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

%I A080696 #33 Mar 16 2021 04:55:40
%S A080696 3,15,165,2805,86955,3565155,210344145,14093057715,1169723790345,
%T A080696 127499893147605,16192486429745835,2542220369470096095,
%U A080696 455057446135147201005,86915972211813115391955,18339270136692567347702505,4419764102942908730796303705
%N A080696 Piptorial numbers = product of first n pips or prime-indexed primes.
%C A080696 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..
%H A080696 Harvey P. Dale, <a href="/A080696/b080696.txt">Table of n, a(n) for n = 1..278</a>
%F A080696 a(n) = Product_{k=1..n} prime(prime(k)). - _Michel Marcus_, Mar 15 2021
%e A080696 prime(prime(1)), prime(prime(1))*prime(prime(2)), ...
%e A080696 pip(1) = 3, pip(2) = 5, pip(3) = 11; piptorial(3) = 3*5*11 = 165.
%t A080696 nn=50;FoldList[Times,1,Transpose[Select[Thread[{Prime[Range[nn]], Range[nn]}], PrimeQ[ Last[#]]&]][[1]]] (* _Harvey P. Dale_, Jul 05 2011 *)
%t A080696 FoldList[Times,Table[Prime[Prime[n]],{n,20}]] (* _Harvey P. Dale_, May 06 2018 *)
%o A080696 (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) }
%o A080696 (PARI) a(n) = prod(k=1, n, prime(prime(k))); \\ _Michel Marcus_, Mar 15 2021
%o A080696 (Python)
%o A080696 from sympy import prime, nextprime
%o A080696 def aupton(terms):
%o A080696   prod, p, alst = 1, 2, []
%o A080696   while len(alst) < terms:
%o A080696     p, prod = nextprime(p), prod * prime(p)
%o A080696     alst.append(prod)
%o A080696   return alst
%o A080696 print(aupton(16)) # _Michael S. Branicky_, Mar 15 2021
%Y A080696 Cf. A006450.
%K A080696 easy,nonn
%O A080696 1,1
%A A080696 _Cino Hilliard_, Mar 04 2003
%E A080696 Name clarified by _Michel Marcus_, Aug 04 2015
%E A080696 More terms from _Harvey P. Dale_, May 06 2018