A006509 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.

1, 3, 6, 11, 4, 15, 2, 19, 38, 61, 32, 63, 26, 67, 24, 71, 18, 77, 16, 83, 12, 85, 164, 81, 170, 73, 174, 277, 384, 275, 162, 35, 166, 29, 168, 317, 468, 311, 148, 315, 142, 321, 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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

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

References

• F. Cald, Problem 356, Franciscan order, J. Rec. Math., 7 (No. 4, 1974), 318; 10 (No. 1, 1977-78), 62-64.
• "Cald's Sequence", Popular Computing (Calabasas, CA), Vol. 4 (No. 41, Aug 1976), pp. 16-17.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Francis Cald and others, Problem 356, Franciscan order, Solution and Guesses, J. Rec. Math., 10 (No. 1, 1977-78), 62-64: Page 62, Page 63, Page 64. [Annotated copy]
• R. G. Wilson, Letter to N. J. A. Sloane with attachment, Jan 1989

Crossrefs

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

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

Cf. A117128, A117129, A064365, A000040.

Programs

Haskell

a006509 n = a006509_list !! (n-1)
a006509_list = 1 : f [1] a000040_list where
   f xs'@(x:_) (p:ps) | x' > 0 && x' `notElem` xs = x' : f (x':xs) ps
                      | x'' `notElem` xs          = x'' : f (x'':xs) ps
                      | otherwise                 = 0 : f (0:xs) ps
                      where x' = x - p; x'' = x + p
-- Reinhard Zumkeller, Oct 17 2011

Maple

M1:=500000; a:=array(0..M1); have:=array(0..M1); a[0]:=1;
for n from 0 to M1 do have[n]:=0; od: have[0]:=1; have[1]:=1;
M2:=2000; nmax:=M2; for n from 1 to M2 do p:=ithprime(n); i:=a[n-1]-p; j:=a[n-1]+p;
if i >= 1 and have[i]=0 then a[n]:=i; have[i]:=1;
elif j <= M1 and have[j]=0 then a[n]:=j; have[j]:=1;
elif j <= M1 then a[n]:=0; else nmax:=n-1; break; fi; od:
# To get A006509:
[seq(a[n],n=0..M2)];
# To get A112877 (off by 1 because of different offset in A006509):
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))];

Mathematica

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 *)

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

Python

from sympy import primerange, prime
def aupton(terms):
  alst = [1]
  for n, pn in enumerate(primerange(1, prime(terms)+1), start=1):
    x, y = alst[-1] - pn, alst[-1] + pn
    if x > 0 and x not in alst: alst.append(x)
    elif y > 0 and y not in alst: alst.append(y)
    else: alst.append(0)
  return alst
print(aupton(130)) # Michael S. Branicky, May 30 2021

Python

from sympy import nextprime
from itertools import islice
def agen(): # generator of terms
    pn, an, aset = 2, 1, {1}
    while True:
        yield an
        an = m if (m:=an-pn) > 0 and m not in aset else p if (p:=an+pn) not in aset else 0
        aset.add(an)
        pn = nextprime(pn)
print(list(islice(agen(), 131))) # Michael S. Branicky, Mar 07 2024

Extensions

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

Many more terms added by N. J. A. Sloane, Apr 20 2006, to show difference from A117128.

Entry revised by N. J. A. Sloane, Mar 06 2024