This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A364902 #54 Jul 28 2024 09:19:10
%S A364902 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 A364902 49,27,13,26,33,44,32,17,34,39,52,65,98,100,56,63,77,80,121,110,105,
%U A364902 140,112,143,154,147,196,245,135,91,130,165,220,160,85,170,195,260,64,19,38,51,68
%N 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.
%C A364902 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 A364902 There are no multiples of 6 in this sequence.
%C A364902 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 A364902 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 A364902 Michael De Vlieger, <a href="/A364902/b364902.txt">Table of n, a(n) for n = 1..16384</a>
%H A364902 Michael De Vlieger, <a href="/A364902/a364902.png">Log log scatterplot of a(n)</a>, n = 1..2^16.
%F A364902 For n > 6, a(A006899(n) + 1) = prime(n-2).
%e A364902 a(n) = n for n <= 4 because all such n are powers of 2 or 3.
%e A364902 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 A364902 17=16+1=9+8, so a(17) = least novel Min{a(1)*p,a(8)*q} = Min{p,8*q} = 11.
%e A364902 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 A364902 1; 1;
%e A364902 2; 2;
%e A364902 3,4; 3;
%e A364902 5,10,15,8,9; 4,5,10,15;
%e A364902 7,14,20,25,35,50,16; 8;
%e A364902 11,22,21,28,55... 9,7,14,20,25,35,50
%t A364902 nn = 120; c[_] = False; s = {1, 2}; w = Length[s]; t = Prime[s]; flag = 0;
%t A364902 Array[Set[{q[#1], p[#1],
%t A364902 r[#1]}, {#1, #2,
%t A364902 Prepend[#2^Range[Floor@Log[#2, nn]], 1]} & @@ {#2,
%t A364902 Prime[#2]}] & @@ {#, s[[#]]} &, w];
%t A364902 Do[If[n == 1,
%t A364902 Set[{a[n], c[1]}, {1, True}],
%t A364902 Array[Set[m[#], 1] &, w];
%t A364902 Array[Set[j[#], n - p[#]^(-1 + LengthWhile[r[#], # < n + 1 &])] &, w];
%t A364902 Array[
%t A364902 If[j[#] == 0,
%t A364902 k[#] = n; flag = #,
%t A364902 While[Set[k[#], Prime[m[#]] a[j[#]]];
%t A364902 Or[MemberQ[s, m[#]], c[k[#]]], m[#]++]] &, w];
%t A364902 If[flag > 0,
%t A364902 Set[{a[n], c[k[flag]]}, {k[flag], True}]; flag = 0,
%t A364902 Set[{a[n], c[#]}, {#, True}] &[Min@ Array[k, w]] ]], {n, nn}];
%t A364902 Array[a, nn] (* _Michael De Vlieger_, Sep 24 2023 *)
%Y A364902 Cf. A005940, A006899, A047253, A108906, A356867, A364611, A364628.
%K A364902 nonn
%O A364902 1,2
%A A364902 _David James Sycamore_ and _Michael De Vlieger_ Sep 21 2023