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 A006509 Cald's sequence: a(n+1)=a(n)-p(n) if new and >0, else a(n)+p(n) if new, otherwise 0, where p(n) = n-th prime. (Formerly M2539) 16

%I M2539

%S 1,3,6,11,4,15,2,19,38,61,32,63,26,67,24,71,18,77,16,83,12,85,164,81,

%T 170,73,174,277,384,275,162,35,166,29,168,317,468,311,148,315,142,321,

%U 140,331,138,335,136,347,124,351,122,355,116,357,106,363,100,369,98,375,94,377,84,391,80,393,76,407,70,417,68,421,62,429,56,435,52,441,44,445,36,455,34,465,898,459,902,453,910,449,912,1379,900,413,904,405,908,399,920,397,938,1485,928,365,934,1505,2082,1495,2088,1489,888,281,894,1511,892,261,0,643,1290,637,1296,635,1308,631,1314,623,1324,615,1334,607,1340

%N Cald's sequence: a(n+1)=a(n)-p(n) if new and >0, else a(n)+p(n) if new, otherwise 0, where p(n) = n-th prime.

%D F. Cald, Problem 356, Franciscan order, J. Rec. Math., 7 (No. 4, 1974), 318; 10 (No. 1, 1974), 62-64.

%D "Cald's Sequence", Popular Computing (Calabasas, CA), Vol. 4 (No. 41, Aug 1976), pp. 16-17.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A006509/b006509.txt">Table of n, a(n) for n=1..10000</a>

%H R. G. Wilson, <a href="/A006509/a006509.pdf">Letter to N. J. A. Sloane with attachment, Jan 1989</a>

%p M1:=500000; a:=array(0..M1); have:=array(0..M1); a[0]:=1;

%p for n from 0 to M1 do have[n]:=0; od: have[0]:=1; have[1]:=1;

%p M2:=2000; nmax:=M2; for n from 1 to M2 do p:=ithprime(n); i:=a[n-1]-p; j:=a[n-1]+p;

%p if i >= 1 and have[i]=0 then a[n]:=i; have[i]:=1;

%p elif j <= M1 and have[j]=0 then a[n]:=j; have[j]:=1;

%p elif j <= M1 then a[n]:=0; else nmax:=n-1; break; fi; od:

%p # To get A006509:

%p [seq(a[n],n=0..M2)];

%p # To get A112877 (off by 1 because of different offset in A006509):

%p zzz:=[]; for n from 0 to nmax do if a[n]=0 then zzz:=[op(zzz),n]; fi; od: [seq(zzz[i],i=1..nops(zzz))];

%t lst = {1}; f := Block[{b = Last@lst, p = Prime@ Length@lst}, If[b > p && !MemberQ[lst, b - p], AppendTo[lst, b - p], If[ !MemberQ[lst, b + p], AppendTo[lst, b + p], AppendTo[lst, 0]] ]]; Do[f, {n, 60}]; lst (* _Robert G. Wilson v_, Apr 25 2006 *)

%o a006509 n = a006509_list !! (n-1)

%o a006509_list = 1 : f [1] a000040_list where

%o f xs'@(x:_) (p:ps) | x' > 0 && x' `notElem` xs = x' : f (x':xs) ps

%o | x'' `notElem` xs = x'' : f (x'':xs) ps

%o | otherwise = 0 : f (0:xs) ps

%o where x' = x - p; x'' = x + p

%o -- _Reinhard Zumkeller_, Oct 17 2011

%Y Cf. A005132, A093903. Zeros are in A112877. Sorted: A111338, A111339 (numbers missing from A006509).

%Y Cf. A117128, A117129, A064365, A005132, A000040.

%K nonn,nice,easy

%O 1,2

%A _N. J. A. Sloane_

%E More terms from Larry Reeves (larryr(AT)acm.org), Jul 20 2001

%E Many more terms added by _N. J. A. Sloane_, Apr 20 2006, to show difference from A117128.

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Last modified August 8 06:27 EDT 2020. Contains 336290 sequences. (Running on oeis4.)