A396079 Product of the least factors in all proper factorizations of n into factors > 1, up to permutation; a(n) = 1 for noncomposite n.

1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 12, 1, 2, 3, 32, 1, 12, 1, 16, 3, 2, 1, 192, 5, 2, 9, 16, 1, 60, 1, 128, 3, 2, 5, 3456, 1, 2, 3, 320, 1, 72, 1, 16, 45, 2, 1, 27648, 7, 20, 3, 16, 1, 432, 5, 448, 3, 2, 1, 34560, 1, 2, 63, 16384, 5, 72, 1, 16, 3, 140, 1, 2654208, 1, 2, 45

Offset

1,4

Comments

A proper factorization means a factorization different from the one-term factorization n.

Links

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

Formula

a(n) = A395579(n)/n.

a(n) = Product_{f in F'(n)} min(f), where F'(n) is the set of factorizations of n into factors > 1, up to permutation, excluding the one-term factorization n.

Example

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

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] := a[n] = Times @@ (First /@ Rest[mp[n, 2]]);
Table[a[n], {n, 1, 75}]

Crossrefs

Cf. A395579, A001055, A046746.

Sequence in context: A395192 A098372 A385999 * A177236 A341827 A138567

Adjacent sequences: A396076 A396077 A396078 * A396081 A396082 A396083

Keyword

nonn,easy

Author

Bartlomiej Pawlik, May 16 2026

Status

approved