OFFSET
1,2
COMMENTS
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.
REFERENCES
W. Sierpinski, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #43.
FORMULA
a(1) = 1, a(n) = (3*k + 1)*(6*k + 1)*(8*k + 1), where k = Product_{i=1..n-1} a(i).
MATHEMATICA
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ivan N. Ianakiev, May 08 2023
STATUS
approved