A258324 -id:A258324 - cp's OEIS frontend

A072513 Product of all n - d, where d < n and d is a divisor of n.

1, 1, 2, 6, 4, 60, 6, 168, 48, 360, 10, 47520, 12, 1092, 1680, 20160, 16, 440640, 18, 820800, 5040, 4620, 22, 734469120, 480, 7800, 11232, 4953312, 28, 3946320000, 30, 9999360, 21120, 17952, 28560, 439723468800, 36, 25308, 35568, 35852544000

Offset: 1

Amarnath Murthy, Jul 28 2002

nonn

			a(6) = (6-1)(6-2)(6-3) = 60.
For n = 16 the divisors d < n are 1,2,4 and 8, so a(16) = (16-1)*(16-2)*(16-4)*(16-8) = 15*14*12*8 = 20160.

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A072512, A080497, A080498, A080500 (similar products), A258324 (LCM instead of product).

Cf. A027751.

a072513 n = product $ map (n -) $ a027751_row n
-- Reinhard Zumkeller, May 27 2015

Table[Times @@ (n - Most[Divisors[n]]), {n, 1, 40}] (* Ivan Neretin, May 26 2015 *)

for(n=1,40,d=divisors(n); print1(prod(j=1,matsize(d)[2]-1,n-d[j]),","))

a(n)=factorback(apply(d->if(dCharles R Greathouse IV, May 26 2015

a(n) = (n-d_1)(n-d_2)...(n-d_k) where d_k is the largest divisor of n less than n (k = tau(n) - 1).
a(p) = p-1, a(pq) = pq(p-1)(q-1)(pq-1), p and q prime.
If n is not a prime or the square of a prime then n divides a(n).

Edited and extended by Klaus Brockhaus, Jul 31 2002

A380600 Irregular table T(n, k), n > 0, k = 1..A000005(n) read by rows: the n-th row lists the numbers of the form n * (d-1) / d with d a positive divisor of n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 2, 3, 0, 4, 0, 3, 4, 5, 0, 6, 0, 4, 6, 7, 0, 6, 8, 0, 5, 8, 9, 0, 10, 0, 6, 8, 9, 10, 11, 0, 12, 0, 7, 12, 13, 0, 10, 12, 14, 0, 8, 12, 14, 15, 0, 16, 0, 9, 12, 15, 16, 17, 0, 18, 0, 10, 15, 16, 18, 19, 0, 14, 18, 20, 0, 11, 20, 21, 0, 22

Offset: 1

Rémy Sigrist, Feb 02 2025

			Table T(n, k) begins:
  n   n-th row
  --  ------------------
   1  0
   2  0, 1
   3  0, 2
   4  0, 2, 3
   5  0, 4
   6  0, 3, 4, 5
   7  0, 6
   8  0, 4, 6, 7
   9  0, 6, 8
  10  0, 5, 8, 9
  11  0, 10
  12  0, 6, 8, 9, 10, 11
  13  0, 12
  14  0, 7, 12, 13

Index entries for sequences related to divisors

Cf. A000005, A027750, A046666, A060681, A072513, A094471 (row sums), A258324.

Table[Map[n*(# - 1)/# &, Divisors[n]], {n, 23}] // Flatten (* Michael De Vlieger, Feb 03 2025 *)

row(n) = apply (d -> n*(d-1)/d, divisors(n))

T(n, k) = n * (A027750(n, k) - 1) / A027750(n, k).
Sum_{k = 1..A000005(n)} T(n, k) = A094471(n).
Product_{k = 2..A000005(n)} T(n, k) = A072513(n).
LCM{k = 2..A000005(n)} T(n, k) = A258324(n).
T(n, 1) = 0.
T(n, 2) = A060681(n) for any n > 1. - Michel Marcus, Feb 03 2025
T(n, A000005(n)-1) = A046666(n) for any n > 1.
T(n, A000005(n)) = n-1.

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A072513 Product of all n - d, where d < n and d is a divisor of n.

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A380600 Irregular table T(n, k), n > 0, k = 1..A000005(n) read by rows: the n-th row lists the numbers of the form n * (d-1) / d with d a positive divisor of n.

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Author

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Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula