A079852 a(1) = 1, a(2) = 2, a(3) = 3 and a(n) is the smallest number such that all a(i)*a(j)*a(k) are different.

1, 2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 210, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 289, 293, 307, 311, 313, 317, 331

Offset

1,2

Comments

Note that a(57) = 210 = 2*3*5*7, while 330 = 2*3*5*11 is not in the sequence. This demonstrates that this sequence is not determined by prime signature alone. - Charles R Greathouse IV, Oct 17 2015

Every prime number appears in this sequence. Proof: the greedy algorithm processes natural numbers in increasing order. When prime p is considered, every element already in the sequence is less than p, so none is divisible by p. By Euclid's lemma, no existing triple product a(i)*a(j)*a(k) is divisible by p. But any new triple product involving p is divisible by p. Therefore no collision is possible, and p is admitted. For collisions among new triples: a(i)*a(j)*p = a(k)*a(l)*p implies a(i)*a(j) = a(k)*a(l), which would make 1*a(i)*a(j) = 1*a(k)*a(l) an existing collision (since 1 is in the sequence), contradicting the B_3 property. The same argument shows every prime appears in the greedy multiplicative B_k sequence for any k >= 2. Lean 4 formalization and computational verification: see Links. - Nova Spivack, Mar 21 2026

Links

Zak Seidov, Table of n, a(n) for n = 1..200

Nova Spivack, Proof, Lean 4 formalization, and computational verification that every prime appears in this sequence, GitHub, 2026.

Mathematica

f[l_List] := Block[{k = 1, p2 = Times @@@ Subsets[l, {2}], p3 = Times @@@ Subsets[l, {3}]}, While[Intersection[p3, p2*k] != {}, k++ ]; Append[l, k]]; Nest[f, {1, 2, 3}, 62] (* Ray Chandler, Feb 12 2007 *)

Crossrefs

Cf. A000045, A079850, A066720, A026477.

Sequence in context: A352870 A009087 A026477 * A084400 A050376 A280257

Adjacent sequences: A079849 A079850 A079851 * A079853 A079854 A079855

Keyword

nonn

Author

Amarnath Murthy, Feb 19 2003

Extensions

Extended by Ray Chandler, Feb 12 2007

Typo in name fixed by Zak Seidov, Jul 06 2013

Status

approved