A006509 - OEIS

%I M2539 #83 Mar 07 2024 15:02:25

%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) - prime(n) if that value is positive and new, otherwise a(n) + prime(n) if new, otherwise 0; start with a(1)=1.

%C The differences between this sequence and A117128 ("Recamán transform of primes") are (i) the offset (0 there) and (ii) there the sum is used in the second case whether it has already occurred or not (so duplicates occur), while here a(n+1) = 0 if the sum already occurred (so there are no duplicates apart from the zeros). - _M. F. Hasler_, Mar 06 2024

%D F. Cald, Problem 356, Franciscan order, J. Rec. Math., 7 (No. 4, 1974), 318; 10 (No. 1, 1977-78), 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 Francis Cald and others, Problem 356, Franciscan order, Solution and Guesses, J. Rec. Math., 10 (No. 1, 1977-78), 62-64: <a href="/A006509/a006509_1.pdf">Page 62</a>, <a href="/A006509/a006509_2.pdf">Page 63</a>, <a href="/A006509/a006509_3.pdf">Page 64</a>. [Annotated copy]

%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 (Haskell)

%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

%o (Python)

%o from sympy import primerange, prime

%o def aupton(terms):

%o   alst = [1]

%o   for n, pn in enumerate(primerange(1, prime(terms)+1), start=1):

%o     x, y = alst[-1] - pn, alst[-1] + pn

%o     if x > 0 and x not in alst: alst.append(x)

%o     elif y > 0 and y not in alst: alst.append(y)

%o     else: alst.append(0)

%o   return alst

%o print(aupton(130)) # _Michael S. Branicky_, May 30 2021

%o (Python)

%o from sympy import nextprime

%o from itertools import islice

%o def agen(): # generator of terms

%o     pn, an, aset = 2, 1, {1}

%o     while True:

%o         yield an

%o         an = m if (m:=an-pn) > 0 and m not in aset else p if (p:=an+pn) not in aset else 0

%o         aset.add(an)

%o         pn = nextprime(pn)

%o print(list(islice(agen(), 131))) # _Michael S. Branicky_, Mar 07 2024

%o (PARI) A006509_upto(N, U=0)=vector(N,i, N=if(i>1, my(p=prime(i-1)); if( N>p && !bittest(U,N-p), N-p, !bittest(U, N+p), N+p), 1); N && U += 1 << N; N) \\ _M. F. Hasler_, Mar 06 2024

%Y Cf. A005132, A093903, A112877 & A370951 (indices of zeros).

%Y A111338 gives (conjecturally) the terms of the present sequence sorted into increasing order, and A111339 gives (conjecturally) the numbers missing from the present sequence.

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

%K nonn,nice

%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.

%E Entry revised by _N. J. A. Sloane_, Mar 06 2024