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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.

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A299971 Lexicographic first sequence of positive integers such that a(n) + a(n+1) has a digit 0, and no term occurs twice.

Original entry on oeis.org

1, 9, 11, 19, 21, 29, 31, 39, 41, 49, 51, 50, 10, 20, 30, 40, 60, 42, 8, 2, 18, 12, 28, 22, 38, 32, 48, 52, 53, 7, 3, 17, 13, 27, 23, 37, 33, 47, 43, 57, 44, 6, 4, 16, 14, 26, 24, 36, 34, 46, 54, 55, 5, 15, 25, 35, 45, 56, 64, 66, 74, 76, 84, 86, 94, 96, 104, 97
Offset: 1

Views

Author

M. F. Hasler and Eric Angelini, Feb 22 2018

Keywords

Comments

It happens that from a(18) = 42 on, the sequence coincides with the "nonnegative variant" A299970. Indeed, n = 18 is the first index for which the same value occurs, and {a(n), 1 <= n < 18} U {0} = {A299970(n), 0 <= n < 18}. - M. F. Hasler, Feb 28 2018

Links

Crossrefs

Cf. A299970 (analog with nonnegative terms), A299957 (analog with digit 1), A299972 .. A299979 (digit 2..9).
Cf. A299980, A299981, A299402, A299403, A298974, A298975, A299996, A299997, A298978, A298979 for the analog using multiplication: a(n)*a(n+1) has a digit 0, resp. 1, ..., resp. 9.

Programs

  • Mathematica
    Nest[Append[#, Block[{k = 1}, While[Nand[FreeQ[#, k], DigitCount[k + #[[-1]], 10, 0] > 0], k++]; k]] &, {1}, 67] (* Michael De Vlieger, Feb 22 2018 *)
  • PARI
    a(n,f=1,d=0,a=1,u=[a])={for(n=2,n,f&&if(f==1,print1(a","),write(f,n-1," "a));for(k=u[1]+1,oo,setsearch(u,k)&&next;setsearch(Set(digits(a+k)),d)&&(a=k)&&break);u=setunion(u,[a]);u[2]==u[1]+1&&u=u[^1]);a}

A299996 Lexicographic first sequence of positive integers such that a(n)*a(n+1) has a digit 6, and no term occurs twice.

Original entry on oeis.org

1, 6, 10, 16, 4, 9, 7, 8, 2, 3, 12, 5, 13, 20, 18, 17, 28, 22, 21, 26, 11, 15, 24, 14, 19, 32, 23, 27, 25, 64, 29, 34, 39, 35, 36, 31, 44, 37, 38, 42, 30, 52, 33, 49, 40, 41, 43, 48, 45, 57, 46, 47, 56, 51, 60, 61, 65, 71, 53, 50, 72, 55, 63, 58, 62, 59, 54, 66, 70, 67, 68, 69, 74, 76, 79, 73, 77, 78, 80, 75, 81, 82, 83, 84, 89, 85, 90, 94, 92
Offset: 1

Views

Author

M. F. Hasler, Feb 22 2018

Keywords

Comments

A permutation of the positive integers.

Examples

			a(1) = 1 is the least positive integer, and a(1) has no other constraint to satisfy.
a(2) = 6 is the least positive integer > a(1) = 1 such that a(2)*a(1) = 6 has a digit 6.
a(3) = 10 is the least positive integer not in {1, 6} such that a(3)*a(2) (= 60) has a digit 6: All smaller choices (2, 3, 4 or 5) do not satisfy this.
a(4) = 16 is the least positive integer not in {1, 6, 10} such that a(4)*a(3) (= 160) has a digit 6: All smaller choices 2,...,15 do not satisfy this.

Crossrefs

Cf. A299402, A299403, A298974, ..., A298979, A299997: analog with digit 2, 3, ..., 9, 7.
Cf. A299957, A299969, ..., A299988 (analog with addition instead of multiplication, and different digits).

Programs

  • PARI
    A299996(n,f=1,d=6,a=1,u=[a])={for(n=2,n,f&&if(f==1,print1(a","),write(f,n-1," "a)); for(k=u[1]+1,oo, setsearch(u,k)&&next;setsearch(Set(digits(a*k)),d)&&(a=k)&&break);u=setunion(u,[a]);while(#u>1&&u[2]==u[1]+1,u=u[^1]));a}

A299997 Lexicographic first sequence of positive integers such that a(n)*a(n+1) has a digit 7, and no term occurs twice.

Original entry on oeis.org

1, 7, 10, 17, 11, 16, 36, 2, 35, 5, 14, 27, 21, 13, 6, 12, 23, 9, 3, 19, 4, 18, 15, 25, 28, 24, 30, 26, 22, 8, 34, 50, 54, 31, 37, 20, 38, 44, 29, 33, 39, 43, 32, 46, 45, 55, 65, 42, 41, 47, 51, 53, 49, 56, 62, 48, 57, 61, 52, 63, 59, 64, 58, 72, 66, 83, 69, 40, 68, 70, 71, 67, 81, 75, 73, 79, 60, 95, 74, 78, 86, 82, 85, 84, 80, 88, 77, 91, 87
Offset: 1

Views

Author

M. F. Hasler, Feb 22 2018

Keywords

Comments

A permutation of the positive integers.

Examples

			a(1) = 1 is the least positive integer, and a(1) has no other constraint to satisfy.
a(2) = 7 is the least positive integer > a(1) = 1 such that a(2)*a(1) = 7 has a digit 7.
a(3) = 10 is the least positive integer not in {1, 7} such that a(3)*a(2) (= 70) has a digit 7: All smaller choices (2, ..., 6) do not satisfy this.
a(4) = 17 is the least positive integer not in {1, 7, 10} such that a(4)*a(3) (= 170) has a digit 7: All smaller choices 2,...,16 do not satisfy this.

Crossrefs

Cf. A299402, A299403, A298974, ..., A298979, A299996: analog with digit 2, 3, 4, ..., 9, 6.
Cf. A299957, A299969, ..., A299988 (analog with addition instead of multiplication, and different digits).

Programs

  • PARI
    A299997(n,f=1,d=7,a=1,u=[a])={for(n=2,n,f&&if(f==1,print1(a","),write(f,n-1," "a)); for(k=u[1]+1,oo, setsearch(u,k)&&next;setsearch(Set(digits(a*k)),d)&&(a=k)&&break);u=setunion(u,[a]);while(#u>1&&u[2]==u[1]+1,u=u[^1]));a}

A298978 Lexicographic first sequence of positive integers such that a(n)*a(n+1) has a digit 8, and no term occurs twice.

Original entry on oeis.org

1, 8, 6, 3, 16, 5, 17, 4, 2, 9, 12, 7, 14, 13, 22, 19, 15, 32, 24, 20, 29, 27, 18, 10, 28, 11, 26, 23, 21, 38, 31, 35, 25, 33, 36, 30, 46, 40, 37, 34, 42, 39, 47, 44, 41, 45, 53, 54, 52, 49, 58, 48, 51, 55, 56, 43, 60, 63, 61, 62, 59, 65, 69, 70, 64, 57, 50, 76, 68, 66, 71, 73, 67, 72, 74, 77, 79, 82, 83, 86, 78, 75, 91, 80, 81, 84, 87, 90, 89
Offset: 1

Views

Author

M. F. Hasler, Feb 22 2018

Keywords

Comments

A permutation of the positive integers.

Examples

			a(1) = 1 is the least positive integer, and a(1) has no other constraint to satisfy.
a(2) = 8 is the least positive integer > a(1) = 1 such that a(2)*a(1) = 8 has a digit 3.
a(3) = 6 is the least positive integer not in {1, 8} such that a(3)*a(2) (= 48) has a digit 8: All smaller choices 2, 4, ..., 7 do not satisfy this.
a(4) = 3 is the least positive integer not in {1, 6, 8} such that a(4)*a(3) (= 18) has a digit 8: The only smaller choice 2 does not satisfy this.

Crossrefs

Cf. A299402, A298974, ..., A298979: analog with digit 2, 4, ..., 9.
Cf. A299957, A299969, ..., A299988 (analog with addition instead of multiplication, and different digits).

Programs

  • PARI
    A298978(n,f=1,d=8,a=1,u=[a])={for(n=2,n,f&&if(f==1,print1(a","),write(f,n-1," "a)); for(k=u[1]+1,oo, setsearch(u,k)&&next;setsearch(Set(digits(a*k)),d)&&(a=k)&&break);u=setunion(u,[a]);while(#u>1&&u[2]==u[1]+1,u=u[^1]));a}

A299982 Lexicographic first sequence of nonnegative integers such that a(n) + a(n+1) has a digit 2, and no term occurs twice.

Original entry on oeis.org

0, 2, 10, 11, 1, 19, 3, 9, 12, 8, 4, 16, 5, 7, 13, 14, 6, 15, 17, 25, 27, 35, 37, 45, 47, 55, 57, 63, 29, 23, 39, 33, 49, 43, 59, 53, 67, 54, 18, 24, 28, 34, 38, 44, 48, 64, 56, 26, 36, 46, 66, 58, 62, 20, 22, 30, 32, 40, 42, 50, 52, 60, 61, 21, 31, 41, 51, 69, 73, 79, 83, 89, 93, 99, 101, 71, 81, 91, 109, 92, 70
Offset: 0

Views

Author

M. F. Hasler and Eric Angelini, Feb 22 2018

Keywords

Comments

A permutation of the nonnegative integers.

Crossrefs

Cf. A299972 (analog with positive terms), A299957 (analog with digit 1), A299970, A299983, ..., A299988, A299969 (digit 0, 3, ..., 9).

Programs

  • PARI
    a(n,f=1,d=2,a=0,u=[a])={for(n=1,n,f&&if(f==1,print1(a","),write(f,n-1," "a));for(k=u[1]+1,oo,setsearch(u,k)&&next;setsearch(Set(digits(a+k)),d)&&(a=k)&&break);u=setunion(u,[a]);u[2]==u[1]+1&&u=u[^1]);a}

A299980 Lexicographic first sequence of positive integers such that a(n)*a(n+1) has a digit 0, and no term occurs twice.

Original entry on oeis.org

1, 10, 2, 5, 4, 15, 6, 17, 12, 9, 20, 3, 30, 7, 29, 14, 22, 23, 35, 8, 13, 16, 19, 11, 28, 18, 25, 24, 21, 40, 26, 27, 38, 37, 46, 44, 32, 33, 31, 34, 45, 36, 39, 50, 41, 49, 42, 43, 47, 60, 48, 55, 51, 53, 57, 54, 52, 58, 65, 56, 59, 68, 70, 61, 64, 63, 62, 66, 75, 67, 76, 79, 71, 80, 69, 73, 74, 82, 72, 84, 81, 87, 90, 77, 78, 85, 83, 88, 91
Offset: 1

Views

Author

M. F. Hasler, Feb 22 2018

Keywords

Comments

A permutation of the positive integers.

Examples

			a(1) = 1 is the least positive integer, and a(1) has no other constraint to satisfy.
a(2) = 10 is the least positive integer > a(1) = 1 such that a(2)*a(1) = 10 has a digit 0.
a(3) = 2 is the smallest available positive integer, and such that a(3)*a(2) (= 20) has a digit 0.
a(4) = 5 is the least positive integer not in {1, 2, 10} such that a(4)*a(3) (= 10) has a digit 0: The smaller choices 2, 3 and 4 do not satisfy this.

Crossrefs

Cf. A299981, A299402, A299403, A298974, A298975, A299996, A299997, A298978, A298979: analog with digit 1, ..., 9.
Cf. A299957, A299969, ..., A299988: analog with addition instead of multiplication, and different digits.

Programs

  • PARI
    A299980(n,f=1,d=0,a=1,u=[a])={for(n=2,n,f&&if(f==1,print1(a","),write(f,n-1," "a)); for(k=u[1]+1,oo, setsearch(u,k)&&next;setsearch(Set(digits(a*k)),d)&&(a=k)&&break);u=setunion(u,[a]);while(#u>1&&u[2]==u[1]+1,u=u[^1]));a}

A299981 Lexicographic first sequence of positive integers such that a(n)*a(n+1) has a digit 1, and no term occurs twice.

Original entry on oeis.org

1, 10, 11, 12, 9, 2, 5, 3, 4, 25, 6, 17, 7, 13, 8, 14, 15, 21, 31, 23, 18, 34, 24, 38, 27, 19, 22, 28, 29, 35, 26, 16, 32, 33, 36, 30, 37, 39, 40, 41, 42, 43, 44, 45, 47, 46, 59, 20, 50, 62, 51, 61, 52, 56, 57, 53, 55, 58, 54, 65, 48, 66, 63, 67, 73, 70, 74, 69, 49, 64, 80, 77, 82, 75, 68, 76, 81, 71, 72, 78, 79, 85, 60, 86, 83, 87, 91, 90, 89
Offset: 1

Views

Author

M. F. Hasler, Feb 22 2018

Keywords

Comments

A permutation of the positive integers.

Examples

			a(1) = 1 is the least positive integer, and a(1) has no other constraint to satisfy.
a(2) = 10 is the least positive integer > a(1) = 1 such that a(2)*a(1) = 10 has a digit 1. (For all small choices 2, ..., 9 this is not the case.)
a(3) = 11 is the smallest positive integer not in {1, 10} such that a(3)*a(2) (= 110) has a digit 1.
a(4) = 12 is the least positive integer not in {1, 10, 11} such that a(4)*a(3) (= 132) has a digit 1: All smaller choices 2, 3, ..., 9 do not satisfy this.

Crossrefs

Cf. A299980, A299402, A299403, A298974, A298975, A299996, A299997, A298978, A298979: analog with digit 0, 2, ..., 9.
Cf. A299957, A299969, ..., A299988: analog with addition instead of multiplication, and different digits.

Programs

  • PARI
    A299981(n,f=1,d=1,a=1,u=[a])={for(n=2,n,f&&if(f==1,print1(a","),write(f,n-1," "a)); for(k=u[1]+1,oo, setsearch(u,k)&&next;setsearch(Set(digits(a*k)),d)&&(a=k)&&break);u=setunion(u,[a]);while(#u>1&&u[2]==u[1]+1,u=u[^1]));a}

A299952 The sum a(n) + a(n+1) is a substring of the concatenation of all terms up to a(n+1). Lexicographic first sequence of positive integers without duplicate terms having this property.

Original entry on oeis.org

1, 10, 99, 11, 80, 19, 61, 30, 31, 49, 12, 2, 4, 5, 3, 6, 7, 15, 9, 13, 17, 14, 8, 16, 20, 25, 23, 22, 26, 27, 18, 34, 28, 21, 24, 29, 32, 35, 36, 44, 37, 43, 38, 33, 41, 39, 42, 40, 51, 45, 46, 47, 52, 57, 53, 56, 54, 55, 63, 59, 50, 60, 58, 64, 66, 65, 74, 48, 62, 68, 71, 77, 72, 67, 78, 70
Offset: 1

Views

Author

Eric Angelini and Lars Blomberg, Feb 22 2018

Keywords

Comments

The sequence starts with a(1) = 1 and is always extended with the smallest integer not yet present that does not lead to a contradiction.
This is probably a permutation of the natural numbers (after 10000 terms, the smallest integer not yet present is 9990).

Examples

			a(1) + a(2) = 1 + 10 = 11 and “11” is visible in [1,10]
a(2) + a(3) = 10 + 99 = 109 and “109” is visible in [10,99]
a(3) + a(4) = 99 + 11 = 110 and “110” is visible in [1,10]
a(4) + a(5) = 11 + 80 = 91 and “91” is visible in [99,11]
a(5) + a(6) = 80 + 19 = 99 and “99” is visible in [99]
a(6) + a(7) = 19 + 61 = 80 and “80” is visible in [80]
...

Links

Crossrefs

Cf. A300000.
For a different constraint on a(n)+a(n+1) (must have a digit '1'), see A299957 and A299970, A299982, ..., A299988, A299969 (nonnegative analog with digit 0, 2, ..., 9), A299971, A299972, ..., A299979 (positive analog with digit 0, 2, ..., 9).
Cf. A299980, A299981, A299402, A299403, A298974, A298975, A299996, A299997, A298978, A298979 for the analog using multiplication: a(n)*a(n+1) has a digit 0, resp. 1, ..., resp. 9.

Programs

  • Mathematica
    Nest[Function[a, Append[a, Block[{k = 1, d}, While[Nand[FreeQ[a, k], SequenceCount[Flatten@ IntegerDigits[Append[a, k]], IntegerDigits[a[[-1]] + k]] > 0], k++]; k]]], {1}, 75] (* Michael De Vlieger, Feb 22 2018 *)
  • PARI
    a(n,show=1,a=1,s=a,u=[a],t,m)={for(n=2,n, show&&print1(a","); for(k=u[1]+1,oo, setsearch(u,k)&&next;m=Mod(a+k,10^#Str(a+k));t=s*10^#Str(k)+k; until(k>=t\=10,t==m&&(a=k)&&break(2)));s=s*10^#Str(a)+a;u=setunion(u,[a]); u[2]==u[1]+1&&u=u[^1]);a} \\ M. F. Hasler, Feb 22 2018

A299973 Lexicographic first sequence of positive integers such that a(n) + a(n+1) has a digit 3, and no term occurs twice.

Original entry on oeis.org

1, 2, 11, 12, 18, 5, 8, 15, 16, 7, 6, 17, 13, 10, 3, 20, 14, 9, 4, 19, 24, 29, 34, 39, 44, 49, 54, 59, 64, 66, 27, 26, 37, 36, 47, 46, 57, 56, 67, 63, 30, 23, 40, 33, 50, 43, 60, 53, 70, 61, 22
Offset: 1

Views

Author

M. F. Hasler and Eric Angelini, Feb 22 2018

Keywords

Comments

A permutation of the positive integers.

Crossrefs

Cf. A299983 (analog with nonnegative terms), A299957 (analog with digit 1), A299970..A299979 (digit 0..9).

Programs

  • PARI
    a(n,f=1,d=3,a=1,u=[a])={for(n=2,n,f&&if(f==1,print1(a","),write(f,n-1," "a));for(k=u[1]+1,oo,setsearch(u,k)&&next;setsearch(Set(digits(a+k)),d)&&(a=k)&&break);u=setunion(u,[a]);u[2]==u[1]+1&&u=u[^1]);a}

A299978 Lexicographic first sequence of positive integers such that a(n) + a(n+1) has a digit 8, and no term occurs twice.

Original entry on oeis.org

1, 7, 11, 17, 21, 27, 31, 37, 41, 39, 9, 19, 29, 49, 32, 6, 2, 16, 12, 26, 22, 36, 42, 38, 10, 8, 20, 18, 30, 28, 40, 43, 5, 3, 15, 13, 25, 23, 35, 33, 45, 44, 4, 14, 24, 34, 46, 52, 56, 62, 66, 72, 76, 82, 86, 92, 88, 50, 48, 60, 58, 70, 68, 80, 78, 90, 91, 47, 51, 57, 61, 67, 71, 77, 81, 87, 93, 55, 53, 65, 63, 75, 73, 85, 83, 95, 89, 59, 69, 79, 99
Offset: 1

Views

Author

M. F. Hasler and Eric Angelini, Feb 22 2018

Keywords

Comments

A permutation of the positive integers.

Crossrefs

Cf. A299988 (analog with nonnegative terms), A299957 (analog with digit 1), A299971, A299972, ..., A299979 (digit 0, 2, ..., 9).

Programs

  • PARI
    a(n,f=1,d=8,a=1,u=[a])={for(n=2,n,f&&if(f==1,print1(a","),write(f,n-1," "a));for(k=u[1]+1,oo,setsearch(u,k)&&next;setsearch(Set(digits(a+k)),d)&&(a=k)&&break);u=setunion(u,[a]);u[2]==u[1]+1&&u=u[^1]);a}
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