A376839 a(1) = 1. For n > 1 if A007947(a(n-1)) is in A002110, a(n) is the smallest prime not already a term. Otherwise a(n) is the least novel multiple of the smallest non divisor prime of a(n-1).

1, 2, 3, 4, 5, 6, 7, 8, 11, 10, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 23, 26, 27, 28, 30, 29, 32, 31, 34, 33, 36, 37, 38, 39, 40, 42, 25, 44, 45, 46, 48, 41, 50, 51, 52, 54, 43, 56, 57, 58, 60, 47, 62, 63, 64, 53, 66, 35, 68, 69, 70, 72, 59, 74, 75

Offset: 1

Author

David James Sycamore, Oct 06 2024

Keywords

• nonn

Comments

A non divisor prime of a(n-1) is any prime p < Gpf(a(n-1)) which does not divide a(n-1). A007947(a(n-1)) is in A002110 iff a(n-1) is a term in A055932. Sequence is conjectured to be a permutation of the natural numbers (A000027) with primes in order.

Scatterplot shows trajectories of numbers whose smallest prime factor is prime p, e.g., for p = 5, numbers in A084967, p = 7, those in A084968, p = 11 those in A084969, etc. - Michael De Vlieger, Oct 09 2024

Examples

			a(1) = 1 = A002110(0) so a(2) = 2 (smallest prime not already a term).
a(2) = 2 = A002110(1) so a(3) = 3.
a(3) = 3 not a term in A002110 so a(4) is least novel multiple of 2, the least non divisor prime of 3. Therefore a(4) = 4 since 2 has occurred earlier.
a(39) = 42, not a term in A002110 so a(40) = 25, the least novel multiple of 5, the smallest non divisor prime of 42.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue or purple, with the latter additionally representing powerful numbers that are not prime powers.

Crossrefs

Cf. A000040, A000027, A002110, A007947, A055932, A084967, A084968, A084969, A372368, A376865.

Programs

Mathematica

nn = 120; c[] := False; m[] := 1; f[x_] := FactorInteger[x][[All, 1]];
  Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2]; j = 2; v = 3;
  Do[If[Or[IntegerQ@ Log2[j], And[EvenQ[j], Union@ Differences@ PrimePi[#] == {1}]],
     k = v; While[c[k*m[k]], m[k]++]; k *= m[k],
     k = 2; While[Divisible[j, k], k = NextPrime[k]];
     While[c[k*m[k]], m[k]++]; k *= m[k]] &[f[j]];
  Set[{a[n], c[k], j}, {k, True, k}];
  If[k == v, While[c[v], v = NextPrime[v]]], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Oct 09 2024 *)

Extensions

More terms from Michael De Vlieger, Oct 09 2024