

A106108


Rowland's primegenerating sequence: a(1) = 7; for n > 1, a(n) = a(n1) + gcd(n, a(n1)).


61



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

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(n1) 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(n1) + gcd(A000265(n), a(n1)).  David Morales Marciel, Sep 14 2016


REFERENCES

Eric S. Rowland, A simple primegenerating recurrence, Abstracts Amer. Math. Soc., 29 (No. 1, 2008), p. 50 (Abstract 103511986).


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 RuizCabello, On Rowland's sequence, Electronic J. Combin., Vol. 18(2), 2011, #P10.
Brian Hayes, Pumping the Primes, bitplayer, 19 August 2015.
Eric S. Rowland, A simple primegenerating recurrence, arXiv:0710.3217 [math.NT], 20072008.
Eric S. Rowland, PrimeGenerating 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(n1)+gcd(n, f(n1))); 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[n1]+gcd(n, a[n1])); a \\ Charles R Greathouse IV, Jul 15 2011
(Haskell)
a106108 n = a106108_list !! (n1)
a106108_list =
7 : zipWith (+) a106108_list (zipWith gcd a106108_list [2..])
 Reinhard Zumkeller, Nov 15 2013
(MAGMA) [n le 1 select 7 else Self(n1) + Gcd(n, Self(n1)): 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: A236683 A120200 A343297 * A120309 A035705 A205698
Adjacent sequences: A106105 A106106 A106107 * A106109 A106110 A106111


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Jan 28 2008


STATUS

approved



