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A006509
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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.
(Formerly M2539)
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20
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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
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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]
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MAPLE
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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:
[seq(a[n], n=0..M2)];
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))];
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MATHEMATICA
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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 *)
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PROG
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(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
(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
(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)
(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
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CROSSREFS
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A111338 gives (conjecturally) the terms of the present sequence sorted into increasing order, and A111339 gives (conjecturally) the numbers missing from the present sequence.
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Jul 20 2001
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STATUS
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approved
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