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A106108 Rowland's prime-generating sequence: a(1) = 7; for n > 1, a(n) = a(n-1) + gcd(n, a(n-1)). 60
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The title refers to the sequence of first differences, A132199.

Setting a(1) = 4 gives A084662.

Rowland proves that the first differences are all 1's or primes. The prime differences form A137613.

See A137613 for additional comments, links and references. - Jonathan Sondow, Aug 14 2008

"This recurrence was discovered at the 2003 NKS Summer School by a group led by Matt Frank. This Demonstration allows initial conditions. a(1) >= 4. For 1 <= a(1) <= 3, a(n) - a(n-1) is 1 for n >= 3." See Wolfram hyperlink. - Robert G. Wilson v, Sep 10 2008

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

REFERENCES

Eric S. Rowland, A simple prime-generating recurrence, Abstracts Amer. Math. Soc., 29 (No. 1, 2008), p. 50 (Abstract 1035-11-986).

LINKS

Indranil Ghosh, Table of n, a(n) for n = 1..25000 (terms 1..1000 from T. D. Noe)

Fernando Chamizo, Dulcinea Raboso and Serafin Ruiz-Cabello, On Rowland's sequence, Electronic J. Combin., Vol. 18(2), 2011, #P10.

Brian Hayes, Pumping the Primes, bit-player, 19 August 2015.

Eric S. Rowland, A simple prime-generating recurrence, arXiv:0710.3217 [math.NT], 2007-2008.

Eric S. Rowland, Prime-Generating Recurrence, Wolfram Demonstrations Project. - Robert G. Wilson v, Sep 10 2008

MAPLE

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)];

MATHEMATICA

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 *)

PROG

(PARI) a=vector(100); a[1]=7; for(n=2, #a, a[n]=a[n-1]+gcd(n, a[n-1])); a \\ Charles R Greathouse IV, Jul 15 2011

(Haskell)

a106108 n = a106108_list !! (n-1)

a106108_list =

   7 : zipWith (+) a106108_list (zipWith gcd a106108_list [2..])

-- Reinhard Zumkeller, Nov 15 2013

(MAGMA) [n le 1 select 7 else Self(n-1) + Gcd(n, Self(n-1)): n in [1..70]]; // Vincenzo Librandi, Jul 19 2015

CROSSREFS

Cf. A084662, A084663, A132199, A134734, A134736, A134743, A134744, A134162, A137613, A221869.

Cf. A230504.

Sequence in context: A065976 A236683 A120200 * A120309 A035705 A205698

Adjacent sequences:  A106105 A106106 A106107 * A106109 A106110 A106111

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jan 28 2008

STATUS

approved

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Last modified January 22 23:00 EST 2019. Contains 319365 sequences. (Running on oeis4.)