A376336 Least positive integer m such that tau(m) is divisible by n, where the Ramanujan tau function is given by A000594.

1, 2, 2, 2, 5, 2, 3, 2, 3, 5, 8, 2, 7, 3, 5, 4, 239, 3, 89, 8, 3, 8, 4, 2, 25, 7, 6, 3, 13, 5, 139, 4, 8, 239, 5, 3, 191, 89, 11, 8, 257, 3, 19, 8, 10, 4, 67, 6, 15, 40, 239, 7, 107, 6, 8, 6, 89, 13, 61, 8, 9, 139, 3, 4, 35, 8, 31, 239, 5, 5, 137, 6, 2069, 191, 40, 178, 19, 11, 25, 8, 9, 257, 227, 3, 239, 19, 26, 8, 59, 10, 7, 4, 278, 67, 89, 6, 863, 15, 24, 40

Offset

1,2

Comments

Conjecture: a(n) exists for any positive integer n. Moreover, a(n) <= n^2 for all n > 0.

It seems that for some primes p the value of a(p) is relatively large. For example, 4327 is a prime with a(4327) = 316717, 9133 is a prime with a(9133) = 789977, and 9643 is a prime with a(9643) = 1001401.

Links

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

Example

a(5) = 5 since tau(5) = 4830 is divisible by 5, but none of tau(1) = 1, tau(2) = -24, tau(3) = 252 and tau(4) = -1472 is a multiple of 5.

Mathematica

f[n_]:=f[n]=RamanujanTau[n]; L={}; Do[m=1; Label[bb]; If[Mod[f[m], n]==0, L=Append[L, m]; Goto[aa]]; m=m+1; Goto[bb]; Label[aa], {n, 1, 100}]; Print[L]
(* Alternative: *)
a[n_] := SelectFirst[Range[1, 30000], Divisible[RamanujanTau[#], n] &]; Array[a, 1000] (* Peter Luschny, Dec 22 2024 *)

Prog

(SageMath)
from itertools import count
tau = delta_qexp(30000)  # adjust search length for n > 1000
a = lambda n: next((k for k in count(1) if n.divides(tau[k])))
print([a(n) for n in srange(1, 1001)])  # Peter Luschny, Dec 22 2024
(PARI) first(n) = {
	my(todo = [1..n], res = vector(n, i, oo));
	for(i = 1, oo,
		c = ramanujantau(i);
		for(j = 1, #todo,
			if(res[todo[j]] > i && c % todo[j] == 0,
				res[todo[j]] = i;
				todo = setminus(todo, Set(todo[j]));
				if(#todo == 0,
					return(res)
				)
			)
		);
); } \\ David A. Corneth, Dec 23 2024

Crossrefs

Cf. A000594.

Sequence in context: A130155 A350403 A113516 * A226525 A120642 A216624

Adjacent sequences: A376333 A376334 A376335 * A376337 A376338 A376339

Keyword

nonn,changed

Author

Zhi-Wei Sun, Dec 22 2024

Status

approved