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A159578 Dividend associated with A116536.

%I A159578 #20 Jun 07 2022 13:46:01
%S A159578 2,30,9699690,304250263527210,267064515689275851355624017992790,
%T A159578 5766152219975951659023630035336134306565384015606066319856068810,
%U A159578 962947420735983927056946215901134429196419130606213075415963491270,29819592777931214269172453467810429868925511217482600306406141434158090
%N A159578 Dividend associated with A116536.
%C A159578 a(2)-a(4) are mentioned by Alladi and Erdős (1977). They conjectured that this sequence is infinite. - _Amiram Eldar_, Nov 02 2020
%H A159578 Amiram Eldar, <a href="/A159578/b159578.txt">Table of n, a(n) for n = 1..80</a>
%H A159578 Krishnaswami Alladi and Paul Erdős, <a href="https://projecteuclid.org/euclid.pjm/1102811427">On an additive arithmetic function</a>, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294, <a href="https://msp.org/pjm/1977/71-2/pjm-v71-n2-p01-s.pdf">alternative link</a>. See p. 290.
%F A159578 a(n) = A002110(A051838(n)). - _Amiram Eldar_, Nov 02 2020
%e A159578 a(2) = 30 because 2*3*5 = 30, 2+3+5 = 10, and 30/10 = 3 in A116536.
%p A159578 # First define t1, the sequence A051838.
%p A159578 t1:=[1,3,8,13,23,38,39,41,43,48,50,53,56,57,58,66,68,
%p A159578 70,73,77,84,90,94,98,126,128,134,140,143,145,149,
%p A159578 151,153,157,160,164,167,168,172,174,176,182,191,
%p A159578 194,196,200,210,212,215,217,218,219,222,225,228,
%p A159578 229];
%p A159578 p:=ithprime;
%p A159578 num:=n->mul(p(i),i=1..t1[n]);
%p A159578 s:=[num(i),i=1..11)];
%t A159578 seq = {}; sum = 0; prod = 1; p = 1; Do[p = NextPrime[p]; prod *= p; sum += p; If[Divisible[prod, sum], AppendTo[seq, prod]], {50}]; seq (* _Amiram Eldar_, Nov 02 2020 *)
%t A159578 Module[{nn=50,s,p},s=Accumulate[Prime[Range[nn]]];p=FoldList[Times,Prime[Range[ nn]]]; Select[Thread[{p,s}],Divisible[#[[1]],#[[2]]]&]][[All,1]] (* _Harvey P. Dale_, Jun 07 2022 *)
%Y A159578 Intersection of A002110 and A036844.
%Y A159578 Cf. A051838, A116536.
%K A159578 easy,nonn
%O A159578 1,1
%A A159578 _Enoch Haga_, Apr 16 2009
%E A159578 Corrected by _N. J. A. Sloane_, Oct 02 2011 (all the terms were wrong).