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

%I A084663 #33 Jul 09 2025 03:58:27
%S A084663 8,10,11,12,13,14,21,22,23,24,25,26,39,40,45,46,47,48,49,50,51,52,53,
%T A084663 54,55,56,57,58,87,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,
%U A084663 105,106,107,108,109,110,111,112,113,114,115,116,117,118,177,180,181,182,189,190
%N A084663 a(1) = 8; a(n) = a(n-1) + gcd(a(n-1), n).
%C A084663 The first 150000000 differences are all primes or 1. Is this true in general?
%C A084663 The proof of the conjecture is identical to the proof in the Rowland link. - _Yifan Xie_, Apr 11 2025
%D A084663 Eric S. Rowland, A simple prime-generating recurrence, Abstracts Amer. Math. Soc., 29 (No. 1, 2008), p. 50 (Abstract 1035-11-986).
%H A084663 Indranil Ghosh, <a href="/A084663/b084663.txt">Table of n, a(n) for n = 1..50000</a>
%H A084663 Eric S. Rowland, <a href="https://arxiv.org/abs/0710.3217">A natural prime-generating recurrence</a>, arXiv:0710.3217 [math.NT], 2007-2008.
%H A084663 Eric S. Rowland, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Rowland/rowland21.html">A natural prime-generating recurrence</a>, JIS 11 (2008) 08.2.8.
%p A084663 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;
%t A084663 a[n_]:= a[n]= If[n==1,8, a[n-1] + GCD[n, a[n-1]]]; Table[a[n], {n,70}]
%t A084663 RecurrenceTable[{a[1]==8,a[n]==a[n-1]+GCD[a[n-1],n]},a,{n,70}] (* _Harvey P. Dale_, Apr 12 2016 *)
%o A084663 (Haskell)
%o A084663 a084663 n = a084663_list !! (n-1)
%o A084663 a084663_list =
%o A084663    8 : zipWith (+) a084663_list (zipWith gcd a084663_list [2..])
%o A084663 -- _Reinhard Zumkeller_, Nov 15 2013
%o A084663 (SageMath)
%o A084663 @CachedFunction
%o A084663 def a(n): # a = A084663
%o A084663     if (n==1): return 8
%o A084663     else: return a(n-1) + gcd(a(n-1), n)
%o A084663 [a(n) for n in range(1, 71)] # _G. C. Greubel_, Mar 22 2023
%Y A084663 Cf. A084662, A106108.
%Y A084663 Cf. A230504, A134744 (first differences), A134736.
%K A084663 nonn
%O A084663 1,1
%A A084663 Matthew Frank (mfrank(AT)wopr.wolfram.com) on behalf of the 2003 New Kind of Science Summer School, Jul 15 2003