A321962 - OEIS

A321962

Where the zeros in A123066 occur.

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

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OFFSET

1,2

COMMENTS

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.

LINKS

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.

MAPLE

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);

MATHEMATICA

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

A321962List[n_] := Position[A123066[n], 0] // Flatten;

A321962List[10000]

CROSSREFS