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

Views

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.

Links

Crossrefs

Cf. A096097, A096098, A241811.

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

A250745 Start with a(1) = 1; then a(n) = smallest number, not already in the sequence, such that a(n) divides concat(a(1), a(2), ..., a(n)).

Original entry on oeis.org

1, 2, 3, 5, 10, 4, 8, 6, 11, 20, 13, 7, 9, 12, 15, 18, 14, 25, 30, 24, 16, 32, 40, 29, 50, 100, 26, 52, 39, 21, 28, 35, 42, 17, 34, 51, 23, 46, 27, 36, 45, 43, 19, 38, 68, 48, 60, 75, 90, 54, 56, 58, 22, 44, 33, 55, 97, 125, 200, 64, 80, 69, 66, 88, 70, 41, 82
Offset: 1

Views

Author

Paolo P. Lava, Nov 27 2014

Keywords

Comments

Like A171785 but without the constraint a(n) > a(n-1).
Among the first 1000 terms, a(n) = n for n = 1, 2, 3, 15, 170, 577, 759, and the numbers not yet found are 149, 298, 347, 401, 447, 454, 457, 467, 487, 509, etc.
Is this sequence a rearrangement of the natural numbers?

Examples

			a(1) = 1;
a(2) = 2 -> 12 /2 = 6;
a(3) = 3 -> 123 / 3 = 41;
Then we cannot use 4 as the next term because 1234 / 4 = 617 / 2.
a(4) = 5 -> 1235 / 5 = 247;
Again, 4, 6, 7, 8 and 9 cannot be used as the next term.
a(5) = 10 -> 123510 / 10 = 12351;
a(6) = 4 -> 1235104 / 4 = 308776;
a(7) = 8 -> 12351048 / 8 = 1543881; etc.

Links

Crossrefs

Cf. A171785, A240588, A241811, A249398, A249399, A250746, A250747.

Programs

  • Maple
    with(numtheory); P:=proc(q) local a,b,k,n; a:=0; b:={};
for k from 1 to q do for n from 1 to q do if nops({n} intersect b)<1
then if type((a*10^(1+ilog10(n))+n)/n,integer)
then a:=a*10^(1+ilog10(n))+n; b:= b union {n}; print(n); break;
fi; fi; od; od; end: P(10^5);

A250746 Start with a(0) = 0; then a(n) = smallest number > a(n-1) such that a(n) divides concat(a(n), a(n-1), ..., a(0)).

Original entry on oeis.org

0, 1, 2, 3, 5, 10, 15, 18, 19, 35, 42, 51, 55, 70, 85, 93, 95, 106, 155, 217, 310, 745, 1210, 1342, 3355, 5185, 6222, 6330, 9495, 10413, 11115, 12070, 13774, 34435, 41322, 61983, 68870, 1601065116264571, 2217993924228622, 2324778503347862, 2325380783693255
Offset: 0

Views

Author

Paolo P. Lava, Nov 27 2014

Keywords

Comments

This sequence is infinite. - Robert G. Wilson v, Dec 09 2014

Examples

			a(0) = 0;
a(1) = 1 -> 10 / 1 = 10;
a(2) = 2 -> 210 / 2 = 105;
a(3) = 3 -> 3210 / 3 = 1070;
Now we cannot use 4 as the next term because 43210 / 4 = 21605 / 2.
a(4) = 5 -> 32105 / 5 = 6421; etc.

Links

Crossrefs

Cf. A171785, A240588, A241811, A249398, A249399, A250745, A250747.

Programs

  • Maple
    with(numtheory); P:=proc(q) local a,k,n; print(0); print(1); a:=10;
for n from 2 to q do if type((n*10^(1+ilog10(a))+a)/n,integer)
then a:=n*10^(1+ilog10(a))+a; print(n);
fi; od; end: P(10^9);
  • Mathematica
    f[lst_List] := Block[{k = lst[[-1]] + 1, id = FromDigits@ Flatten@ IntegerDigits@ Reverse@ lst}, While[ Mod[ id, k] > 0, k++]; Append[lst, k]]; Nest[f, {0}, 36] (* or *)
f[lst_List] := Block[{mn = lst[[-1]], id = FromDigits@ Flatten@ IntegerDigits@ Reverse@ lst}, d = Divisors@ id; Append[lst, Min@ Select[d, # > mn &]]]; Nest[f, {0, 1}, 36] (* Robert G. Wilson v, Dec 08 2014 *)

Extensions

a(37)-a(40) from Robert G. Wilson v, Dec 08 2014

A250747 Start with a(0) = 0; then a(n) = smallest number not already in the sequence such that a(n) divides concat(a(n), a(n-1), ..., a(0)).

Original entry on oeis.org

0, 1, 2, 3, 5, 10, 6, 9, 13, 26, 15, 18, 30, 431, 73, 67, 134, 7, 14, 21, 35, 29, 58, 127, 27, 39, 43, 70, 11, 22, 19, 38, 95, 190, 2748070932534311, 2768821759897, 5537643519794, 787, 191, 382, 955, 17, 31, 45, 54, 90, 101, 202, 303, 57, 114, 47, 55, 33, 66
Offset: 0

Views

Author

Paolo P. Lava, Nov 28 2014

Keywords

Comments

Like A250746, but without the constraint a(n) > a(n-1).

Examples

			a(0) = 0;
a(1) = 1 -> 10 / 1 = 10;
a(2) = 2 -> 210 / 2 = 105;
a(3) = 3 -> 3210 / 3 = 1070;
Now we cannot use 4 as the next term because 43210 / 4 = 21605 / 2.
a(4) = 5 -> 32105 / 5 = 6421;
Again, we cannot use 4, 6, 7, 8 or 9.
a(5) = 10 -> 1053210 / 10 = 105321.
We still cannot use 4, but 6 is ok.
a(6) = 6 -> 61053210 / 6 = 10175535. Etc.

Links

Crossrefs

Cf. A171785, A240588, A241811, A249398, A249399, A250745, A250746.

Programs

  • Maple
    with(numtheory); P:=proc(q) local a,b,k,n; print(0); print(1); a:=10; b:={0,1};
for k from 1 to q do for n from 1 to q do if nops({n} intersect b)<1
then if type((n*10^(1+ilog10(a))+a)/n,integer)
then a:=n*10^(1+ilog10(a))+a; b:= b union {n}; print(n); break; fi; fi;
od; od; end: P(10^5);

Extensions

More terms from Jon E. Schoenfield, Nov 29 2014
Showing 1-4 of 4 results.

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