A064365 a(0) = 0; thereafter a(n) = a(n-1)-prime(n) if positive and new, otherwise a(n) = a(n-1)+prime(n), where prime(n) is the n-th prime.

0, 2, 5, 10, 3, 14, 1, 18, 37, 60, 31, 62, 25, 66, 23, 70, 17, 76, 15, 82, 11, 84, 163, 80, 169, 72, 173, 276, 383, 274, 161, 34, 165, 28, 167, 316, 467, 310, 147, 314, 141, 320, 139, 330, 137, 334, 135, 346, 123, 350, 121, 354, 115, 356, 105, 362, 99, 368, 97, 374, 93

Offset: 0

Author

Neil Fernandez, Sep 25 2001

Keywords

• nonn
• easy
• nice
• look
• hear

Comments

'Recamán transform' (see A005132) of the prime sequence. Note that the definition permits repeated terms [though only by addition] (and there are many repeated terms, just as there are in A005132).

Does every positive integer appear in the sequence? This seems unlikely, since 4 has not appeared in 70000 terms.

Note: this is similar to Clark Kimberling's A022831, except in the latter sequence the words 'and new' have been omitted.

The smallest numbers not occurring in the first million terms: 4, 6, 7, 12, 13, 16, 19, 20, 21, 22, 24, 26, 27, 29, 30, 32, 36, 39, 41, 42. - Reinhard Zumkeller, Apr 26 2012

Examples

			To find a(9) we try subtracting the 9th prime, which is 23, from a(8), which is 37. 37 - 23 = 14, but 14 is already in the sequence (it is a(5)), so we must add. a(9) = 37 + 23 = 60.
		

Links

• N. J. A. Sloane, First 70000 terms
• Index entries for sequences related to Recamán's sequence

Crossrefs

Cf. A005132, A022831, A117128.

Programs

Haskell

import Data.Set (singleton, notMember, insert)
a064365 n = a064365_list !! n
a064365_list = 0 : f 0 a000040_list (singleton 0) 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

Mathematica

a = {0}; Do[ If[ a[ [ -1 ] ] - Prime[ n ] > 0 && Position[ a, a[ [ -1 ] ] - Prime[ n ] ] == {}, a = Append[ a, a[ [ -1 ] ] - Prime[ n ] ], a = Append[ a, a[ [ -1 ] ] + Prime[ n ] ] ], {n, 1, 70} ]; a (* Modified by Ivan N. Ianakiev, Aug 05 2019, to accommodate the new initial term of a(0). *)

PARI

A064365(N,s/*=1 to print all terms*/)={ my(a=0,u=0); N & forprime(p=1,prime(N), s & print1(a","); u=bitor(u,2^a+=if(a<=p || bittest(u,a-p),p,-p)));a}  \\ M. F. Hasler, Mar 07 2012

Python

from sympy import primerange, prime
def aupton(terms):
  alst = [0]
  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(60)) # Michael S. Branicky, May 30 2021

Formula

a(n) = A117128(n) - 1. - Thomas Ordowski, Dec 05 2016

Extensions

More terms from Robert G. Wilson v, Sep 26 2001

Further terms from N. J. A. Sloane, Feb 10 2002

Added initial term a(0)=0, in analogy with A128204, A005132, A053461, A117073/A078783. - M. F. Hasler, Mar 07 2012