A123173 Least semiprime s for which the Mertens function M(s) = n.

33, 9, 4, 39, 94, 95, 218, 219, 221, 554, 586, 1357, 1389, 1393, 1403, 1405, 1418, 3227, 3233, 3235, 3239, 3241, 3242, 3277, 3281, 3293, 3295, 8201, 8413, 8486, 8489, 8495, 8491, 8503, 8506, 8507, 8509, 8519, 8511, 11759, 11761, 11762, 11769, 11785, 11771

Offset

-3,1

Links

Jean-François Alcover, Table of n, a(n) for n = -3..100

Formula

a(n) = min{s in A001358 and A002321(s) = n}.

Example

a(-3) = 33 = 3 * 11 = the first semiprime s for which the Mertens function M(s) = -3.
a(-2) = 9 = 3^2 = the first semiprime s for which the Mertens function M(s) = -2.
a(-1) = 4 = 2^2 = the first semiprime s for which the Mertens function M(s) = -1.
a(0) = 39 = 3 * 13 = min{A001358 INTERSECTION A028442} = the first semiprime s for which the Mertens function M(s) = 0.
a(1) = 94 = 2 * 47 = min{A001358 INTERSECTION A118684} = the first semiprime s for which the Mertens function M(s) = 1.
a(2) = 95 = 5 * 19 = the first semiprime s for which the Mertens function M(s) = 2.
a(3) = 341 = 11 * 31 = the first semiprime s for which the Mertens function M(s) = 3.

Mathematica

M[k_] := M[k] = MoebiusMu[Range[k]] // Total;
PO[k_] := PO[k] = PrimeOmega[k];
a[n_] := a[n] = Module[{k}, For[k = 4, True, k++, If[PO[k] == 2 && M[k] == n, Return[k]]]];
Table[Print[n, " ", a[n]]; a[n], {n, -3, 100}] (* Jean-François Alcover, Jan 22 2024 *)

Prog

(PARI) nextsemip(n)=x = n; while (bigomega(x) != 2, x++); x;
a(n) = {sp = 4; while (mertens(sp) != n, sp = nextsemip(sp+1)); sp; } \\ Michel Marcus, Sep 24 2013
(PARI) a(n)=my(s, start=0, step=10^8); while(1, forsquarefree(k=start+1, start+step, s+=moebius(k); if(s==n&&bigomega(k)==2, return(k[1]))); start+=step) \\ Charles R Greathouse IV, Apr 19 2023

Crossrefs

Cf. A001358, A002321, A028442, A118684.

Sequence in context: A113458 A120584 A236177 * A147021 A200897 A138839

Adjacent sequences: A123170 A123171 A123172 * A123174 A123175 A123176

Keyword

nonn

Author

Jonathan Vos Post, Oct 02 2006

Extensions

a(3) corrected and a(4)-a(41) added by Michel Marcus, Sep 24 2013

Status

approved