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

Author

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

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

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

Programs

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 *)