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A137613
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Omit the 1's from Rowland's sequence f(n) - f(n-1) = gcd(n,f(n-1)), where f(1) = 7.
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11
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5, 3, 11, 3, 23, 3, 47, 3, 5, 3, 101, 3, 7, 11, 3, 13, 233, 3, 467, 3, 5, 3, 941, 3, 7, 1889, 3, 3779, 3, 7559, 3, 13, 15131, 3, 53, 3, 7, 30323, 3, 60647, 3, 5, 3, 101, 3, 121403, 3, 242807, 3, 5, 3, 19, 7, 5, 3, 47, 3, 37, 5, 3, 17, 3, 199, 53, 3, 29, 3, 486041, 3, 7, 421, 23
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OFFSET
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1,1
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COMMENTS
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Rowland proves that each term is prime. He says it is likely that all odd primes occur.
In the first 5000 terms, there are 965 distinct primes and 397 is the least odd prime that does not appear. - T. D. Noe, Mar 01 2008
In the first 10000 terms, the least odd prime that does not appear is 587, according to Rowland. - Jonathan Sondow, Aug 14 2008
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LINKS
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Eric S. Rowland, A simple prime-generating recurrence, Abstracts Amer. Math. Soc. 29 (No. 1, 2008), p. 50 (Abstract 1035-11-986).
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FORMULA
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Denote by Lpf(n) the least prime factor of n. Then a(n) = Lpf( 6-n+Sum_{i=1..n-1} a(i) ). - Vladimir Shevelev, Mar 03 2010
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EXAMPLE
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f(n) = 7, 8, 9, 10, 15, 18, 19, 20, ..., so f(n) - f(n-1) = 1, 1, 1, 5, 3, 1, 1, ... and a(n) = 5, 3, ... .
a(1) = Lpf(6-1) = 5;
a(2) = Lpf(6-2+5) = 3;
a(3) = Lpf(6-3+5+3) = 11;
a(4) = Lpf(6-4+5+3+11) = 3;
a(5) = Lpf(6-5+5+3+11+3) = 23. (End)
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MAPLE
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local a, c, k, L;
L := NULL; a := 7;
for k from 2 to n do
c := igcd(k, a);
a := a + c;
if c > 1 then L:=L, c fi;
od;
L end:
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MATHEMATICA
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f[1] = 7; f[n_] := f[n] = f[n - 1] + GCD[n, f[n - 1]]; DeleteCases[Differences[Table[f[n], {n, 10^6}]], 1] (* Alonso del Arte, Nov 17 2011 *)
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PROG
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(Haskell)
a137613 n = a137613_list !! (n-1)
a137613_list = filter (> 1) a132199_list
(PARI)
ub=1000; n=3; a=9; while(n<ub, m=a\n; d=factor((m-1)*n-1)[1, 1]; print1(d, ", "); n=n+((d-1)\(m-1)); a=m*n; ); \\ Daniel Constantin Mayer, Aug 31 2014
(Python)
from itertools import count, islice
from math import gcd
def A137613_gen(): # generator of terms
a = 7
for n in count(2):
if (b:=gcd(a, n)) > 1: yield b
a += b
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CROSSREFS
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f(n) = f(n-1) + gcd(n, f(n-1)) = A106108(n) and f(n) - f(n-1) = A132199(n-1).
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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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