A089451 a(n) = mu(prime(n)-1), where mu is the Moebius function (A008683).

1, -1, 0, 1, 1, 0, 0, 0, 1, 0, -1, 0, 0, -1, 1, 0, 1, 0, -1, -1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, 0, 1, 0, -1, 0, 0, 0, 1, -1, 1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 1, 0, 1, 0, 0, 1, -1, 0, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, -1, -1, 0, 0, 0, 1, 1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, -1, 0, 0

Offset: 1

Author

T. D. Noe, Nov 03 2003

Keywords

• sign

Comments

Note that A049092 lists prime(n) such that a(n) = 0. Similarly, A078330 lists prime(n) such that a(n) = -1. See A088179 for prime(n) such that a(n) = 1. Also note that a(n) == A088144(n) (mod prime(n)).

References

• J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 236.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Ed Pegg, Jr., Moebius Function (and squarefree numbers)
• Eric Weisstein's World of Mathematics, Moebius Function

Crossrefs

Cf. A089495 (mu(p+1) for prime p), A089496 (mu(p+1)+mu(p-1) for prime p), A089497 (mu(p+1)-mu(p-1) for prime p).

Cf. A049092, A078330, A088144, A088179.

Programs

Magma

[MoebiusMu(NthPrime(n)-1): n in [1..100]]; // Vincenzo Librandi, Dec 23 2018

Mathematica

Table[MoebiusMu[Prime[n]-1], {n, 150}]

PARI

a(n)=moebius(prime(n)-1)

Formula

a(n) = A067460(n) - 1. - Benoit Cloitre, Nov 04 2003

If p = prime(n), then a(n) is congruent modulo p to the sum of all primitive roots modulo p. [Uspensky and Heaslet]. - Michael Somos, Feb 16 2020