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

%I A376336 #38 Dec 26 2024 20:09:21
%S A376336 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,
%T A376336 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,
%U A376336 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
%N A376336 Least positive integer m such that tau(m) is divisible by n, where the Ramanujan tau function is given by A000594.
%C A376336 Conjecture: a(n) exists for any positive integer n. Moreover, a(n) <= n^2 for all n > 0.
%C A376336 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.
%H A376336 Zhi-Wei Sun, <a href="/A376336/b376336.txt">Table of n, a(n) for n = 1..10000</a>
%e A376336 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.
%t A376336 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]
%t A376336 (* Or: *)
%t A376336 a[n_] := SelectFirst[Range[1, 30000], Divisible[RamanujanTau[#], n] &]; Array[a, 1000] (* _Peter Luschny_, Dec 22 2024 *)
%o A376336 (SageMath)
%o A376336 from itertools import count
%o A376336 tau = delta_qexp(30000)  # adjust search length for n > 1000
%o A376336 a = lambda n: next((k for k in count(1) if n.divides(tau[k])))
%o A376336 print([a(n) for n in srange(1, 1001)])  # _Peter Luschny_, Dec 22 2024
%o A376336 (PARI) first(n) = {
%o A376336 	my(todo = [1..n], res = vector(n, i, oo));
%o A376336 	for(i = 1, oo,
%o A376336 		c = ramanujantau(i);
%o A376336 		for(j = 1, #todo,
%o A376336 			if(res[todo[j]] > i && c % todo[j] == 0,
%o A376336 				res[todo[j]] = i;
%o A376336 				todo = setminus(todo, Set(todo[j]));
%o A376336 				if(#todo == 0,
%o A376336 					return(res)
%o A376336 				)
%o A376336 			)
%o A376336 		);
%o A376336 );} \\ _David A. Corneth_, Dec 23 2024
%Y A376336 Cf. A000594.
%K A376336 nonn
%O A376336 1,2
%A A376336 _Zhi-Wei Sun_, Dec 22 2024