A376865 a(1) = 1. Thereafter, if A007947(a(n-1)) is in A002110, a(n) = smallest prime not already a term (condition [A]). Otherwise, if q is the greatest prime < Gpf(a(n-1)) which does not divide a(n-1), a(n) is the least novel multiple of q,(condition [B]).

1, 2, 3, 4, 5, 6, 7, 10, 9, 8, 11, 14, 15, 12, 13, 22, 21, 20, 18, 17, 26, 33, 28, 25, 24, 19, 34, 39, 44, 35, 27, 16, 23, 38, 51, 52, 55, 42, 30, 29, 46, 57, 68, 65, 66, 49, 40, 36, 31, 58, 69, 76, 85, 78, 77, 45, 32, 37, 62, 87, 92, 95, 102, 91, 88, 56, 50, 48

Offset: 1

Author

David James Sycamore, and Michael De Vlieger, Oct 07 2024

Keywords

• nonn

Comments

A sequence with 3 distinct phases, similar to A372368.

Define cycle c(i) to be a run of consecutive terms beginning with a prime a(n) = prime(i) resulting from condition [A], which ends when a(n) is a term in A055932.

Phase I consists of consecutive closed cycles c(i) that start with a(n) = prime(i) via condition [A] and end with a term in A055932. As n increases through cycle c(i), G = gpf(a(n)) strictly decreases, and g = gpf(m(q)) is small compared to G. This phase ends at n = 4318.

Phase II consists of closed cycles c(i) that start with a(n) = prime(j), j > i, via condition [A] and end with a term in A055932. As n increases through cycle c(i), at times, g > G and we have a rejuvenated cycle. We may see multiple condition [B] primes, as well as runs of composite a(n) for n = 99528..155219 and n = 222811..262605. The humps in scatterplot are associated with these particular runs of composite terms. Rejuvenation of a cycle has G increment m(q) each time. A "ridge" of high m(q) values builds and grows increasingly difficult to traverse to reach G = 11, where we might have a number in A055932 and close the cycle. This phase likely ends with n = 2048704.

Phase III consists of condition [A] prime a(2048704) = prime(742) = 5647 and terms that follow, starting cycle c(135). As n increases, there are repeated rejuvenations and regular entry of primes through condition [B]. The repeated rejuvenations increase and expand a bank of high values of m(q) across many primes q only a few dozen iterations after new primes appear. New primes mean that prime(i) increases while in order to find a(n) in A055932, we need numbers with G = 11. Therefore, the circumstance that needs to arise to close the cycle becomes harder to achieve as n increases.

It is unlikely that this sequence is a permutation of natural numbers.

A full description of the phased behavior of this sequence is given in the link.

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 in A002110 therefore a(4) = 4, the least novel multiple of 2.
rad(4) = 2 so a(5) = 5 the smallest prime not already a term.
		

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 G(n) = gpf(a(n)), n = 1..3000000, showing the confinement of G(n) to large primes for n > 2048704.
• Michael De Vlieger, Notes on this sequence.
• Michael De Vlieger, Tipping Point Sequence A365865, ResearchGate, 2024.

Crossrefs

Cf. A000040, A000027, A002110, A007947, A055932, A376839, A372368.

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 = Last[#]; While[And[k > 1, Divisible[j, k]], k = NextPrime[k, -1]];
     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