A322665 Partial sums of A089451.

1, 0, 0, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 1, 2, 2, 3, 3, 2, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, -1, -1, -1, -1, -1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, -1, -1, -1, 0, 0, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 0, 0, -1

Offset

1,5

Comments

a(n) is the number of primes p among the first n ones such that the sum of primitive roots is congruent to +1 modulo p minus the number of primes p among the first n ones such that the sum of primitive roots is congruent to -1 modulo p. Here, the prime number 2 is counted in the minuends but not in the subtrahends.

Although there are more positive terms among the first few ones, there are 5887 negative ones among the first 10000 terms, along with 237 zeros.

The largest terms among the first 10000 ones are a(n) = 41 for n in {8389, 8749, 8750, 8751, 8752, 8753}, and the smallest being a(n) = -41 for n in {4037, 4038, 4039, 4040, 4041, 4043, 4044, 4045, 4063, 4064, 4065, 4081, 4082, 4083, 4086, 4098, 4099, 4100}. What is the rate of growth for sup_{i=1..n} a(i) and inf_{i=1..n} a(i)?

Links

Table of n, a(n) for n=1..85.

Example

prime(11) = 31, mu(1) = mu(6) = mu(10) = mu(22) = +1, mu(2) = mu(30) = -1, so a(11) = 4 - 2 = 2.
prime(22) = 79, mu(1) = mu(6) = mu(10) = mu(22) = mu(46) = mu(58) = +1, mu(2) = mu(30) = mu(42) = mu(66) = mu(70) = mu(78) = -1, so a(22) = 6 - 6 = 0.

Prog

(PARI) a(n) = sum(i=1, n, moebius(prime(i)-1))

Crossrefs

Cf. A089451.

Cf. Also A049092, A078330, A088144, A088179.

Sequence in context: A342881 A292137 A292138 * A368857 A273632 A347387

Adjacent sequences: A322662 A322663 A322664 * A322666 A322667 A322668

Keyword

sign

Author

Jianing Song, Dec 22 2018

Status

approved