A003681 a(n) = min { p +- q : p +- q > 1 and p*q = Product_{k=1..n-1} a(k) }. (Formerly M0659)

2, 3, 5, 7, 11, 13, 17, 107, 197, 3293, 74057, 1124491, 1225063003, 48403915086083, 229199690093487791653, 139394989871393443893426292667, 2310767115930351361890156080500119173238113, 521722354210765171422123515738862106081757768167379798858040637

Offset

1,1

Comments

a(n) = A082120(Product_{k=1..n-1} a(k)) for n >= 3. - Robert Israel, Aug 12 2015

The + sign in the definition applies only for n = 1 and n = 2, thereafter only the - sign is relevant and will yield the minimum. The definition could be reformulated in a way similar to that of A056737. - M. F. Hasler, Aug 17 2015

References

J. H. Conway, personal communication.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Robert G. Wilson v and Sean A. Irvine, Table of n, a(n) for n = 1..19 [a(19) added by Sean A. Irvine, Jul 29 2015.]

Example

a(4) = 7 because 2*3*5 = 30 whose divisors are 1, 2, 3, 5, 6, 10, 15 and 30. The closest p and q are 5 and 6 but its difference is 1 so the next closest p and q are 3 and 10 whose difference is 7.

Mathematica

a[1] = 2; a[2] = 3; a[n_] := a[n] = Block[{d, l, t, p = Product[a[i], {i, n - 1}]}, d = Divisors[p]; l = Length[d]; t = Take[d, {l/2 - 1, l/2 + 2}]; If[t[[3]] - t[[2]] == 1, t[[4]] - t[[1]], t[[3]] - t[[2]]]]; Array[a, 16] (* Robert G. Wilson v, May 27 2012 *)

Prog

(PARI) A003681(N, a=[2, 3])={while(#a<N, my(d=divisors(prod(i=1, #a, a[i]))); for(i=(#d)\2, #d, d[i+1]-d[#d-i]>1||next; a=concat(a, d[i+1]-d[#d-i]); break)); a} \\ May require allocatemem() for N >= 15. - M. F. Hasler, Aug 17 2015

Crossrefs

Cf. A082120, A082125.

Cf. A033676, A033677, A056737, A219695.

Sequence in context: A053434 A241716 A061166 * A217147 A029732 A037950

Adjacent sequences: A003678 A003679 A003680 * A003682 A003683 A003684

Keyword

nonn,hard,nice

Author

N. J. A. Sloane, Mira Bernstein

Extensions

a(15) from Robert G. Wilson v, Feb 26 1996

a(16) from Naohiro Nomoto, Jun 25 2001

a(17) from Robert G. Wilson v, Sep 15 2006

a(18) from Robert G. Wilson v, Jul 20 2009

a(19) from Sean A. Irvine, Jul 29 2015

Status

approved