A181391 -id:A181391 - cp's OEIS frontend

A171897 Numbers in the van Eck sequence A181391 in order of first appearance.

0, 1, 2, 6, 5, 4, 3, 9, 14, 15, 17, 11, 8, 42, 20, 32, 18, 7, 31, 33, 56, 19, 37, 46, 23, 21, 25, 52, 13, 62, 40, 36, 16, 27, 10, 92, 51, 131, 39, 12, 44, 34, 97, 82, 43, 98, 152, 170, 24, 29, 45, 84, 70, 58, 144, 94, 78, 26, 38, 76, 166, 111, 145, 206, 194

Offset: 1

N. J. A. Sloane, Oct 19 2010

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..1702

Cf. A181391.

A171915 Van Eck sequence (cf. A181391) starting with a(1) = 5.

Original entry on oeis.org

5, 0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 12, 0, 3, 0, 2, 9, 0, 3, 5, 9, 4, 0, 5, 4, 3, 7, 0, 5, 5, 1, 23, 0, 5, 4, 10, 0, 4, 3, 13, 0, 4, 4, 1, 13, 5, 12, 35, 0, 8, 0, 2, 36, 0, 3, 16, 0, 3, 3, 1, 16, 5, 16, 2, 12, 18, 0, 10, 32, 0, 3, 12, 7, 46, 0, 5, 14, 0, 3, 8, 30, 0, 4, 40, 0, 3, 7

Offset: 1

N. J. A. Sloane, Oct 22 2010

nonn

A van Eck sequence is defined recursively by a(n+1) = min { k > 0 | a(n-k) = a(n) } or 0 if this set is empty, i.e., a(n) does not appear earlier in the sequence. - M. F. Hasler, Jun 15 2019

Chai Wah Wu, Table of n, a(n) for n = 1..10001

Cf. A181391, A171911, ..., A171918 (same but starting with 0, 1, ..., 8).

A171915_vec(N, a=5, i=Map())={vector(N, n, a=if(n>1, iferr(n-mapget(i, a), E, 0)+mapput(i, a, n), a))} \\ M. F. Hasler, Jun 15 2019

A171915_list, l = [5, 0], 0
for n in range(1, 10**4):
    for m in range(n-1, -1, -1):
        if A171915_list[m] == l:
            l = n-m
            break
    else:
        l = 0
    A171915_list.append(l) # Chai Wah Wu, Jan 02 2015

Name and other sections edited by M. F. Hasler, Jun 15 2019

A171917 Van Eck sequence (cf. A181391) starting with a(1) = 7.

Original entry on oeis.org

7, 0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4, 0, 5, 3, 0, 3, 2, 9, 0, 4, 9, 3, 6, 14, 0, 6, 3, 5, 15, 0, 5, 3, 5, 2, 17, 0, 6, 11, 0, 3, 8, 0, 3, 3, 1, 42, 0, 5, 15, 20, 0, 4, 32, 0, 3, 11, 18, 0, 4, 7, 66, 0, 4, 4, 1, 20, 16, 0, 6, 32, 17, 36, 0, 5, 26, 0, 3, 22, 0, 3, 3, 1, 17

Offset: 1

N. J. A. Sloane, Oct 22 2010

nonn

A van Eck sequence is defined recursively by a(n+1) = min { k > 0 | a(n-k) = a(n) } or 0 if this set is empty. - M. F. Hasler, Jun 15 2019

Cf. A181391, A171911, ..., A171918 (same but starting with 0, 1, ..., 8).

A171917_vec(N, a=7, i=Map())={vector(N, n, a=if(n>1, iferr(n-mapget(i, a), E, 0)+mapput(i, a, n), a))} \\ M. F. Hasler, Jun 15 2019

from itertools import count, islice
def A171917gen(): # generator of terms
    b, bdict = 7, {7:(1,)}
    for n in count(2):
        yield b
        if len(l := bdict[b]) > 1:
            b = n-1-l[-2]
        else:
            b = 0
        if b in bdict:
            bdict[b] = (bdict[b][-1],n)
        else:
            bdict[b] = (n,)
A171917_list = list(islice(A171917gen(),20)) # Chai Wah Wu, Dec 21 2021

a(n+1) = A181391(n) until the first occurrence of a(1) = 7 in A181391. - M. F. Hasler, Jun 15 2019

Name edited and cross-references added by M. F. Hasler, Jun 15 2019

A171957 Partial sums of A181391.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 5, 7, 8, 14, 14, 19, 19, 21, 27, 32, 36, 36, 41, 44, 44, 47, 49, 58, 58, 62, 71, 74, 80, 94, 94, 100, 103, 108, 123, 123, 128, 131, 136, 138, 155, 155, 161, 172, 172, 175, 183, 183, 186, 189, 190, 232, 232, 237, 252, 272, 272, 276, 308, 308, 311, 322, 340, 340

Offset: 1

N. J. A. Sloane, Nov 02 2010

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..100000

A171958 First differences of A181391.

Original entry on oeis.org

0, 1, -1, 2, -2, 2, 0, -1, 5, -6, 5, -5, 2, 4, -1, -1, -4, 5, -2, -3, 3, -1, 7, -9, 4, 5, -6, 3, 8, -14, 6, -3, 2, 10, -15, 5, -2, 2, -3, 15, -17, 6, 5, -11, 3, 5, -8, 3, 0, -2, 41, -42, 5, 10, 5, -20, 4, 28, -32, 3, 8, 7, -18, 4, 3, -7, 3, 4, -4, -1, 29, -31, 6, 25, -28, 3, -3, -1, 6, 25, -33, 9, 47, -56, 3, 5, -1, 12, -19, 5, 32, -37, 3

Offset: 1

N. J. A. Sloane, Nov 02 2010

sign

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C++ program for A171958

Cf. A181391.

A175041 Length of longest A181391-suffix other than 11...1 with entries (0 <= a(n) <= d).

Original entry on oeis.org

2, 4, 9, 12, 15, 19, 24, 28, 33, 41, 45, 49, 54, 61

Offset: 0

N. J. A. Sloane, Dec 02 2010

A sequence is an "A181391-suffix" if it satisfies the following definition, which is less stringent than that of A181391. For n>=1, if there exists an m < n such that a(m) = a(n), take the largest such m and set a(n+1) = n-m; otherwise set a(n+1) either to 0 or to a number >= n.
The motivation for calling this an "A181391-suffix" is that we treat n <= 0 as a kind of unknown prefix - each entry has to be consistent with some prefix, but we don't require the same prefix for all values.
This sequence arises when searching for possible cycles in sequences generated by the rule in A181391.
For example, 1 2 2 1 3 5 is an A181391-suffix, since the sample prefixes below justify the *'d entries:
....0.0.|.1*.2.2.1.3.5
....1.x.|.1.2*.2.1.3.5
......2.|.1.2.2*.1.3.5
......3.|.1.2.2.1.3.5*
Clearly, any continuation, including any cycle, from any starting point, is an A181391-suffix.

			d  length  lex-min seq
0   2      0 0
1   4      0 0 1 0
2   9      0 0 1 0 2 0 2 2 1
3  12      1 0 0 1 3 0 3 2 0 3 3 1
4  15      0 2 3 0 3 2 4 0 4 2 4 2 2 1 0
5  19      0 1 3 5 4 0 5 3 5 2 0 5 3 5 2 5 2 2 1
6  24      2 1 0 3 0 2 5 0 3 5 3 2 6 0 6 2 4 0 4 2 4 2 2 1
7  28      0 0 1 0 2 7 0 3 0 2 5 0 3 5 3 2 6 0 6 2 4 0 4 2 4 2 2 1
8  33      3 7 2 5 6 7 4 7 2 6 5 7 4 6 4 2 7 5 7 2 4 6 8 0 0 1 0 2 8 6 8 2 4
9  41      2 0 2 2 1 5 0 5 2 5 2 2 1 8 0 8 2 5 8 3 0 6 0 2 7 0 3 7 3 2 6 9 0 7 6 4 0 4 2 9 8
10 45      9 5 0 7 6 6 1 0 5 7 6 5 3 0 6 4 0 3 5 7 10 0 5 4 8 0 4 3 10 8 5 8 2 0 8 3 8 2 5 8 3 5 3 2 6
11 49      7 4 6 7 3 5 0 7 4 7 2 11 0 6 11 3 11 2 7 9 0 8 0 2 6 11 9 7 9 2 6 6 1 0 11 9 7 9 2 9 2 2 1 10 0 11 11 1 5
12 54      7 4 6 7 3 12 0 7 4 7 2 11 0 6 11 3 11 2 7 9 0 8 0 2 6 11 9 7 9 2 6 6 1 0 11 9 7 9 2 9 2 2 1 10 0 11 11 1 5 0 5 2 10 9
13 61      4 5 2 0 12 5 4 6 10 12 5 5 1 0 10 6 8 0 4 12 10 6 6 1 11 0 8 10 7 0 4 12 12 1 10 7 7 1 4 8 13 0 12 10 9 0 4 8 8 1 12 8 3 0 8 3 3 1 8 4 13

Email from David Applegate, Oct 19 2010

Cf. A181391, A175100.

A200780 a(n) = number of i in the range 1 <= i <= n such that b(i)=b(n), where b is the sequence A181391 taken with offset 1.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 2, 3, 2, 1, 5, 1, 6, 4, 2, 2, 1, 7, 3, 1, 8, 2, 5, 1, 9, 2, 2, 3, 3, 1, 10, 4, 4, 4, 1, 11, 5, 5, 6, 6, 1, 12, 5, 1, 13, 6, 1, 14, 7, 8, 3, 1, 15, 7, 2, 1, 16, 3, 1, 17, 9, 2, 1, 18, 4, 1, 19, 10, 2, 11, 7, 1, 20, 6, 2, 12, 7, 13, 8, 2, 1, 21, 3, 1, 22, 14, 3, 3, 1, 23, 8, 1, 24, 15, 4, 5, 4

Offset: 1

N. J. A. Sloane, Nov 22 2011

nonn

Equivalently, this is the ORDINAL transform of Van Eck's sequence A181391. - N. J. A. Sloane, Apr 09 2020

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C++ program for A200780

Cf. A181391.

# The ORDINAL transform of a sequence a[0], a[1], a[2], ... is the sequence b[0], b[1], b[2], ... where b[n] is the number of times a[n] has occurred in [a[0], ..., a[n]].
ORDINAL:=proc(a)  local b,t1,tlist,clist,n,t,nt;
if whattype(a) <> list then RETURN([]); fi:
t1:=nops(a);
tlist:=[];
clist:=Array(1..t1,0);
b:=[]; nt:=0;
for n from 1 to t1 do t:=a[n];
if member(t,tlist,'p') then clist[p] := clist[p]+1; b:=[op(b),clist[p]];
else nt:=nt+1; tlist:=[op(tlist),t]; clist[nt]:=1; b:=[op(b),1]; fi;
od: b; end: # N. J. A. Sloane, Apr 09 2020
See also A200779.

A308721 Locations of the first occurrence of pair (0,n) in the van Eck sequence (A181391).

Original entry on oeis.org

0, 1, 3, 20, 24, 10, 30, 276, 388, 81, 225, 726, 2935, 1408, 7718, 5624, 5680, 85998, 26706, 546290, 1112929, 702575, 3425417, 10537360, 21301906, 217230900, 108698091, 32381774, 846522986, 851764846, 11692311325, 46163898987

Offset: 0

Tomas Rokicki, Jun 19 2019

a(33) = 118456929919 and a(34) = 250327022558 but we do not yet know a(32).

			For a(5)=10, the pair (0,5) first occurs in A181391 at element 10.

Cf. A181391.

With[{s = Nest[# /. {{Longest[p___], a_, q___, a_} :> {p, a, q, a, Length[{a, q}]}, {a___} :> {a, 0}} &, {}, 10^3]}, TakeWhile[#, # > -1 &] &@ Array[If[Length@ # == 0, -1, #[[1, 1]] - 1 ] &@ SequencePosition[s, {0, #}] &, Max@ s, 0]] (* Michael De Vlieger, Jul 08 2019, after JungHwan Min at A181391 *)

A309363 Van Eck's sequence (cf. A181391), but outputting 2 for a new number, not 0.

Original entry on oeis.org

0, 2, 2, 1, 2, 2, 1, 3, 2, 3, 2, 2, 1, 6, 2, 3, 6, 3, 2, 4, 2, 2, 1, 10, 2, 3, 8, 2, 3, 3, 1, 8, 5, 2, 6, 18, 2, 3, 8, 7, 2, 4, 22, 2, 3, 7, 6, 12, 2, 5, 17, 2, 3, 8, 15, 2, 4, 15, 3, 6, 13, 2, 6, 3, 5, 15, 8, 13, 7, 23, 2, 9, 2, 2, 1, 44

Offset: 1

Nicholas FitzGerald, Jul 25 2019

nonn

After the initial value, the sequence is extended by a(n+1) = min { k > 0: a(n-k) = a(n) } or 2 if no such k exists, i.e., if a(n) did not appear earlier.

Although the sequence has properties that are superficially similar to the original A181391, there is an important difference. Using a positive number m instead of 0 to mark a new value means there is no 1-to-1 correspondence between the occurrence of a new value and the occurrence of m. - Jan Ritsema van Eck, Aug 14 2019

Rémy Sigrist, Table of n, a(n) for n = 1..25000

Cf. A181391, A171911, ..., A171918 (same but starting with 0, 1, 2, ..., 8 and returning 0 when a new term appears).

from itertools import count, islice
def A309363gen(): # generator of terms
    b, bdict = 0, {0:(1,)}
    for n in count(2):
        yield b
        if len(l := bdict[b]) > 1:
            b = n-1-l[-2]
        else:
            b = 2
        if b in bdict:
            bdict[b] = (bdict[b][-1],n)
        else:
            bdict[b] = (n,)
A309363_list = list(islice(A309363gen(),20)) # Chai Wah Wu, Dec 21 2021

A333211 Variation of Van Eck's sequence A181391: a(n+1) = the minimum positive offset m from a(n) such that a(n-m-1)a(n-m) = a(n-1)a(n); a(n+1)=0 if no such m exists. Start with a(1) = a(2) = 0.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 2, 1, 0, 2, 1, 3, 0, 3, 1, 3, 1, 1, 13, 0, 6, 1, 0, 2, 1, 14, 0, 3, 1, 12, 0, 3, 1, 4, 0, 3, 1, 4, 4, 0, 4, 1, 4, 1, 1, 27, 0, 6, 1, 27, 4, 0, 4, 1, 10, 0, 3, 1, 21, 0, 3, 1, 4, 9, 0, 4, 1, 4, 1, 1, 25, 0, 6, 1, 25, 4, 0, 4, 1, 10, 25

Offset: 1

Scott R. Shannon, Mar 11 2020

nonn

After 100 million terms the smallest number not appearing is 179549, while the smallest product of adjacent terms not appearing is 2969.

			a(3) = 0 as a(1)*a(2) = 0*0 = 0, which has not previously appeared as the product of two adjacent terms.
a(4) = 1 as a(2)*a(3) = 0*0 = 0, which equals the product a(1)*a(2), one term back from a(3).
a(5) = 1 as a(3)*a(4) = 0*1 = 0, which equals the product a(2)*a(3), one term back from a(3).
a(6) = 0 as a(4)*a(5) = 1*1 = 1, which has not previously appeared as the product of two adjacent terms.
a(19) = 13 as a(17)*a(18) = 1*1 = 1, which equals the product a(4)*a(5), thirteen terms back from a(18).

Scott R. Shannon, Table of n, a(n) for n = 1..10000.
Brady Haran and N. J. A. Sloane, Don't Know (the Van Eck Sequence), Numberphile video (2019).

Cf. A181391, A171898, A308721, A333210.

cp's OEIS Frontend

A171897 Numbers in the van Eck sequence A181391 in order of first appearance.

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A171915 Van Eck sequence (cf. A181391) starting with a(1) = 5.

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A171917 Van Eck sequence (cf. A181391) starting with a(1) = 7.

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A171957 Partial sums of A181391.

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A171958 First differences of A181391.

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A175041 Length of longest A181391-suffix other than 11...1 with entries (0 <= a(n) <= d).

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A200780 a(n) = number of i in the range 1 <= i <= n such that b(i)=b(n), where b is the sequence A181391 taken with offset 1.

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Maple

A308721 Locations of the first occurrence of pair (0,n) in the van Eck sequence (A181391).

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Mathematica

A309363 Van Eck's sequence (cf. A181391), but outputting 2 for a new number, not 0.

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Python

A333211 Variation of Van Eck's sequence A181391: a(n+1) = the minimum positive offset m from a(n) such that a(n-m-1)*a(n-m) = a(n-1)*a(n); a(n+1)=0 if no such m exists. Start with a(1) = a(2) = 0.

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A333211 Variation of Van Eck's sequence A181391: a(n+1) = the minimum positive offset m from a(n) such that a(n-m-1)a(n-m) = a(n-1)a(n); a(n+1)=0 if no such m exists. Start with a(1) = a(2) = 0.