A112877 -id:A112877 - cp's OEIS frontend

A370951 First differences of A112877 (zero terms in Cald's sequence A006509).

82, 182, 46, 94, 200, 430, 846, 1628, 2982, 5662, 10940, 17924, 34308, 65768, 125760, 240732, 460672, 883598, 1697502, 3268008, 6297778, 12152690, 23482980, 45422208, 87949242, 170465380, 330760622, 642315104, 1094147916, 2132023868, 4153355532, 8093060816, 15777058876

Offset: 1

N. J. A. Sloane, Mar 07 2024

nonn

The terms essentially double at each step. The ratios of successive terms are 2.219512195, 0.2527472527, 2.043478261, 2.127659574, 2.150000000, 1.967441860, 1.924349882, 1.831695332, 1.898725687, 1.932179442, 1.638391225, 1.914081678, 1.916987292, 1.912176134, 1.914217557, 1.913630095, 1.918063177, 1.921124765, 1.925186539, 1.927099934, 1.929679007, 1.932327740, 1.934260814, 1.936260826, 1.938224550, 1.940338983, 1.941933414, 1.703444165, 1.948570058, 1.948081161, 1.948559605, 1.949455124...

Cf. A006509, A112877.

nn = 2^20; c[_] := False; a[1] = j = 1; c[1] = True;
Differences@ Monitor[Reap[
    Do[p = Prime[n - 1];
     If[And[# > 0, ! c[#]], k = #,
        If[c[#], k = 0; Sow[n], k = #] &[j + p]] &[j - p];
Set[{c[k], j}, {True, k}], {n, 2, nn}]][[-1, 1]], n] (* Michael De Vlieger, Mar 07 2024 *)

from itertools import count, islice
from sympy import nextprime
def A370951_gen(): # generator of terms
    a, aset, p, q = 1, {1}, 2, 0
    for c in count(2):
        if (b:=a-p) > 0 and b not in aset:
            a = b
        elif (b:=a+p) not in aset:
            a = b
        else:
            a = 0
            if q:
                yield c-q
            q = c
        aset.add(a)
        p = nextprime(p)
A370951_list = list(islice(A370951_gen(),10)) # Chai Wah Wu, Mar 07 2024

a(29)-a(33) from Martin Ehrenstein, Mar 07 2024

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.

Original entry on oeis.org

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

N. J. A. Sloane

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

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).

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

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.

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

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))];

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 *)

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

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

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

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

A117128 Recamán transform of primes (another version): a(0)=1; for n>0, a(n) = a(n-1) - prime(n) if that number is positive and not already in the sequence, otherwise a(n) = a(n-1) + prime(n).

Original entry on oeis.org

Offset: 0

N. J. A. Sloane, Apr 20 2006

nonn

Differs from Cald's sequence A006509 for first time at n=116 (or 117, depending on offset).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Recamán's sequence

Cf. A064365, A006509, A112877.

import Data.Set (singleton, notMember, insert)
a117128 n = a117128_list !! n
a117128_list = 1 : f 1 a000040_list (singleton 1) where
   f x (p:ps) s | x' > 0 && x' `notMember` s = x' : f x' ps (insert x' s)
                | otherwise                  = xp : f xp ps (insert xp s)
                where x' = x - p; xp = x + p
-- Reinhard Zumkeller, Apr 26 2012

M1:=500000; a:=array(0..M1); have:=array(1..M1); a[0]:=1; for n from 1 to M1 do have[n]:=0; od: 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 then a[n]:=j; have[j]:=1;
else nmax:=n-1; break; fi; od: [seq(a[n],n=0..M2)];

a = {1}; Do[If[And[#1 > 0, ! MemberQ[a, #1]], AppendTo[a, #1], AppendTo[a, #2]] & @@ {#1 - #2, #1 + #2} & @@ {a[[n - 1]], Prime[n - 1]}, {n, 2, 62}]; a (* Michael De Vlieger, Dec 05 2016 *)

from sympy import primerange, prime
def aupton(terms):
  alst = [1]
  for n, pn in enumerate(primerange(1, prime(terms)+1), start=1):
    x = alst[-1] - pn
    alst += [x if x > 0 and x not in alst else alst[-1] + pn]
  return alst
print(aupton(61)) # Michael S. Branicky, May 30 2021

a(n) = A064365(n) + 1. - Thomas Ordowski, Dec 05 2016

A117129 Primes not occurring as |differences| in Cald's sequence A006509.

Original entry on oeis.org

641, 1213, 2617, 2957, 3727, 5443, 9283, 17359, 33829, 66173, 131303, 264763, 494743, 957547, 1888157, 3753647, 7490797, 14961157, 29899357, 59773871, 119551489, 239106347, 478234723, 956607929, 1913366111, 3826828409, 7653840367, 15308666783, 30619196381, 57415599151

Offset: 1

N. J. A. Sloane, Apr 20 2006

nonn

In other words, primes that do not occur as absolute values of differences of successive terms in Cald's sequence A006509.

Cf. A006509, A112877.

from itertools import count, islice
from sympy import nextprime
def A117129_gen(): # generator of terms
    a, aset, p = 1, {1}, 2
    for c in count(2):
        if (b:=a-p) > 0 and b not in aset:
            a = b
        elif (b:=a+p) not in aset:
            a = b
        else:
            a = 0
            yield p
        aset.add(a)
        p = nextprime(p)
A117129_list =  list(islice(A117129_gen(),10)) # Chai Wah Wu, Mar 04 2024

a(n) = prime(A112877(n) - 1).

a(19)-a(26) from Donovan Johnson, Feb 18 2010
a(27)-a(29) from Chai Wah Wu, Mar 04 2024
a(30) from Chai Wah Wu, Mar 10 2024

cp's OEIS Frontend

A370951 First differences of A112877 (zero terms in Cald's sequence A006509).

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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.

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A117128 Recamán transform of primes (another version): a(0)=1; for n>0, a(n) = a(n-1) - prime(n) if that number is positive and not already in the sequence, otherwise a(n) = a(n-1) + prime(n).

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A117129 Primes not occurring as |differences| in Cald's sequence A006509.

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A112878 A100298(n) = 0.

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