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A321962 Where the zeros in A123066 occur.

%I A321962 #22 Oct 09 2024 09:11:19
%S A321962 1,51,53,7955,7959,7961,7985,7987,8245,8805,8807,8809,8813,8815,8817,
%T A321962 8819,8821,8825,8829,8847,8851,8853,8855,8857,8859,8873,8877,8879,
%U A321962 8969,8973,8975,9063,9079,9081,9083,9089,9091,9093,9095,9097,9163,9165,9215,9219
%N A321962 Where the zeros in A123066 occur.
%C A321962 Let pp(n) be the prime parity of n, defined as 1 if the number of distinct primes dividing n is odd and -1 if it is even; by convention pp(1) = 0. The cumulative sum of pp is A123066. We call the initial segment of the integers [1..n] balanced with respect to prime parity if the cumulative sum of pp(n) is 0. [1..a(n)] give the balanced segments.
%H A321962 Amiram Eldar, <a href="/A321962/b321962.txt">Table of n, a(n) for n = 1..8908</a> (terms below 10^10)
%H A321962 Harald Helfgott and Adrián Ubis, <a href="https://pmu.uy/pmu18/pmu18-0205.pdf">Primos, paridad y análisis</a>, Publicaciones Matemáticas del Uruguay, Vol. 18 (2023), pp. 205-283; <a href="https://arxiv.org/abs/1812.08707">arXiv preprint</a>, arXiv:1812.08707 [math.NT], 2018-2019.
%p A321962 a_list := proc(len) local omega, c, L, j; c := 0; L := 1;
%p A321962 omega := n -> nops(numtheory[factorset](n));
%p A321962 for j from 2 to len do
%p A321962    c := c + (-1)^omega(j);
%p A321962    if c = 0 then L := L,j fi
%p A321962 od; L end: a_list(10000);
%t A321962 A123066[n_] := Join[{0}, Accumulate[Table[-(-1)^PrimeNu[j], {j,2,n}]]];
%t A321962 A321962List[n_] := Position[A123066[n], 0] // Flatten;
%t A321962 A321962List[10000]
%Y A321962 Cf. A001222, A123066, A030230, A030231.
%K A321962 nonn
%O A321962 1,2
%A A321962 _Peter Luschny_, Dec 21 2018