A396079 - OEIS

%I #9 May 20 2026 22:22:15

%S 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,

%T 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,

%U 448,3,2,1,34560,1,2,63,16384,5,72,1,16,3,140,1,2654208,1,2,45

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

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

%H Bartlomiej Pawlik, <a href="/A396079/b396079.txt">Table of n, a(n) for n = 1..1000</a>

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

%F 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.

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

%t 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]];

%t a[1] = 1;

%t a[n_Integer?Positive] := a[n] = Times @@ (First /@ Rest[mp[n, 2]]);

%t Table[a[n], {n, 1, 75}]

%Y Cf. A395579, A001055, A046746.

%K nonn,easy

%O 1,4

%A _Bartlomiej Pawlik_, May 16 2026