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

1,1

Comments

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

Links

Table of n, a(n) for n=1..20.

Michael S. Branicky, Python program

Example

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.

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 *)

Crossrefs

Cf. A180278.

Sequence in context: A148148 A148149 A034769 * A285216 A212422 A189832

Adjacent sequences: A219014 A219015 A219016 * A219018 A219019 A219020

Keyword

nonn,hard,more

Author

Michel Lagneau, Nov 09 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

Status

approved