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A304752
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Beginning with a(1) = 1, for n > 1, a(n) = the least divisor of a(n-1) not included earlier, otherwise a(n) = the least multiple m*a(n-1) such that m is not a divisor of a(n-1) and m*a(n-1) is not included earlier.
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2
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1, 2, 6, 3, 12, 4, 20, 5, 10, 30, 15, 60, 420, 7, 14, 42, 21, 84, 28, 140, 35, 70, 210, 105, 630, 9, 18, 72, 8, 24, 120, 40, 240, 16, 48, 336, 56, 168, 840, 280, 1680, 80, 480, 32, 96, 672, 112, 560, 3360, 160, 960, 64, 192, 1344, 224, 1120, 6720, 320, 1920, 128, 384, 2688, 448, 2240, 13440, 640, 3840, 256, 768, 5376, 896, 4480, 26880, 1280, 7680, 512
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OFFSET
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1,2
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COMMENTS
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In the table below, we note a cycle that subtends for 41 <= n <= 2^14.
Let e = floor(n/8). We write multiple k if the condition is false, or the parity of divisor d if d does not occur in a. We can express a(n) as the product of the smallest four primes as shown below.
n (mod 8) k or d 2 3 5 7
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0 5 2^(e-2) 5 7
1 6 2^(e-1) 3 5 7
2 EVEN 2^(e-1) 5
3 6 2^(e-1) 3 5
4 EVEN 2^e
5 3 2^e 3
6 7 2^e 3 7
7 EVEN 2^(e-1) 7
Conjectures:
1. All terms are divisible only by some combination of the smallest 4 primes.
2. Powers 2^e, positive integer e, are at n = {1, 2, 6, 29, 34, 44, 52, 60, 68, ...}; first differences are {1, 4, 23, 5, 10, 8, 8, 8, ...}, and 8 thereafter.
3. For n > 41 such that n (mod 8) = 4, a(n) = 2^((n-4)/8).
4. For n > 26 all terms are even. Odd terms are {1, 3, 5, 15, 7, 21, 35, 105, 9} at indices {1, 4, 8, 11, 14, 17, 21, 24, 26}. (End)
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LINKS
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EXAMPLE
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After a(27) = 18 = 2 * 3^2, the next term a(28) is neither 2*18 = 2^2 * 3^2, nor 3*18 = 2 * 3^3 as both 2 and divide 18. But 4 does not divide 18, and 4*18 = 72 haven't yet been used in the sequence, thus a(28) = 72.
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MAPLE
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lim:=60: with(numtheory): membera := proc(val) global a, n: local j: for j from 1 to n-1 do if(a[j]=val)then return true: fi: od: return false: end: a[1]:=1:for n from 2 to lim do d:=sort([divisors(a[n-1])[]]): s:=true: for k from 1 to nops(d) do if(not membera(d[k]))then a[n]:=d[k]:s:=false: break:fi:od: if(s)then for j from 2 do if(not member(j, d) and not membera(j*a[n-1]))then a[n]:=j*a[n-1]:break: fi:od:fi:od: seq(a[n], n=1..lim); # Nathaniel Johnston, May 10 2011, given originally for A113552
# second Maple program:
b:= proc(n) is(n=1) end:
a:= proc(n) option remember; local j, l, i, m;
j:= a(n-1): l:= sort([numtheory[divisors](j)[]]);
for i to nops(l) do if not b(l[i])
then b(l[i]):=true; return l[i]
fi od;
for m while m in l or b(m*j) do od;
b(m*j):=true; m*j
end: a(1):=1:
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MATHEMATICA
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f[s_] := Append[s, d = Divisors[ s[[ -1]]]; If[ Complement[d, s] != {}, Complement[d, s][[1]], k = 2; While[ Mod[ s[[ -1]], k] == 0 || MemberQ[s, k*s[[ -1]]], k++ ]; k*s[[ -1]] ]]; Nest[f, {1}, 60] (* Robert G. Wilson v, Aug 20 2006, given originally for A113552 *)
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PROG
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(PARI)
up_to = (2^14)+1;
v304752 = vector(up_to);
m_occurrences = Map();
k=0; prev=1; for(n=1, up_to, fordiv(prev, d, if(!mapisdefined(m_occurrences, d), v304752[n] = d; mapput(m_occurrences, d, n); break)); if(!v304752[n], m = 1; try = prev; while(!(prev%m) || mapisdefined(m_occurrences, try), m++; try = prev*m); mapput(m_occurrences, v304752[n] = try, n)); prev = v304752[n]);
(PARI) A304752(n, a=1, list=List(a)/*set to 0 to get just a(n)*/, U=[])={ for(i=2, n, U=setunion(U, [a]); fordiv(a, d, setsearch(U, d)||[a=-d, break]); if(a>0, for(m=2, oo, a%m && !setsearch(U, m*a)&& (a*=m)&& break), a=-a); list&& listput(list, a); /*a%2&&printf("a(%d)=%d, ", i, a)*/); if(list, list, a)} \\ M. F. Hasler, Dec 26 2020
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CROSSREFS
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Differs from A113552 for the first time at n=28, where a(28) = 72, while A113552(28) = 90.
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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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