A113880 Variation on Recamán's sequence utilizing the four basic operations (/,-,+,*) in that order.

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 18, 432, 407, 381, 354, 326, 297, 267, 236, 204, 171, 137, 102, 66, 29, 67, 28, 68, 27, 69, 26, 70, 115, 161, 114, 162, 113, 163, 112, 60, 3180, 3126, 3071, 3015, 2958, 51, 110, 50, 111, 49

Offset

0,3

Comments

More precisely:

a(n) = a(n-1)/n if a(n-1)/n is integer and not already in the sequence. Else:

a(n) = a(n-1)-n if a(n-1)-n is positive and not already in the sequence. Else:

a(n) = a(n-1)+n if a(n-1)+n is not already in the sequence. Else:

a(n) = a(n-1)*n if a(n-1)*n is not already in the sequence. Else STOP.

In other words, divide if you can, else subtract, else add, else multiply.

By a(1000) there are 3 division steps, 928 subtraction steps, 59 addition steps and 10 multiplication steps. It is unlikely that every number belongs to the sequence since there are many "holes". It is an open question if there are any repetitions after a multiplication step. Can anybody expand the series?

At a(2500000000) = 2285684529311288243, there have been 44 divisions, 2499821613 subtractions, 178253 additions, and 90 multiplications. The largest value seen was a(1926305697) = 3555357710450807490. No multiplication step has produced a duplicate term. - Benjamin Chaffin, Sep 22 2016

Links

Robert G. Wilson v, Table of n, a(n) for n = 0..10000

Index entries for sequences related to Recamán's sequence

Example

a(24) = 432 because:
- a(23) = 18,
- 18/24 is not an integer,
- 18 - 24 is negative,
- 18 + 24 = 42 is already in the sequence,
- 18 * 24 = 432 is not already in the sequence.

Mathematica

f[s_List] := Block[{l = s[[-1]], n = Length@ s}, If[ IntegerQ[l/n] && !MemberQ[s, l/n], Append[s, l/n], If[l > n && !MemberQ[s, l - n], Append[s, l - n], If[ !MemberQ[s, l + n], Append[s, l + n], Append[s, l*n]]]]]; Nest[f, {0}, 62] (* Robert G. Wilson v, Sep 10 2016 *)

Prog

(Python)
from itertools import count, islice
def A113880(): # generator of terms
    aset, an = {0, 1}, 1
    yield from [0, 1]
    for n in count(2):
        q, r = divmod(an, n)
        if r == 0 and q not in aset: an = q
        elif (d:=an-n) > 0 and d not in aset: an = d
        elif (s:=an+n) not in aset: an = s
        elif (m:=an*n) not in aset: an = m
        else: return
        yield an
        aset.add(an)
print(list(islice(A113880(), 63))) # Michael S. Branicky, May 26 2023

Crossrefs

Cf. A005132.

Sequence in context: A064389 A118201 A274647 * A339192 A171884 A339557

Adjacent sequences: A113877 A113878 A113879 * A113881 A113882 A113883

Keyword

nonn

Author

Sergio Pimentel, Jan 27 2006

Extensions

a(0) = 0 prepended by Robert G. Wilson v, Sep 10 2016

Status

approved