A219017 Smallest number k such that k^2 - 1 has exactly n distinct prime factors.

2, 4, 11, 29, 131, 419, 1429, 14629, 77141, 509081, 1456729, 22486309, 117048931, 1625292241, 10326137821, 117440297701, 1110819807371, 8678298841211, 138645880242871, 980010587880169

Offset: 1

Author

Michel Lagneau, Nov 09 2012

Keywords

• nonn
• hard
• more

Comments

a(14) <= 1625292241. - Donovan Johnson, Nov 10 2012

Examples

			a(3) = 11 is the smallest number of the set {k(i)} = {11, 13, 14, 16, 19, 20, ...} where k(i)^2 - 1 contains 3 distinct prime factors.
a(10) = 509081 because 509081^2-1 = 2 ^ 4 * 3 * 5 * 7 * 11 * 13 * 17 * 23 * 31 * 89 with 10 distinct prime factors.
		

Links

• Michael S. Branicky, Python program

Crossrefs

Cf. A180278.

Programs

Maple

with(numtheory) :for n from 1 to 11 do:ii:=0:for k from 1 to 10^10 while(ii=0) do:x:=k^2-1:y:=factorset(x):n1:=nops(y):if n1=n then ii:=1: printf ( "%d %d \n",n,k):
else fi:od:od:

Mathematica

L = {}; Do[n = 2; While[Length[FactorInteger[n^2 - 1]] != k, n++];  Print@AppendTo[L, n], {k, 15}] (* Giovanni Resta, Nov 10 2012 *)

Extensions

a(12)-a(13) from Donovan Johnson, Nov 10 2012

a(14)-a(17) from Giovanni Resta, May 10 2017

a(18) from Michael S. Branicky, Feb 08 2023

a(19) from Michael S. Branicky, Feb 15 2023

a(20) from Michael S. Branicky, Feb 19 2023

Name clarified by Pontus von Brömssen, Sep 12 2023