cp's OEIS Frontend

A379654 Least positive integer k <= n such that |tau(k)|*n + 1 is prime, or 0 if such k does not exist, where the tau function is given by A000594.

%I A379654 #15 Dec 30 2024 21:52:48
%S A379654 1,1,2,1,5,1,5,2,3,1,4,1,2,2,11,1,2,1,2,5,13,1,4,2,2,3,5,1,3,1,5,2,3,
%T A379654 6,3,1,21,15,2,1,3,1,2,11,5,1,2,2,6,2,3,1,4,2,2,11,7,1,3,1,3,2,3,5,3,
%U A379654 1,2,3,2,1,4,1,2,2,2,6,10,1,15,3,4,1,2,2,5,3,2,1,2,2,6,12,4,3,2,1,7,3,2,1
%N A379654 Least positive integer k <= n such that |tau(k)|*n + 1 is prime, or 0 if such k does not exist, where the tau function is given by A000594.
%C A379654 Conjecture 1: a(n) > 0 for all n > 0. In other words, for each positive integer n, there is a number k among 1,...,n such that |tau(k)|*n + 1 is prime.
%C A379654 Conjecture 2: For each integer n > 1 not equal to 22, there is a number k among 1,...,n such that |tau(k)|*n - 1 is prime.
%C A379654 We have verified both conjectures for n up to 10^8.
%H A379654 Zhi-Wei Sun, <a href="/A379654/b379654.txt">Table of n, a(n) for n = 1..10000</a>
%e A379654 a(1) = 1 since 1*|tau(1)| + 1 = 2 is a prime.
%e A379654 a(5) = 5 since 5*|tau(5)| + 1 = 5*4830 + 1 = 24151 is prime, and 5*|tau(k)| + 1 is composite for every k = 1, 2, 3, 4.
%t A379654 t[n_]:=t[n]=Abs[RamanujanTau[n]];
%t A379654 L={};Do[Do[If[PrimeQ[t[k]n+1],L=Append[L,k];Goto[aa]],{k,1,n}];L=Append[L,0];Label[aa],{n,1,100}];Print[L]
%Y A379654 Cf. A000040, A000594, A034693.
%K A379654 nonn
%O A379654 1,3
%A A379654 _Zhi-Wei Sun_, Dec 28 2024