A319502 Indices of Mertens's function M(n) (A002321) whose nearest neighbors have value 0.

165, 237, 330, 354, 357, 365, 402, 406, 421, 426, 794, 797, 813, 885, 894, 897, 905, 914, 1257, 1281, 1290, 1298, 1301, 1337, 1522, 1526, 1545, 1842, 1865, 2094, 2098, 2118, 2121, 2137, 4602, 4609, 4621, 4629, 4726, 4729, 4738, 5106, 5109, 5198, 5206, 5221

Offset: 1

Author

Torlach Rush, Sep 20 2018

Keywords

• nonn

Comments

This sequence records the shortest intervals where M(n) leaves 0 before returning to 0.

a(n) - 1 and a(n) + 1 are both terms of A028442.

Both A045882 and A028442 are infinite and this allows for the possibility that this sequence is also infinite (for A028442 see comment of A002321).

Examples

			165 is a term because A002321(164) = A002321(166) = 0.
237 is a term because A002321(236) = A002321(238) = 0.
		

Crossrefs

Cf. A002321, A008683, A028442, A045882.

Programs

Maple

with(numtheory): a:=n->add(mobius(k),k=1..n): select(n->a(n-1)=0 and a(n+1)=0,[$2..2200]); # Muniru A Asiru, Sep 20 2018

Mathematica

With[{s = Partition[Accumulate@ Array[MoebiusMu, 5300], 3, 1]}, 1 + First /@ Position[s, {0, k_, 0} /; k != 0]] (* Michael De Vlieger, Sep 24 2018 *)

PARI

isok(n) = {if (n > 1, x = sum(k=1, n-1, moebius(k)); if (x == 0, if (x + moebius(n) + moebius(n+1) == 0, return (1)););); return (0);} \\ Michel Marcus, Sep 27 2018

Formula

(A002321(a(n)) - A008683(a(n))) = (A002321(a(n)) + A008683(a(n+1))) = (A008683(a(n)) + A008683(a(n+1))) = 0.