A089496 a(n) = mu(prime(n)+1) + mu(prime(n)-1), where mu is the Moebius function.

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

Offset

1

Comments

This sum is always -1, 0 or 1 because for odd prime p, both p-1 and p+1 cannot be squarefree; one of them will be divisible by 4. This also implies that terms in this sequence are zero only for 2 and odd primes p such that mu(p-1) = mu(p+1) = 0, which is A075432.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Moebius Function

Eric Weisstein's World of Mathematics, Legendre Symbol

Formula

Let p = prime(n), then a(n) = mu(p+(-1/p)), where (-1/p) is the Legendre symbol, A070750. (Pieter Moree). (This is true for n > 1) - Antti Karttunen, Jul 23 2017

Mathematica

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

Prog

(PARI) A089496(n) = (moebius(prime(n)-1)+moebius(prime(n)+1)); \\ Antti Karttunen, Jul 23 2017

Crossrefs

Cf. A000040, A008683, A089451 (mu(p-1) for prime p), A089495 (mu(p+1) for prime p), A089497 (mu(p+1)-mu(p-1) for prime p).

Sequence in context: A080110 A286419 A257799 * A182067 A196147 A097325

Adjacent sequences: A089493 A089494 A089495 * A089497 A089498 A089499

Keyword

sign

Author

T. D. Noe, Nov 04 2003

Extensions

Term a(1) = 0 prepended by Antti Karttunen, Jul 23 2017

Status

approved