A360826 a(1) = 1, a(n) = (k+1)*(2*k+1), where k = Product_{i=1..n-1} a(i).

1, 6, 91, 597871, 213122969971321411, 9680343693975641657052402556458789711774336036960631

Offset: 1

Author

Ivan N. Ianakiev, Feb 22 2023

Keywords

• nonn

Comments

A sequence of pairwise relatively prime triangular (and also hexagonal) numbers.

As a clarification to the problem definition by Sierpinski, here we show that only one triangular (hexagonal) seed is needed to produce such a sequence.

This sequence can be used for proving the infinitude of primes.

In general: Let m = 2*q, for any q > 0. There are infinitely many sequences of pairwise coprime m-gonal numbers, whose first term is any positive m-gonal number and whose general term is of the form a(n) = (k + 1)*((q - 1)*k + 1), where k = Product_{i=1..n-1} a(i).

References

• W. Sierpinski, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #42.

Links

• Michel Marcus, Table of n, a(n) for n = 1..8

Crossrefs

Cf. A000217, A000384, A004019, A034792, A115590.

Programs

Mathematica

a[1]=1; a[n_]:=Module[{k=Product[a[i],{i,1,n-1}]},(k+1)*(2*k+1)];
a/@Range[6]
Join[{1}, RecurrenceTable[{a[2] == 6, a[n+1] == (1 + a[n]*(Sqrt[1 + 8*a[n]] - 3)/4) * (1 + 2*a[n]*(Sqrt[1 + 8*a[n]] - 3)/4)}, a, {n, 2, 8}]] (* Vaclav Kotesovec, May 05 2023 *)

PARI

a(n) = if (n==1, 1, my(k = prod(i=1,n-1, a(i))); (k+1)*(2*k+1)); \\ Michel Marcus, Mar 25 2025

Formula

a(1) = 1, a(n) = (k+1)*(2*k+1), where k = Product_{i=1..n-1} a(i).

a(n) ~ c^(3^n), where c = 1.1784502032269064445225839284451956694752084180050932315805089054871825498... - Vaclav Kotesovec, May 05 2023