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A321962 Where the zeros in A123066 occur. 0
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 (list; graph; refs; listen; history; text; internal format)
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 anĂ¡lisis, 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

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Last modified October 24 18:08 EDT 2021. Contains 348233 sequences. (Running on oeis4.)