A392667 - OEIS

%I #17 Jan 20 2026 22:34:04

%S 1,7,11,13,13,19,29,29,29,43,47,89,89,89,89,89,97,97,173,173,173,173,

%T 173,173,173,173,257,257,257,257,257,257,257,397,397,433,625,641,641,

%U 641,641,641,641,641,641,641,641,641,641,641,641,751,751,751,751,751,751,751,751,751

%N Least positive integer m such that tau(k)*(tau(k)-1) (k = 1..n) are incongruent modulo m, where tau(.) is Ramanujan's tau function.

%C Conjecture: a(n) is prime except for n = 1, 37.

%C This has been verified for n <= 13585. Note that a(1) = 1 and a(37) = 5^4.

%H Zhi-Wei Sun, <a href="/A392667/b392667.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="https://doi.org/10.1016/j.jnt.2013.02.003">On functions taking only prime values</a>, J. Number Theory, 133 (2013), no. 8, 2794-2812.

%e a(2) = 7 since tau(1)*(tau(1)-1) = 1*(1-1) = 0 and tau(2)*(tau(2)-1) = -24*(-24-1) = 600 are incongruent modulo 7.

%t t[n_]:=t[n]=RamanujanTau[n]; f[n_]:=f[n]=t[n]*(t[n]-1);

%t tab={};m=1;Do[Label[bb]; If[Length[Union[Table[Mod[f[k], m], {k, 1, n}]]]==n,tab=Append[tab,m]; Goto[aa]]; m=m+1; Goto[bb];

%t Label[aa], {n, 1, 60}];Print[tab]

%o (PARI) a(n) = my(v=vector(n, k, ramanujantau(k)*(ramanujantau(k)-1)), m=1); while (#Set(apply(x->Mod(x,m), v)) != #v, m++); m; \\ _Michel Marcus_, Jan 19 2026

%Y Cf. A000040, A000594, A390228.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Jan 18 2026