A391884 Numbers k such that k^2 - 1 is one of Ramanujan's largely composite numbers (A067128).

2, 3, 5, 7, 11, 13, 19, 29, 41, 71, 181, 209, 701, 1189, 4159, 43471

Offset

1,1

Comments

Is this a finite sequence?

Ramanujan's largely composite numbers (A067128) are numbers m, such that there is no number from 1 to m - 1 with more divisors than m.

Also numbers k such that (k - 1)*(k + 1) is a Ramanujan's largely composite number.

There is no number k such that 208 < k < 10000 such that A067128(k) + 1 is a square number.

Links

Table of n, a(n) for n=1..16.

Example

13 is in this sequence because 13^2 - 1 = 168 which is a Ramanujan's largely composite number, which is because 168 has 16 divisors, and there is no number between 1 and 167 which has more than 16 divisors.
23 is not in this sequence because 23^2 - 1 = 528 which is not a Ramanujan's largely composite number, which is because 528 has 20 divisors, and between 1 and 527, the numbers 360, 420, 480 and 504 all have more than 20 divisors, and all of them have 24 divisors.

Mathematica

r = 0; Reap[Do[d = DivisorSigma[0, n]; If[d >= r, If[IntegerQ[#], Sow[#]] &[Sqrt[n + 1]]; If[d > r, r = d]], {n, 2^22}] ][[-1, 1]] (* Michael De Vlieger, Jan 31 2026 *)

Prog

(PARI) is_A067128(n) = my(nd=numdiv(n)); for(k=1, n-1, if(numdiv(k) > nd, return(0))); return(1)
for(n=2, 1e5, is_A067128(n^2-1) && print1(n", "))

Crossrefs

Cf. A000290, A067128.

Sequence in context: A113188 A358718 A242738 * A079153 A020616 A275272

Adjacent sequences: A391881 A391882 A391883 * A391885 A391886 A391887

Keyword

nonn,more

Author

Zhicheng Wei, Jan 26 2026

Status

approved