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A106108
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Rowland's prime-generating sequence: a(1) = 7; for n > 1, a(n) = a(n-1) + gcd(n, a(n-1)).
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61
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7, 8, 9, 10, 15, 18, 19, 20, 21, 22, 33, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 69, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 141, 144, 145, 150, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168
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OFFSET
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1,1
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COMMENTS
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The title refers to the sequence of first differences, A132199.
Rowland proves that the first differences are all 1's or primes. The prime differences form A137613.
Not all starting values generate differences of all 1's or primes. The following a(1) generate composite differences: 532, 533, 534, 535, 698, 699, 706, 707, 708, 709, 712, 713, 714, 715, ... - Dmitry Kamenetsky, Jul 18 2015
The same results are obtained if 2's are removed from n when gcd is performed, so the following is also true: a(1) = 7; for n > 1, a(n) = a(n-1) + gcd(A000265(n), a(n-1)). - David Morales Marciel, Sep 14 2016
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REFERENCES
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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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LINKS
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Fernando Chamizo, Dulcinea Raboso and Serafin Ruiz-Cabello, On Rowland's sequence, Electronic J. Combin., Vol. 18(2), 2011, #P10.
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MAPLE
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S:=7; f:= proc(n) option remember; global S; if n=1 then RETURN(S); else RETURN(f(n-1)+gcd(n, f(n-1))); fi; end; [seq(f(n), n=1..200)];
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MATHEMATICA
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a[1] = 7; a[n_] := a[n] = a[n - 1] + GCD[n, a[n - 1]]; Array[a, 66] (* Robert G. Wilson v, Sep 10 2008 *)
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PROG
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(Haskell)
a106108 n = a106108_list !! (n-1)
a106108_list =
7 : zipWith (+) a106108_list (zipWith gcd a106108_list [2..])
(Magma) [n le 1 select 7 else Self(n-1) + Gcd(n, Self(n-1)): n in [1..70]]; // Vincenzo Librandi, Jul 19 2015
(Python)
from itertools import count, islice
from math import gcd
def A106108_gen(): # generator of terms
yield (a:=7)
for n in count(2):
yield (a:=a+gcd(a, n))
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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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