A392667 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.

1, 7, 11, 13, 13, 19, 29, 29, 29, 43, 47, 89, 89, 89, 89, 89, 97, 97, 173, 173, 173, 173, 173, 173, 173, 173, 257, 257, 257, 257, 257, 257, 257, 397, 397, 433, 625, 641, 641, 641, 641, 641, 641, 641, 641, 641, 641, 641, 641, 641, 641, 751, 751, 751, 751, 751, 751, 751, 751, 751

Offset

1,2

Comments

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

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

Links

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

Zhi-Wei Sun, On functions taking only prime values, J. Number Theory, 133 (2013), no. 8, 2794-2812.

Example

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.

Mathematica

t[n_]:=t[n]=RamanujanTau[n]; f[n_]:=f[n]=t[n]*(t[n]-1);
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];
Label[aa], {n, 1, 60}]; Print[tab]

Prog

(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

Crossrefs

Cf. A000040, A000594, A390228.

Sequence in context: A200327 A206546 A275516 * A084451 A091901 A072823

Adjacent sequences: A392664 A392665 A392666 * A392668 A392669 A392670

Keyword

nonn

Author

Zhi-Wei Sun, Jan 18 2026

Status

approved