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A364902
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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.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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For n > 6, a(A006889(n) + 1) = prime(n-2).
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EXAMPLE
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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
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MATHEMATICA
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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}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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