A364902 Let x, y be the greatest exponents of 2, 3 respectively such that 2^x, 3^y do not exceed n and let k_2, k_3 be n - 2^x, and n - 3^y respectively. Then for n such that k_2 = 0 or k_3 = 0, a(n) = n, else a(n) is the least novel number Min{p*a(k_2), q*a(k_3)}, where p, q are primes not equal to either 2 or 3.

1, 2, 3, 4, 5, 10, 15, 8, 9, 7, 14, 20, 25, 35, 50, 16, 11, 22, 21, 28, 55, 70, 75, 40, 45, 49, 27, 13, 26, 33, 44, 32, 17, 34, 39, 52, 65, 98, 100, 56, 63, 77, 80, 121, 110, 105, 140, 112, 143, 154, 147, 196, 245, 135, 91, 130, 165, 220, 160, 85, 170, 195, 260, 64, 19, 38, 51, 68

Offset: 1

Author

David James Sycamore and Michael De Vlieger Sep 21 2023

Keywords

• nonn

Comments

Motivated by the recursion D(2) known to reproduce A005940, this sequence uses a compound version based on a squarefree semiprime (6) rather than a prime, in which the terms are generated by a greedy algorithm related to the distances between n and the greatest powers of 2, and 3 not exceeding n. After a(9) = 9 each power of 2 or 3 is followed by the smallest prime not yet in the sequence. (e.g. 11 follows 16, 13 follows 27, etc).

There are no multiples of 6 in this sequence.

For k > 2, if a(i) = prime(k) = p and a(j) = p^2 then j-i is a term in A006899 (e.g. a(17) = 11, a(44) = 121 and 44 - 17 = 27 = 3^3).

Conjectures: (i). This is a permutation of A047253 with primes in order; (ii). All terms between consecutive prime terms, prime(k), prime(k+1) are prime(k)-smooth.

Examples

			a(n) = n for n <= 4 because all such n are powers of 2 or 3.
a(5) = least novel Min{a(1)*p,a(2)*q} = Min{p,2*q} for o,q prime != 2 or 3, so a(5) = 5.
17=16+1=9+8, so a(17) = least novel Min{a(1)*p,a(8)*q} = Min{p,8*q} = 11.
Data can be shown in tabular form in two distinct ways: First row starts with 1 and then rows start with a prime; alternatively each row starts with 2^i or 3^j:
 1;                       1;
 2;                       2;
 3,4;                     3;
 5,10,15,8,9;             4,5,10,15;
 7,14,20,25,35,50,16;     8;
 11,22,21,28,55...        9,7,14,20,25,35,50
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..16384
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.

Crossrefs

Cf. A005940, A006899, A047253, A108906, A356867, A364611, A364628.

Programs

Mathematica

nn = 120; c[_] = False; s = {1, 2}; w = Length[s]; t = Prime[s]; flag = 0;
Array[Set[{q[#1], p[#1],
      r[#1]}, {#1, #2,
        Prepend[#2^Range[Floor@Log[#2, nn]], 1]} & @@ {#2,
       Prime[#2]}] & @@ {#, s[[#]]} &, w];
Do[If[n == 1,
   Set[{a[n], c[1]}, {1, True}],
   Array[Set[m[#], 1] &, w];
   Array[Set[j[#], n - p[#]^(-1 + LengthWhile[r[#], # < n + 1 &])] &, w];
   Array[
    If[j[#] == 0,
      k[#] = n; flag = #,
      While[Set[k[#], Prime[m[#]] a[j[#]]];
       Or[MemberQ[s, m[#]], c[k[#]]], m[#]++]] &, w];
   If[flag > 0,
    Set[{a[n], c[k[flag]]}, {k[flag], True}]; flag = 0,
    Set[{a[n], c[#]}, {#, True}] &[Min@ Array[k, w]] ]], {n, nn}];
Array[a, nn] (* Michael De Vlieger, Sep 24 2023 *)

Formula

For n > 6, a(A006899(n) + 1) = prime(n-2).