A057449 Product of differences between consecutive positive divisors of n.

1, 1, 2, 2, 4, 3, 6, 8, 12, 15, 10, 12, 12, 35, 40, 64, 16, 81, 18, 100, 112, 99, 22, 192, 80, 143, 216, 588, 28, 600, 30, 1024, 352, 255, 224, 1944, 36, 323, 520, 2400, 40, 3087, 42, 3388, 2880, 483, 46, 12288, 252, 5625, 952, 6084, 52, 19683, 1056, 14112, 1216

Offset

1,3

Comments

Row products of triangle A193829, if n >= 2. - Omar E. Pol, Aug 31 2011

a(n) is the determinant of the symmetric tau(n) X tau(n) matrix M(n) defined by M(n) = [min(d_i,d_j)], where d_i, d_j run through the divisors of n. - Ridouane Oudra, Apr 26 2026

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

a(n) = Product_{k=1..A000005(n)-1} A027750(n,k+1)-A027750(n,k). - Reinhard Zumkeller, Jun 23 2013

a(p) = p-1 for prime p. - Michel Marcus, May 15 2014

Example

Since positive divisors of 12 are 1, 2, 3, 4, 6 and 12, a(12) = (2-1) *(3-2) *(4-3) *(6-4) *(12-6) = 1 *1 *1 *2 *6 = 12.

Mathematica

Times@@Differences[Divisors[#]]&/@Range[80]  (* Harvey P. Dale, Apr 20 2011 *)

Prog

(Haskell)
a057449 = product . a193829_row  -- Reinhard Zumkeller, Jun 23 2013
(PARI) a(n) = {my(d = divisors(n)); prod(i=1, #d-1, d[i+1] - d[i]); } \\ Michel Marcus, May 15 2014

Crossrefs

Cf. A000005, A000040, A027750, A193829.

Sequence in context: A341469 A101200 A122585 * A308601 A286931 A374405

Adjacent sequences: A057446 A057447 A057448 * A057450 A057451 A057452

Keyword

nonn,look

Author

Leroy Quet, Sep 25 2000

Status

approved