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

2, 6, 11, 21, 35, 51, 73, 98, 130, 167, 204, 249, 296, 347, 406, 471, 538, 608, 686, 768, 855, 950, 1050, 1156, 1266, 1377, 1495, 1621, 1755, 1898, 2049, 2194, 2347, 2504, 2670, 2837, 3013, 3194, 3380, 3573, 3771, 3974, 4187, 4401, 4625, 4856

Offset

1,1

Comments

Subscript of the smallest primorial number that when divided by the (n-1)-th primorial number gives an abundant number.

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.

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

H. W. Gould, A class of binomial sums and a series transform, 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]

Carlos Rivera, Puzzle 329. Odd abundant numbers not divided by 2 or 3, The Prime Puzzles and Problems Connection.

Formula

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).

Example

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);
n=3: a(n)=11 because prime(3)=5, primorial(11) = 2*3*5*...*29*31, primorial(3-1) = 2*3 = 6.
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.

Mathematica

spr[x_, y_] :=Apply[Times, Table[(Prime[w]+1)/(Prime[w]), {w, x, y}]];
Table[Min[Flatten[Position[Table[Floor[spr[n, w]], {w, 1, 1000}], 2]]], {n, 1, 20}] (* Labos Elemer, Sep 19 2005 *)

Prog

(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

Crossrefs

Cf. A005100, A007686, A007702, A007707 (an essentially identical sequence).

Cf. A002110, A064001, A112640, A005101, A005231.

Sequence in context: A171516 A081691 A085571 * A296557 A241564 A135348

Adjacent sequences: A007681 A007682 A007683 * A007685 A007686 A007687

Keyword

nonn

Author

Walter Nissen

Extensions

Additional comments from Labos Elemer, Sep 19 2005

More terms from Don Reble, Nov 10 2005

Edited by N. J. A. Sloane, Dec 22 2006

Status

approved