A361593 a(1) = 1, a(2) = 2, a(3) = 3; for n > 3, a(n) is the smallest positive number which has not appeared such that all the distinct prime factors of a(n-3) + a(n-2) + a(n-1) are factors of a(n).

1, 2, 3, 6, 11, 10, 9, 30, 7, 46, 83, 34, 163, 70, 267, 20, 357, 322, 699, 1378, 2399, 2238, 6015, 5326, 13579, 6230, 25135, 106, 31471, 14178, 45755, 15234, 75167, 68078, 8341, 151586, 228005, 193966, 573557, 248882, 1016405, 306474, 1571761, 361830, 2240065, 1043414, 3645309, 3464394

Offset

1,2

Comments

This is a variation of A359557 where the previous three terms are added instead of two. Unlike A359557 the terms here do no rapidly reach a regime where all terms share one or more prime factors, and it is unknown if this ever occurs.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..750

Michael De Vlieger, Scatterplot of Log_10(a(n)), n = 1..750, showing records in red.

Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..600, showing primes in red, prime powers in gold, and squarefree composites in green.

Example

a(6) = 10 as a(3) + a(4) + a(5) = 3 + 6 + 11 = 20 = 2*2*5, and the smallest unused number containing 2 and 5 as factors is 10.

Mathematica

nn = 120; c[_] = False; q[_] = 1;
f[n_] := Times @@ FactorInteger[n][[All, 1]]; t = 3;
Array[Set[{a[#], c[#]}, {#, True}] &, t]; Set[{i, j, k, x}, {a[t - 2],
   a[t - 1], a[t], f[a[t - 2] + a[t - 1] + a[t]]}];
Do[m = q[x];
  While[c[x m], m++];
  m *= x; While[c[x q[x]], q[x]++];
  Set[{a[n], c[m], i, j, k, x}, {m, True, j, k, m, f[j + k + m]}], {n,
t + 1, nn}]; Array[a, nn] (* Michael De Vlieger, Mar 20 2023 *)

Crossrefs

Cf. A359557, A359256, A027748, A064413, A352867, A007947.

Sequence in context: A296444 A038752 A125714 * A247953 A284091 A004038

Adjacent sequences: A361590 A361591 A361592 * A361594 A361595 A361596

Keyword

nonn

Author

Scott R. Shannon and Michael De Vlieger, Mar 16 2023

Status

approved