A121940 Product of the first n primes of the form 6k+1.

7, 91, 1729, 53599, 1983163, 85276009, 5201836549, 348523048783, 25442182561159, 2009932422331561, 194963444966161417, 20081234831514625951, 2188854596635094228659, 277984533772656967039693, 38639850194399318418517327, 5834617379354297081196116377

Offset: 1

Author

Jonathan Vos Post, Sep 03 2006

Keywords

• easy
• nonn

Comments

For n>1, a(n) is the least positive integer that can be primitively represented as m^2+mn+n^2 with 0<=m<=n and gcd(m,n)=1 in exactly 2^(n-1) ways. - Ray Chandler, Oct 01 2007

From Bernard Schott, Mar 30 2021: (Start)

Also, for n >= 1, a(n) is the smallest positive integer m such that m^2 can be primitively represented as k^2-k*q+q^2 with 1 <= k < q and gcd(k,q)= 1 in exactly 2^n ways. For example (a(1))^2 = 7^2 = 3^2 - 3*8 + 8^2 = 5^2 - 5*8 + 8^2.

It follows that a(n) is the smallest middle side b that appears exactly 2^n times consecutively in the data of A335895, for integer-sided triangles whose angles A < B < C are in arithmetic progression. (End)

Also, a(n) is the smallest largest side c that appears exactly 2^(n-1) times consecutively in the data of A357277 for integer-sided triangles with angles A < B < C = 2*Pi/3 = 120 degrees. - Bernard Schott, Oct 21 2022

Examples

			a(4) = 53599 = 7 * 13 * 19 * 31.
		

Links

• Ray Chandler, Table of n, a(n) for n=1..100

Crossrefs

Cf. A002476, A002110, A006278, A078586, A335895.

Greedy inverse of A005088.

Programs

Mathematica

Rest@FoldList[Times, 1, Select[6 Range[100] + 1, PrimeQ]] (* Ray Chandler, Oct 01 2007 *)

PARI

lista(nn) = {my(pr=1, list=List()); forprime(p=1, nn, if ((p%3) == 1, listput(list,pr *= p));); Vec(list);} \\ Michel Marcus, Jul 17 2020

Formula

a(n) = Product_{i=1..n} A002476(i).

Extensions

Extended by Ray Chandler, Oct 01 2007