A084663 a(1) = 8; a(n) = a(n-1) + gcd(a(n-1), n).

8, 10, 11, 12, 13, 14, 21, 22, 23, 24, 25, 26, 39, 40, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 87, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 177, 180, 181, 182, 189, 190

Offset: 1

Author

Matthew Frank (mfrank(AT)wopr.wolfram.com) on behalf of the 2003 New Kind of Science Summer School, Jul 15 2003

Keywords

• nonn

Comments

The first 150000000 differences are all primes or 1. Is this true in general?

The proof of the conjecture is identical to the proof in the Rowland link. - Yifan Xie, Apr 11 2025

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..50000
• Eric S. Rowland, A natural prime-generating recurrence, arXiv:0710.3217 [math.NT], 2007-2008.
• Eric S. Rowland, A natural prime-generating recurrence, JIS 11 (2008) 08.2.8.

Crossrefs

Cf. A084662, A106108.

Cf. A230504, A134744 (first differences), A134736.

Programs

Haskell

a084663 n = a084663_list !! (n-1)
a084663_list =
   8 : zipWith (+) a084663_list (zipWith gcd a084663_list [2..])
-- Reinhard Zumkeller, Nov 15 2013

Maple

S := 8; f := proc(n) option remember; global S; if n=1 then S else f(n-1)+igcd(n,f(n-1)); fi; end;

Mathematica

a[n_]:= a[n]= If[n==1,8, a[n-1] + GCD[n, a[n-1]]]; Table[a[n], {n,70}]
RecurrenceTable[{a[1]==8,a[n]==a[n-1]+GCD[a[n-1],n]},a,{n,70}] (* Harvey P. Dale, Apr 12 2016 *)

SageMath

@CachedFunction
def a(n): # a = A084663
    if (n==1): return 8
    else: return a(n-1) + gcd(a(n-1), n)
[a(n) for n in range(1, 71)] # G. C. Greubel, Mar 22 2023