A378689 a(n) = product of divisors d of n that are not coreful.

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 24, 1, 14, 15, 1, 1, 54, 1, 40, 21, 22, 1, 192, 1, 26, 1, 56, 1, 27000, 1, 1, 33, 34, 35, 216, 1, 38, 39, 320, 1, 74088, 1, 88, 135, 46, 1, 3072, 1, 250, 51, 104, 1, 1458, 55, 448, 57, 58, 1, 25920000, 1, 62, 189, 1, 65, 287496

Offset

1,6

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log log scatterplot of log_10(a(n)) for n = 1..2^20.

Michael De Vlieger, Log log scatterplot of log_10(a(n)) for n = 1..2^16 (ignoring a(n) = 1, i.e., n that is a power of a prime), showing a(n) such that n is in A286708 in purple, n in A332785 in blue, n in A120944 in green, highlighting n in A002110 in large green points.

Formula

a(n) = A007955(n) / A308360(n).

a(n) = 1 for powers of primes n (i.e., n in A000961), since d | n such that d > 1 are coreful.

Example

Table of n, a(n), and divisors that are not coreful that produce a(n) for select n:
   n     a(n)
  -----------------------------
   1       1   (empty product)
   2       1 = 1
   3       1 = 1
   4       1 = 1
   5       1 = 1
   6       6 = 1*2*3
  10      10 = 1*2*5
  12      24 = 1*2*3*4
  14      14 = 1*2*7
  15      15 = 1*3*5
  18      54 = 1*2*3*9
  20      40 = 1*2*4*5
  21      21 = 1*3*7
  22      22 = 1*2*11
  24     192 = 1*2*3*4*8
  30   27000 = 1*2*3*5*6*10*15
  36     216 = 1*2*3*4*9

Mathematica

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Times @@ Select[Divisors[n], rad[#] != r &], {n, 120}]

Prog

(PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
a(n) = my(d=divisors(n), c=rad(n), p=1); for (i=1, #d~, if (rad(d[i]) != c, p *= d[i])); p; \\ Michel Marcus, Feb 07 2025

Crossrefs

Cf. A007955, A027750, A308135 (sums), A308360 (product of coreful divisors of n).

Sequence in context: A290479 A304404 A290480 * A183092 A050449 A316623

Adjacent sequences: A378686 A378687 A378688 * A378690 A378691 A378692

Keyword

nonn,easy

Author

Michael De Vlieger, Feb 05 2025

Status

approved