A030231 -id:A030231 - cp's OEIS frontend

1, 51, 53, 7955, 7959, 7961, 7985, 7987, 8245, 8805, 8807, 8809, 8813, 8815, 8817, 8819, 8821, 8825, 8829, 8847, 8851, 8853, 8855, 8857, 8859, 8873, 8877, 8879, 8969, 8973, 8975, 9063, 9079, 9081, 9083, 9089, 9091, 9093, 9095, 9097, 9163, 9165, 9215, 9219

Offset: 1

Peter Luschny, Dec 21 2018

nonn

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.

Amiram Eldar, Table of n, a(n) for n = 1..8908 (terms below 10^10)
Harald Helfgott and Adrián Ubis, Primos, paridad y análisis, Publicaciones Matemáticas del Uruguay, Vol. 18 (2023), pp. 205-283; arXiv preprint, arXiv:1812.08707 [math.NT], 2018-2019.

Cf. A001222, A123066, A030230, A030231.

a_list := proc(len) local omega, c, L, j; c := 0; L := 1;
omega := n -> nops(numtheory[factorset](n));
for j from 2 to len do
   c := c + (-1)^omega(j);
   if c = 0 then L := L,j fi
od; L end: a_list(10000);

A123066[n_] := Join[{0}, Accumulate[Table[-(-1)^PrimeNu[j], {j,2,n}]]];
A321962List[n_] := Position[A123066[n], 0] // Flatten;
A321962List[10000]

cp's OEIS Frontend

A321962 Where the zeros in A123066 occur.

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