A348846 a(1) = 1. For n >=2 the number k in n-th position becomes a(n) only if all terms a(1)..a(n-1) have already been defined, and if the smallest number m, greater than k, not already defined and sharing greatest prime factor (gpf) p with k is reduced to m/p.

1, 2, 3, 2, 5, 2, 7, 2, 9, 2, 11, 4, 13, 2, 15, 2, 17, 18, 19, 4, 21, 2, 23, 8, 25, 2, 27, 4, 29, 6, 31, 2, 33, 2, 35, 12, 37, 2, 39, 40, 41, 6, 43, 4, 9, 2, 47, 16, 49, 50, 51, 4, 53, 18, 55, 8, 57, 2, 59, 12, 61, 2, 63, 2, 65, 6, 67, 4, 69, 10, 71, 24, 73, 2

Offset: 1

Author

David James Sycamore, Nov 07 2021

Keywords

• nonn

Comments

A limiting sequence using greatest prime factor. Each number in A000027, in natural order, is considered for admittance to the sequence. A number in n-th position at the start may be reduced several times prior to being admitted as a(n), or may not be reduced at all. Every power 2^k of 2 is reduced eventually to 2, by reduction of A007053(2^(k-1)) even semiprimes, plus 2s from reductions of smaller powers of 2.

Let [p] = {m: m a fixed point with gpf = p}, then [2] = {2}, [3] = {3,9,18,27}, [5] = {5,15,25,40,50,90}, etc. Every odd multiple of odd prime p, up to and including p^2, is necessarily a fixed point. The number of terms in [p] is limited by reduction of q-smooth numbers (q>p) to those having gpf p. Conjecture: For odd prime p, [p] is a finite set with greatest term > p^2, and <= p^3. A variant based on least prime divisors is also possible.

Examples

			After a(1) = 1, the next eligible number is 2, which becomes a(2) when 4 is reduced to 4/2 = 2.
a(3) = 3 because 6 is reduced to 2.
Next in line is 2 (previously 4), which enters as a(4) when 8 is reduced to 4.
a(5) = 5 when 10 is reduced to 2.
Next in line is 2 (previously 6) which enters as a(6) when 4 (previously 8) is reduced to 2.
a(7) = 7, and so on.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1800
• Michael De Vlieger, Log-log scatterplot of a(n) for n = 1..1800.

Crossrefs

Cf. A000027, A136119, A007053.

Programs

Mathematica

{1}~Join~Reap[Do[If[! IntegerQ[r[i]], Set[r[i], i]]; Which[PrimeQ[i], Set[m, 2 #],IntegerQ@ Log2[#], Block[{j = 1, k = Log2[#]}, While[r[Set[m, 2^(k + j)]] <= #, j++]], True, Block[{n = #1, k = #1/#2, j = 1}, p = #2; While[Nand[FactorInteger[#][[-1, 1]] <= p, r[#] > #] &@ Set[m, (j + k) p], j++]] & @@ {#, FactorInteger[#][[-1, 1]]}] &@ r[i]; If[IntegerQ[r[m]], r[m] /= FactorInteger[r[m]][[-1, 1]], Set[r[m], m/(FactorInteger[m][[-1, 1]])]]; Sow[r[i]], {i, 2, 120}]][[-1, -1]] (* Michael De Vlieger, Nov 07 2021 *)

Formula

a((2*m+1)*p) is a fixed point for all primes p, with m = 0,1,...,(p-1)/2.

a(2*p) = 2 for all primes p.

a(2^k) = 2 for all k >= 1.

Extensions

More terms from Michael De Vlieger, Nov 07 2021