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A364934
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a(n+1) = 1 + number of previous terms that share a factor > 1 with a(n); a(1) = 2.
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1
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2, 2, 3, 2, 4, 5, 2, 6, 8, 8, 9, 4, 10, 12, 14, 13, 2, 14, 15, 8, 16, 17, 2, 18, 22, 20, 23, 2, 22, 23, 3, 8, 24, 29, 2, 26, 28, 28, 29, 3, 10, 32, 31, 2, 32, 33, 13, 4, 34, 36, 42, 43, 2, 38, 39, 17, 4, 40, 43, 3, 15, 21, 21, 22, 43, 4, 43, 5, 9, 19, 3, 20, 48, 58, 48, 60, 63, 28, 52, 53, 2, 51, 28
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OFFSET
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1,1
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COMMENTS
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There are prominent lines that have more terms, their coefficients are approximately: 0.519, 0.329, 0.689, 0.188, 0.615, ... (see the frequency link). They seem to be distorted prime harmonic lines: 1/2, 1/3, 2/3, 1/5, 3/5, ... from A016035.
It appears limsup a(n)/n is approximately 0.83.
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LINKS
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EXAMPLE
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[2,*] 1 term shares a factor with 2, so a(2) = 1+1 = 2.
[2,2,*] 2 terms share a factor with 2, so a(3) = 1+2 = 3.
[2,2,3,*] 1 term shares a factor with 3, so a(4) = 1+1 = 2.
[2,2,3,2,*] 3 terms share a factor with 2, so a(5) = 1+3 = 4.
[2,2,3,2,4,*] 4 terms share a factor with 4, so a(6) = 1+4 = 5.
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MATHEMATICA
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nn = 120; c[_] := 0; s = {}; a[1] = j = 2; Do[c[j]++; If[c[j] == 1, AppendTo[s, j]]; k = 1 + Total@ Map[c[#] Boole[GCD[#, j] > 1] &, s]; Set[{a[n], j}, {k, k}], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Aug 16 2023 *)
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PROG
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(Python)
from sympy.ntheory import primefactors
a=[2]; p = [{2}];
for n in range(1, 1000):
a.append(sum(1 if p[-1].intersection(p[i]) else 0 for i in range(n))+1)
p.append(set(primefactors(a[-1])))
(Python)
from math import gcd, prod
from sympy import factorint
from itertools import islice
from collections import Counter
def agen(): # generator of terms
a=[2]; f=2; c=Counter([f])
while True:
yield a[-1]
a.append(1 + sum(c[fi] for fi in c if gcd(f, fi)>1))
f=prod(factorint(a[-1]).keys())
c[f] += 1
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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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