A395579 Product of the least factors in all factorizations of n, up to permutation of factors, into factors > 1, a(1) = 1 by convention.

1, 2, 3, 8, 5, 12, 7, 32, 27, 20, 11, 144, 13, 28, 45, 512, 17, 216, 19, 320, 63, 44, 23, 4608, 125, 52, 243, 448, 29, 1800, 31, 4096, 99, 68, 175, 124416, 37, 76, 117, 12800, 41, 3024, 43, 704, 2025, 92, 47, 1327104, 343, 1000, 153, 832, 53, 23328, 275, 25088, 171, 116, 59, 2073600

Offset

1,2

Comments

a(n) is divisible by n.

Links

Bartlomiej Pawlik, Table of n, a(n) for n = 1..1000

Formula

a(n) = Product_{f in F(n)} min(f), where F(n) is the set of factorizations of n into factors > 1.

a(p^k) = p^A046746(k-1), p prime.

a(n) = n iff n = 1 or n is prime.

If n = p*q with p <= q primes, then a(n) = p^2*q.

Example

The factorizations of 12 are 12, 2*6, 3*4, 2*2*3, so a(12) = 12*2*3*2 = 144.

Mathematica

mp[n_Integer?Positive, m_Integer?Positive]:=mp[n, m]=
 Join[{{n}}, Flatten[Table[
  If[Mod[n, d]==0, Prepend[#, d]&/@ mp[n/d, d], Nothing],
  {d, m, Floor[Sqrt[n]]}], 1
  ]];
a[1] = 1;
a[n_Integer?Positive]:=Times@@(First/@ mp[n, 2]);
Table[a[n], {n, 1, 60}]

Crossrefs

Cf. A001055, A046746.

Sequence in context: A344368 A342768 A340514 * A086471 A328846 A249154

Adjacent sequences: A395576 A395577 A395578 * A395580 A395581 A395582

Keyword

nonn,easy

Author

Bartlomiej Pawlik, Apr 29 2026

Status

approved