cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A240588 a(1) = 1, a(2) = 2; for n >= 3, a(n) = least number not included earlier that divides the concatenation of all previous terms.

Original entry on oeis.org

1, 2, 3, 41, 7, 9, 137131, 61, 2023244487101, 13, 19, 11, 143, 142733, 21, 17, 193, 37, 3907, 1290366811360047359, 1805030483980039, 3803623, 123, 369, 27, 23, 58271, 47609, 523, 79, 307, 179, 73, 57, 18032419296851, 29, 31, 3281881401611107, 69, 171, 60244474373, 197, 97
Offset: 1

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Author

Paolo P. Lava, Apr 29 2014

Keywords

Comments

From Scott R. Shannon, Dec 19 2019: (Start)
The next unknown term a(131) requires the factorization of a 517-digit composite number 46297...2963. (End)

Examples

			a(1)=1 and a(2)=2. a(1) U a(2) = 12 and its divisors are 1, 2, 3, 4, 6, 12. Therefore 3 is the least number not yet present in the sequence which divides 12. Again, a(1) U a(2) U a(3) = 123 and its divisors are 1, 3, 41, 123. Therefore a(4)=41. Etc.
		

Crossrefs

Programs

  • Maple
    with(numtheory):
    T:=proc(t) local x, y; x:=t; y:=0; while x>0 do x:=trunc(x/10); y:=y+1; od; end:
    P:=proc(q) local a,b,c,k,n; b:=12; print(1); print(2); c:=[1,2];
    for n from 1 to q do a:=sort([op(divisors(b))]); for k from 2 to nops(a) do
    if not member(a[k],c) then c:=[op(c),a[k]]; b:=a[k]+b*10^T(a[k]); print(a[k]); break;
    fi; od; od; end: P(19);
  • Mathematica
    a = {1, 2}; While[Length[a] < 22,
      n = ToExpression[StringJoin[ToString /@ a]];
      AppendTo[a, SelectFirst[Sort[Divisors[n]], FreeQ[a, #] &]]
    ]; a

Extensions

a(20)-a(40) from Alois P. Heinz, May 08 2014
a(22) corrected by Ryan Hitchman, Sep 14 2017
a(23)-a(25) from Robert Price, May 16 2019
a(23)-a(25) corrected, and a(26)-a(43) added by Scott R. Shannon, Dec 10 2019

A096098 a(1) = 2, a(2) = 1; for n >= 3, a(n) = least number not included earlier that divides the concatenation of all previous terms.

Original entry on oeis.org

2, 1, 3, 71, 7, 21, 599, 173, 11, 23, 161, 49, 13, 9, 131, 19, 33, 17, 1489, 331, 3989, 69, 3097350956401900335673788279883089441874368101, 349387, 5651, 443, 29, 51, 479470832244949, 661, 1129, 1873, 181, 1544577973887516219070997863, 521
Offset: 1

Views

Author

Amarnath Murthy, Jun 24 2004

Keywords

Comments

Conjecture (1) Every concatenation is squarefree.
Conjecture (2) This is a rearrangement of the squarefree numbers not divisible by 5. False! (The a(n) are not always squarefree, since a(12)=49 and a(14)=9.)
Fact: All a(n) for n >= 2 are odd, since a(2) = 1 and odd a(n) => odd concatenation => odd a(n+1). - Wolfdieter Lang, May 08 2014 (editing an earlier statement).
Conjecture (3) the sequence for n>=2 is a permutation of the positive integers not divisible by 2 or 5.
a(29) is probably 479470832244949, in which case the sequence continues 479470832244949, 661, 1129, 1873, 181. - Martin Fuller, Nov 21 2007
Factorization for a(29): 479470832244949*3*17*43217123024009614997922599713504735424547343*P51. - Sean A. Irvine, May 25 2010
Assuming Conjecture (3), the smallest number yet to appear is 89. - Sean A. Irvine, May 11 2014
The factorization given by Sean A. Irvine above is not for the prime a(29) = 479470832244949 but for the concatenation of a(1), a(2), ..., a(29), and P51 means a prime with 51 digits, namely 202232656574589264871780464738430216507933940172343. - Wolfdieter Lang, May 11 2014

Examples

			a(6) = 21 as 213717 = 3*7*10177, and 3 = a(3) and 7 = a(4), hence 3*7 = 21 is the least number dividing 213717 not included earlier in the sequence.
		

Crossrefs

Cf. A096097.

Extensions

More terms from R. J. Mathar, Aug 03 2007
a(23)-a(26) from N. J. A. Sloane, Nov 10 2007
Corrected and extended by Martin Fuller, Nov 21 2007
More terms from Sean A. Irvine, May 25 2010
Example detailed. - Wolfdieter Lang, May 08 2014

A241811 a(1) = 1, a(2) = 0; for n >= 3, a(n) = least number not included earlier that divides the concatenation of all previous terms.

Original entry on oeis.org

1, 0, 2, 3, 11, 71, 29, 9, 683, 67, 7, 743, 739, 1933, 23, 161, 21, 37, 19, 17, 119, 49, 332534262883, 13, 39, 13739483941387, 83, 111, 79853560395691, 5431567, 70610371, 69, 51, 4112497, 28384496881337963, 353, 77, 1531, 42787, 63, 27, 41, 709, 33, 81, 487, 139697
Offset: 1

Views

Author

Paolo P. Lava, Apr 29 2014

Keywords

Examples

			a(1)=1 and a(2)=0. a(1) U a(2) = 10 and its divisors are 1, 2, 5, 10. Therefore 2 is the least number not yet present in the sequence which divides 10. Again, a(1) U a(2) U a(3) = 102 and its divisors are 1, 2, 3, 6, 17, 34, 51, 102. Therefore a(4)=3, etc.
		

Crossrefs

Programs

  • Maple
    with(numtheory):
    T:=proc(t) local x, y; x:=t; y:=0; while x>0 do x:=trunc(x/10); y:=y+1; od; end:
    P:=proc(q) local a,b,c,k,n; b:=10; print(1); print(0); c:=[0,1];
    for n from 1 to q do a:=sort([op(divisors(b))]); for k from 2 to nops(a) do
    if not member(a[k],c) then c:=[op(c),a[k]]; b:=a[k]+b*10^T(a[k]); print(a[k]); break;
    fi; od; od; end: P(30);

Extensions

a(23)-a(28) from Zak Seidov, May 08 2014
a(29)-a(47) from Giovanni Resta, Aug 15 2019

A240229 a(n) is the shortest concatenation of the Fibonacci numbers F(1), F(2), ..., divisible by F(n) = A000045(n), n >= 1. a(n) = 0 if there is no such concatenation.

Original entry on oeis.org

1, 1, 112, 11235, 11235, 112, 1123581321345589144233, 11235, 11235813213455891442333776109871597258, 11235813213455891442333776109871597258441816765, 1123581321345589144233377610987159725844181676, 1123581321345589144233377610987159725844181676510946177112865746368
Offset: 1

Views

Author

Wolfdieter Lang, May 10 2014

Keywords

Comments

The corresponding numbers a(n)/F(n) are 1, 1, 56, 3745, 2247, 14, 86429332411199164941, 535, 330465094513408571833346356172694037, 204287512971925298951523201997665404698942123, 12624509228602125216105366415586064335327884, 7802648064899924612731788965188609207251261642437126229950456572, ...
The author's opinion is that this is an example of a not-so-interesting sequence. I call this a WOTS (waste of time sequence). But because I had to write a program to test similar proposed sequences I thought I would apply it to this prominent example.
The next entry a(13) has 324 digits for the divisibility by F(13) = 233 with a(13)/F(13) a 321 digit composite. The given a(n) are all nonprimes.
Question: is there an n with a(n) = 0?

Examples

			a(3) = 112 because neither 1 nor 11 are divisible by F(3) = 2, but 112, the concatenation of F(1), F(2) and F(3) is.
		

Crossrefs

Formula

See the name.
Showing 1-4 of 4 results.