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

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

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• hard
• nice

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

Examples

			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.
		

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.]

Crossrefs

Cf. A082120, A082125.

Cf. A033676, A033677, A056737, A219695.

Programs

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 *)

PARI

A003681(N,a=[2,3])={while(#a1||next;a=concat(a,d[i+1]-d[#d-i]);break));a} \\ May require allocatemem() for N >= 15. - M. F. Hasler, Aug 17 2015

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