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A322665
Partial sums of A089451.
0
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)?
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.
Sequence in context: A342881 A292137 A292138 * A368857 A273632 A347387
KEYWORD
sign
AUTHOR
Jianing Song, Dec 22 2018
STATUS
approved