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

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

Table of n, a(n) for n=1..44.

H. Helfgott and A. Ubis, Primos, paridad y análisis, arXiv:1812.08707 [math.NT], Dec. 2018.

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

Cf. A001222, A123066, A030230, A030231.

Sequence in context: A144162 A169984 A042299 * A060033 A273199 A095525

Adjacent sequences: A321959 A321960 A321961 * A321963 A321964 A321965

Keyword

nonn

Author

Peter Luschny, Dec 21 2018

Status

approved