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A070826 One half of product of first n primes A000040.

%I A070826 #52 Jul 10 2025 10:58:51
%S A070826 1,3,15,105,1155,15015,255255,4849845,111546435,3234846615,
%T A070826 100280245065,3710369067405,152125131763605,6541380665835015,
%U A070826 307444891294245705,16294579238595022365,961380175077106319535,58644190679703485491635,3929160775540133527939545,278970415063349480483707695
%N A070826 One half of product of first n primes A000040.
%C A070826 Also, with offset 0, product of first n odd primes. - _N. J. A. Sloane_, Feb 26 2017
%C A070826 Identical to A002110(n)/2, n>=1.
%C A070826 a(n+1) is the least odd number with exactly n distinct prime divisors. - _Labos Elemer_, Mar 24 2003
%C A070826 Also, odd numbers n for which sigma(n)*phi(n)/n^2 reaches a new record low, monotonically decreasing to the lower bound 8/Pi^2. - _M. F. Hasler_, Jul 08 2025
%H A070826 Vincenzo Librandi, <a href="/A070826/b070826.txt">Table of n, a(n) for n = 1..200</a>
%F A070826 a(n) = A002110(n)/2.
%F A070826 From _Antti Karttunen_, Feb 06 2024: (Start)
%F A070826 a(1) = 1, and for n > 1, a(n) = A276086(A060389(n-1)).
%F A070826 a(n) = A024451(n) - 2*A203008(n-1).
%F A070826 (End)
%F A070826 a(n) = A000040(n)*a(n-1) for n > 1, a(1) = 1. - _M. F. Hasler_, Jul 09 2025
%p A070826 a:=n->mul(ithprime(j), j=2..n):seq(a(n), n=1..17); # _Zerinvary Lajos_, Aug 24 2008
%t A070826 Rest[ FoldList[ Times, 1, Prime[ Range[ 18]] ]]/2 (* _Robert G. Wilson v_, Feb 17 2004 *)
%t A070826 FoldList[Times, 1, Prime[Range[2, 18]]] (* _Zak Seidov_, Jan 26 2009 *)
%o A070826 (PARI) a(n) = prod(k=2, n, prime(k)) \\ _Michel Marcus_, Mar 25 2017, simplified by _M. F. Hasler_, Jul 09 2025
%o A070826 (Python)
%o A070826 from sympy import primorial
%o A070826 def A070826(n): return primorial(n)>>1 # _Chai Wah Wu_, Jul 21 2022
%Y A070826 Cf. A003266 (for Fibonacci), A070825 (for Lucas), A003046 (for Catalan).
%Y A070826 Cf. also A002110, A024451, A060389, A091852, A276086, A203008 [= A003415(a(1+n))].
%Y A070826 Range of A196529.
%K A070826 nonn,easy
%O A070826 1,2
%A A070826 _Wolfdieter Lang_, May 10 2002
%E A070826 Formula corrected by _Gary Detlefs_, Dec 07 2011