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A007684 Prime(n)*...*prime(a(n)) is the least product of consecutive primes that is non-deficient.

%I A007684 #45 Jul 16 2022 14:20:31
%S A007684 2,6,11,21,35,51,73,98,130,167,204,249,296,347,406,471,538,608,686,
%T A007684 768,855,950,1050,1156,1266,1377,1495,1621,1755,1898,2049,2194,2347,
%U A007684 2504,2670,2837,3013,3194,3380,3573,3771,3974,4187,4401,4625,4856
%N A007684 Prime(n)*...*prime(a(n)) is the least product of consecutive primes that is non-deficient.
%C A007684 Subscript of the smallest primorial number that when divided by the (n-1)-th primorial number gives an abundant number.
%C A007684 Products of consecutive primes started with prime(a) up to prime(b) result in abundant squarefree numbers if b is large enough and provides perhaps the least squarefree solutions to Rivera Puzzle 329 and its generalization.
%C A007684 Adding a new prime p to the product increases the relative abundancy sigma(N)/N by a factor 1+1/p. This leads to a simple and fast algorithm, see the PARI code. - _M. F. Hasler_, Jul 30 2016
%H A007684 Amiram Eldar, <a href="/A007684/b007684.txt">Table of n, a(n) for n = 1..1000</a>
%H A007684 H. W. Gould, <a href="/A007680/a007680.pdf">A class of binomial sums and a series transform</a>, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy) [Annotation on p. 82 references this A-number, but the triangle with that annotation is apparently A046900, unrelated to this entry. - _Andrey Zabolotskiy_, Jul 16 2022]
%H A007684 Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_329.htm">Puzzle 329. Odd abundant numbers not divided by 2 or 3</a>, The Prime Puzzles and Problems Connection.
%F A007684 a(n) is the minimal x such that floor(sigma(p#(x)/p#(n-1)) / (p#(x)/p#(n-1))) = 2, where p#(w) is the w-th primorial number, the product of first w prime numbers. For a>b, the p#(a)/p#(b)=A002110(a)/A002110(b) quotients are prime(b+1)*prime(b+2)*...*prime(a).
%e A007684 n=1: a(1)=2 means that primorial(2)=6 divided by primorial(1-1)=1 gives the quotient 6/1=6 which is just non-deficient (being a perfect number);
%e A007684 n=3: a(n)=11 because prime(3)=5, primorial(11) = 2*3*5*...*29*31, primorial(3-1) = 2*3 = 6.
%e A007684 p#(11)/p#(2) = 3*5*7*11*13*17*19*23*29*31 = 33426748355 = q and sigma(q)/q = 2.00097 > 2 so q is an abundant number. Also p#(10)/p#(3-1) is not yet abundant.
%t A007684 spr[x_, y_] :=Apply[Times, Table[(Prime[w]+1)/(Prime[w]), {w, x, y}]];
%t A007684 Table[Min[Flatten[Position[Table[Floor[spr[n, w]], {w, 1, 1000}], 2]]], {n, 1, 20}] (* _Labos Elemer_, Sep 19 2005 *)
%o A007684 (PARI) a=1;i=0;for(n=1,99,while(2>a*=1+1/prime(i++),);print1(i",");a/=1+1/prime(n)) \\ _M. F. Hasler_, Jul 30 2016
%Y A007684 Cf. A005100, A007686, A007702, A007707 (an essentially identical sequence).
%Y A007684 Cf. A002110, A064001, A112640, A005101, A005231.
%K A007684 nonn
%O A007684 1,1
%A A007684 _Walter Nissen_
%E A007684 Additional comments from _Labos Elemer_, Sep 19 2005
%E A007684 More terms from _Don Reble_, Nov 10 2005
%E A007684 Edited by _N. J. A. Sloane_, Dec 22 2006