A364902 - OEIS

%I #54 Jul 28 2024 09:19:10

%S 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,

%T 49,27,13,26,33,44,32,17,34,39,52,65,98,100,56,63,77,80,121,110,105,

%U 140,112,143,154,147,196,245,135,91,130,165,220,160,85,170,195,260,64,19,38,51,68

%N 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.

%C 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).

%C There are no multiples of 6 in this sequence.

%C 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).

%C 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.

%H Michael De Vlieger, <a href="/A364902/b364902.txt">Table of n, a(n) for n = 1..16384</a>

%H Michael De Vlieger, <a href="/A364902/a364902.png">Log log scatterplot of a(n)</a>, n = 1..2^16.

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

%e a(n) = n for n <= 4 because all such n are powers of 2 or 3.

%e 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.

%e 17=16+1=9+8, so a(17) = least novel Min{a(1)*p,a(8)*q} = Min{p,8*q} = 11.

%e 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:

%e  1;                       1;

%e  2;                       2;

%e  3,4;                     3;

%e  5,10,15,8,9;             4,5,10,15;

%e  7,14,20,25,35,50,16;     8;

%e  11,22,21,28,55...        9,7,14,20,25,35,50

%t nn = 120; c[_] = False; s = {1, 2}; w = Length[s]; t = Prime[s]; flag = 0;

%t Array[Set[{q[#1], p[#1],

%t       r[#1]}, {#1, #2,

%t         Prepend[#2^Range[Floor@Log[#2, nn]], 1]} & @@ {#2,

%t        Prime[#2]}] & @@ {#, s[[#]]} &, w];

%t Do[If[n == 1,

%t    Set[{a[n], c[1]}, {1, True}],

%t    Array[Set[m[#], 1] &, w];

%t    Array[Set[j[#], n - p[#]^(-1 + LengthWhile[r[#], # < n + 1 &])] &, w];

%t    Array[

%t     If[j[#] == 0,

%t       k[#] = n; flag = #,

%t       While[Set[k[#], Prime[m[#]] a[j[#]]];

%t        Or[MemberQ[s, m[#]], c[k[#]]], m[#]++]] &, w];

%t    If[flag > 0,

%t     Set[{a[n], c[k[flag]]}, {k[flag], True}]; flag = 0,

%t     Set[{a[n], c[#]}, {#, True}] &[Min@ Array[k, w]] ]], {n, nn}];

%t Array[a, nn] (* _Michael De Vlieger_, Sep 24 2023 *)

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

%K nonn

%O 1,2

%A _David James Sycamore_ and _Michael De Vlieger_ Sep 21 2023