x\10)); x\=10; a=-a);
%o A065081   return(s + a*x)
%o A065081 }
%o A065081 { for (n=1, 1000, p=prime(n);
%o A065081   s=SumAD(baseE(p, 2)); write("b065081.txt", n, " ", s) )
%o A065081 } \\ _Harry J. Smith_, Oct 06 2009
%o A065081 (PARI)
%o A065081 f(p)=
%o A065081 {
%o A065081   v=binary(p);
%o A065081   L=#v; u=1; s=0;
%o A065081   forstep(k=L,1,-1, if(v[k]==1,s+=u); u=-u;);
%o A065081   return(s)
%o A065081 };
%o A065081 for(n=1,100,p=prime(n); an=f(p);print1(an,", ")) \\ _Washington Bomfim_, Jan 16 2011
%o A065081 (Python)
%o A065081 from sympy.ntheory import digits, prime
%o A065081 def A065081(n): return sum((0,1,-1,0)[i] for i in digits(prime(n),4)[1:]) # _Chai Wah Wu_, Jul 19 2024
%Y A065081 Cf. A065359.
%K A065081 base,easy,sign
%O A065081 1,3
%A A065081 _Robert G. Wilson v_, Nov 09 2001