A360826 -id:A360826 - cp's OEIS frontend

A362888 a(1) = 1, a(n) = (3k + 1)(6k + 1)(8*k + 1), where k = Product_{i=1..n-1} a(i).

1, 252, 2310152797, 28410981127871160285705816883937448685

Offset: 1

Ivan N. Ianakiev, May 08 2023

nonn

A sequence of pairwise relatively prime hexagonal pyramidal numbers. Its infinitude implies, by the Fundamental theorem of arithmetic, the infinitude of primes.
Building on an idea by Sierpinsky (see References): For m > 5, the general term of the sequence of m-gonal pyramidal numbers is a(n) = n*(n+1)*((m-2)*n - (m-5))/6. Therefore, for m > 5, there are infinitely many sequences of pairwise relatively prime m-gonal pyramidal numbers, with first term any positive m-gonal pyramidal
number and general term of the form a(n) = (3*k + 1)*(6*k + 1)*(2*k*(m - 2) + 1), where k = Product_{i=1..n-1} a(i). Corollary: There are infinitely many sequences of m-gonal pyramidal numbers to base the proof of the infinitude of primes on.

W. Sierpinski, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #43.

Cf. A002412, A360826.

a[1]=1; a[n_]:=Module[{k=Product[a[i],{i,1,n-1}]},(3*k+1)*(6*k+1)*(8*k+1)];
a/@Range[5]

a(1) = 1, a(n) = (3*k + 1)*(6*k + 1)*(8*k + 1), where k = Product_{i=1..n-1} a(i).

cp's OEIS Frontend

A362888 a(1) = 1, a(n) = (3k + 1)(6k + 1)(8*k + 1), where k = Product_{i=1..n-1} a(i).

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A362888 a(1) = 1, a(n) = (3*k + 1)*(6*k + 1)*(8*k + 1), where k = Product_{i=1..n-1} a(i).

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

Formula

A362888 a(1) = 1, a(n) = (3k + 1)(6k + 1)(8*k + 1), where k = Product_{i=1..n-1} a(i).