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.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Dec 28 2024

Keywords

• nonn

Comments

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.

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.

We have verified both conjectures for n up to 10^8.

Examples

			a(1) = 1 since 1*|tau(1)| + 1 = 2 is a prime.
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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A000594, A034693.

Programs

Mathematica

t[n_]:=t[n]=Abs[RamanujanTau[n]];
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]