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A089496 a(n) = mu(prime(n)+1) + mu(prime(n)-1), where mu is the Moebius function. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A204437 A286419 A257799 * A182067 A196147 A242647

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

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Last modified November 18 21:51 EST 2017. Contains 294912 sequences.