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a(0)=0; thereafter a(n) = a(n-1) + prime(n) if a(n-1) < prime(n), otherwise a(n) = a(n-1) - prime(n).
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%I #51 Sep 13 2024 15:48:41

%S 0,2,5,0,7,18,5,22,3,26,55,24,61,20,63,16,69,10,71,4,75,2,81,164,75,

%T 172,71,174,67,176,63,190,59,196,57,206,55,212,49,216,43,222,41,232,

%U 39,236,37,248,25,252,23,256,17,258,7,264,1,270,541,264,545,262,555

%N a(0)=0; thereafter a(n) = a(n-1) + prime(n) if a(n-1) < prime(n), otherwise a(n) = a(n-1) - prime(n).

%C a(n) < 2*prime(n). Conjecture: a(n) > 0 for n > 3. - _Thomas Ordowski_, Dec 03 2016 [This conjecture is false, because a(369019)=0. The next counterexample occurs at n = 22877145. - _Dmitry Kamenetsky_, Feb 14 2017. (Cf. A309225.)]

%H Robert Israel, <a href="/A008348/b008348.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = c(1)p(1) + ... + c(n)p(n), where c(i) = 1 if a(i-1) > p(i) and c(i) = -1 if a(i-1) <= p(i) (p(i) = primes). - _Clark Kimberling_

%p A008348 := proc(n) option remember; if n = 0 then 0 elif A008348(n-1)>=ithprime(n) then A008348(n-1)-ithprime(n); else A008348(n-1)+ithprime(n); fi; end;

%p # Maple from _N. J. A. Sloane_, Aug 31 2019 (Start)

%p # Riecaman transform

%p Riecaman := proc(a,s,M)

%p # Start with s, add or subtract a[n], get M terms. If a has w terms, can get M=w+1 terms.

%p local b,M2,n,t;

%p if whattype(a) <> list then ERROR("First argument should be a list"); fi;

%p if a[1]=0 then ERROR("a[1] should not be zero"); fi;

%p M2 := min(nops(a),M-1);

%p b:=[s]; t:=s;

%p for n from 1 to M2 do

%p if a[n]>t then t:=t+a[n] else t:=t-a[n]; fi; b:=[op(b),t]; od:

%p b; end;

%p # Riecaman transform of primes, starting at s=0

%p p1:=[seq(ithprime(i),i=1..100)];

%p q0:=Riecaman(p1,0,99);

%p # End

%t a := {0}; For[n = 2, n < 100, n++, If[a[[n - 1]] >= Prime[n - 1], b := a[[n - 1]] - Prime[n - 1], b := a[[n - 1]] + Prime[n - 1];]; a = Append[a, b]]; a (* _Stefan Steinerberger_, May 02 2006 *)

%t nxt[{n_,a_}]:={n+1,If[a<Prime[n+1],a+Prime[n+1],a-Prime[n+1]]}; NestList[nxt,{0,0},70][[;;,2]] (* _Harvey P. Dale_, Sep 13 2024 *)

%o (PARI) lista(nn) = {print1(a=0, ", "); for (n=1, nn, if (a < (p=prime(n)), a += p, a -= p); print1(a, ", "););} \\ _Michel Marcus_, Dec 04 2016

%Y Cf. A008344, A022831, A022837, A309225.

%K nonn,look

%O 0,2

%A _N. J. A. Sloane_ and _J. H. Conway_

%E More terms from _Clark Kimberling_

%E Name edited by _Dmitry Kamenetsky_, Feb 14 2017