A080696 Piptorial numbers = product of first n pips or prime-indexed primes.

3, 15, 165, 2805, 86955, 3565155, 210344145, 14093057715, 1169723790345, 127499893147605, 16192486429745835, 2542220369470096095, 455057446135147201005, 86915972211813115391955, 18339270136692567347702505, 4419764102942908730796303705

Offset: 1

Author

Cino Hilliard, Mar 04 2003

Keywords

• easy
• nonn

Comments

The numbers after the first always end in 5. This is obvious since all pips are odd and their product (excluding 5) = 2k+1 and 5*(2k+1) = 10k+5. Sum of reciprocals converges to 0.4064288978193657814428353009..

Examples

			prime(prime(1)), prime(prime(1))*prime(prime(2)), ...
pip(1) = 3, pip(2) = 5, pip(3) = 11; piptorial(3) = 3*5*11 = 165.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..278

Crossrefs

Cf. A006450.

Programs

Mathematica

nn=50;FoldList[Times,1,Transpose[Select[Thread[{Prime[Range[nn]], Range[nn]}], PrimeQ[ Last[#]]&]][[1]]] (* Harvey P. Dale, Jul 05 2011 *)
FoldList[Times,Table[Prime[Prime[n]],{n,20}]] (* Harvey P. Dale, May 06 2018 *)

PARI

piptorial(n) = {sr=0; pr=1; for(x=1,n, y=prime(prime(x)); pr*=y; print1(pr" "); sr+=1.0/pr; ); print(); print(sr) }

PARI

a(n) = prod(k=1, n, prime(prime(k))); \\ Michel Marcus, Mar 15 2021

Python

from sympy import prime, nextprime
def aupton(terms):
  prod, p, alst = 1, 2, []
  while len(alst) < terms:
    p, prod = nextprime(p), prod * prime(p)
    alst.append(prod)
  return alst
print(aupton(16)) # Michael S. Branicky, Mar 15 2021

Formula

a(n) = Product_{k=1..n} prime(prime(k)). - Michel Marcus, Mar 15 2021

Extensions

Name clarified by Michel Marcus, Aug 04 2015

More terms from Harvey P. Dale, May 06 2018