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%I A143293 #43 Feb 23 2022 12:17:05
%S A143293 1,3,9,39,249,2559,32589,543099,10242789,233335659,6703028889,
%T A143293 207263519019,7628001653829,311878265181039,13394639596851069,
%U A143293 628284422185342479,33217442899375387209,1955977793053588026279,119244359152460559009549,7977565910232727614888639
%N A143293 Partial sums of A002110, the primorial numbers.
%C A143293 After 3, this is never prime because all values thereafter are multiples of 3. Starting from a(6) all are also multiples of 17. - _Jonathan Vos Post_, Feb 10 2010
%C A143293 Starting from a(162) all are also multiples of 967. - _Alex Ratushnyak_, May 14 2013
%C A143293 Repunits in primorial base, A049345. - _Antti Karttunen_, Aug 21 2016
%H A143293 Soumyadeep Dhar, <a href="/A143293/b143293.txt">Table of n, a(n) for n = 0..350</a> (terms up to a(100) from T. D. Noe)
%H A143293 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F A143293 a(n) = Sum_{k=0..n} prime(k)#, where prime(n)# = A002110(n).
%F A143293 a(n) = A276085(A002110(1+n)). - _Antti Karttunen_, Aug 21 2016
%e A143293 a(3) = 39 = (1 + 2 + 6 + 30), where A002110 = (1, 2, 6, 30, 210, 2310,...).
%p A143293 b:= proc(n) option remember; `if`(n=0, [1$2], (h->
%p A143293       (p-> [p, p+h[2]])(ithprime(n)*h[1]))(b(n-1)))
%p A143293     end:
%p A143293 a:= n-> b(n)[2]:
%p A143293 seq(a(n), n=0..19);  # _Alois P. Heinz_, Feb 23 2022
%t A143293 Table[s = 1; Do[s = 1 + s*Prime[i], {i, n, 1, -1}]; s, {n, 0, 20}] (* _T. D. Noe_, May 03 2013 *)
%t A143293 Accumulate[FoldList[Times,1,Prime[Range[20]]]] (* _Harvey P. Dale_, Feb 05 2015 *)
%o A143293 (PARI) a(n)=if(n==0,return(1)); my(P=1,s=1); forprime(p=2,prime(n), s+=P*=p); s \\ _Charles R Greathouse IV_, Feb 05 2014
%o A143293 (Python)
%o A143293 from itertools import chain, accumulate, count, islice
%o A143293 from operator import mul
%o A143293 from sympy import prime
%o A143293 def A143293_gen(): # generator of terms
%o A143293     return accumulate(accumulate(chain((1,),(prime(n) for n in count(1))), mul))
%o A143293 A143293_list = list(islice(A143293_gen(),20)) # _Chai Wah Wu_, Feb 23 2022
%Y A143293 Cf. A002110, A049345, A276085.
%K A143293 nonn
%O A143293 0,2
%A A143293 _Gary W. Adamson_, Aug 05 2008
%E A143293 a(11)-a(19) from _Jonathan Vos Post_, Feb 10 2010

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