A299403 -id:A299403 - cp's OEIS frontend

A299957 The sum a(n) + a(n+1) always has at least one digit "1". Lexicographically first such sequence of nonnegative integers without duplicate term.

0, 1, 9, 2, 8, 3, 7, 4, 6, 5, 10, 11, 20, 21, 30, 31, 40, 41, 50, 51, 49, 12, 19, 22, 29, 32, 39, 42, 58, 13, 18, 23, 28, 33, 38, 43, 48, 52, 53, 47, 14, 17, 24, 27, 34, 37, 44, 56, 15, 16, 25, 26, 35, 36, 45, 46, 54, 55, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 0

Eric Angelini, Feb 22 2018

The sequence starts with a(0) = 0 and is always extended with the smallest integer not yet present that does not lead to a contradiction. The sequence is a permutation of the natural numbers.

Originally the sequence was defined starting with a(1) = 1 and using only positive integers. This leads to the same sequence restricted to positive indices, which yields a permutation of the positive integers. - M. F. Hasler, Feb 28 2018

			1 + 9 = 10; 9 + 2 = 11; 2 + 8 = 10; 8 + 3 = 11; 3 + 7 = 10; 7 + 4 = 11; 4 + 6 = 10; 6 + 5 = 11; etc.

M. F. Hasler, Table of n, a(n) for n = 0..10000

Cf. A299952 (different constraint: a(n) + a(n+1) must be substring of concatenation of a(1..n+1)).
Cf. 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.

Nest[Append[#, Block[{k = 1}, While[Nand[FreeQ[#, k], DigitCount[k + #[[-1]], 10, 1] > 0], k++]; k]] &, {1}, 98] (* Michael De Vlieger, Feb 22 2018 *)

a(n, f=1, a=0, u=[a])={for(n=a+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)),1)&&(a=k)&&break); u=setunion(u, [a]); u[2]==u[1]+1&&u=u[^1]); a} \\ M. F. Hasler, Feb 22 2018

Extended to a(0) = 0 by M. F. Hasler, Feb 28 2018

A299969 Lexicographic first sequence of nonnegative integers such that a(n) + a(n+1) has a digit 9, and no term occurs twice.

Original entry on oeis.org

0, 9, 10, 19, 20, 29, 30, 39, 40, 49, 41, 8, 1, 18, 11, 28, 21, 38, 31, 48, 42, 7, 2, 17, 12, 27, 22, 37, 32, 47, 43, 6, 3, 16, 13, 26, 23, 36, 33, 46, 44, 5, 4, 15, 14, 25, 24, 35, 34, 45, 50, 59, 60, 69, 70, 79, 80, 89, 90, 99, 91, 58, 51, 68, 61, 78, 71, 88, 81, 98, 92, 57, 52, 67, 62, 77, 72, 87, 82, 97, 93, 56, 53, 66, 63, 76, 73, 86, 83, 96, 94

Offset: 0

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

A permutation of the nonnegative integers.

It happens that from a(50) = 50 on, this sequence coincides with the variant A299979 (starting at 1 and having only positive terms). Indeed the two sequences have the property that the terms a(0..49) resp. A299979(1..49) exactly contain all numbers from 0 to 49, respectively 1 to 49. - M. F. Hasler, Feb 28 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Cf. A299979 (analog with positive terms), A299957 (analog with digit 1), A299970, A299982, ..., A299988 (digit 0, 2, ..., 8).

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.

Nest[Append[#, Block[{k = 1}, While[Nand[FreeQ[#, k], DigitCount[#[[-1]] + k, 10, 9] > 0], k++]; k]] &, {0}, 90] (* Michael De Vlieger, Mar 01 2018 *)

a(n,f=1,d=9,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}

A298974 Lexicographic first sequence of positive integers such that a(n)*a(n+1) has a digit 4, and no term occurs twice.

Original entry on oeis.org

1, 4, 6, 7, 2, 12, 17, 20, 21, 14, 3, 8, 5, 9, 16, 15, 23, 18, 13, 11, 22, 19, 24, 10, 34, 26, 29, 36, 39, 32, 27, 35, 40, 31, 37, 38, 28, 30, 47, 42, 44, 33, 43, 48, 50, 49, 46, 51, 54, 41, 45, 52, 57, 25, 56, 58, 53, 65, 62, 55, 59, 60, 64, 61, 63, 66, 67, 68, 69, 70, 71, 74, 73, 75, 72, 76, 79, 82, 77, 84, 81, 80, 78, 83, 88, 85, 97, 86, 87

Offset: 1

M. F. Hasler, Feb 22 2018

A permutation of the positive integers.

			a(1) = 1 is the least positive integer, and a(1) has no other constraint to satisfy.
a(2) = 4 is the least positive integer > a(1) = 1 such that a(2)*a(1) = 4 has a digit 4.
a(3) = 6 is the least positive integer not in {1, 4} such that a(3)*a(2) (= 24) has a digit 4: The smaller choices 2, 3 and 5 do not satisfy this.
a(4) = 7 is the least positive integer not in {1, 4, 6} such that a(4)*a(3) (= 42) has a digit 4: All available smaller choices do not satisfy this.

Cf. A299402, A299403, A298975, ..., A298979: analog with digit 2, 3; ..., 9.

Cf. A299957, A299969, ..., A299988 (analog with addition instead of multiplication, and different digits).

A298974(n,f=1,d=4,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}

A298979 Lexicographic first sequence of positive integers such that a(n)*a(n+1) has a digit 9, and no term occurs twice.

Original entry on oeis.org

1, 9, 10, 19, 5, 18, 11, 27, 7, 13, 3, 23, 4, 24, 8, 12, 16, 6, 15, 26, 35, 14, 21, 29, 17, 37, 25, 36, 22, 41, 34, 28, 32, 30, 31, 39, 46, 2, 45, 20, 47, 42, 38, 50, 58, 33, 43, 44, 59, 49, 40, 48, 52, 56, 53, 55, 54, 61, 64, 62, 63, 57, 51, 69, 68, 72, 82, 60, 65, 66, 75, 79, 67, 70, 71, 76, 78, 73, 81, 74, 77, 83, 84, 88, 90, 91, 87, 80, 99

Offset: 1

M. F. Hasler, Feb 22 2018

A permutation of the positive integers.

			a(1) = 1 is the least positive integer, and a(1) has no other constraint to satisfy.
a(2) = 9 is the least positive integer > a(1) = 1 such that a(2)*a(1) = 9 has a digit 9.
a(3) = 10 is the least positive integer not in {1, 9} such that a(3)*a(2) (= 90) has a digit 9: The smaller choices 2, ..., 8 does not satisfy this.
a(4) = 19 is the least positive integer not in {1, 9, 10} such that a(4)*a(3) (= 190) has a digit 5: All available smaller choices do not satisfy this.

Cf. A299980, A299981, A299402, A299403, A298974, A298975, A299996, A299997, A298978 : analog with digit 0, 1,..., 8.

Cf. A299957, A299969, ..., A299988 (analog with addition instead of multiplication, and different digits).

A298979(n,f=1,d=9,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}

A299402 Lexicographic first sequence of positive integers such that a(n)*a(n+1) has a digit 2, and no term occurs twice.

Original entry on oeis.org

1, 2, 6, 4, 3, 7, 16, 8, 9, 14, 13, 17, 12, 10, 20, 11, 19, 15, 18, 24, 5, 25, 21, 22, 26, 27, 23, 34, 28, 29, 32, 31, 33, 37, 35, 36, 42, 30, 40, 38, 39, 48, 43, 44, 46, 45, 47, 41, 49, 50, 51, 52, 53, 54, 55, 59, 58, 56, 57, 60, 62, 61, 66, 64, 63, 67, 69, 68, 65, 80, 74, 71, 72, 73, 77, 76, 70, 75, 79, 78, 84, 83, 81, 82, 86, 87, 95, 96, 92

Offset: 1

M. F. Hasler, Feb 22 2018

A permutation of the positive integers.

			a(1) = 1 is the least positive integer, it has no other requirement to satisfy.
a(2) = 2 is the least positive integer > a(1) = 1, and a(2)*a(1) = 2 has a digit 2.
a(3) = 6 is the least positive integer > a(2) = 2 such that a(3)*a(2) (= 12) has a digit 2: The smaller choices 3, 4 or 5 do not satisfy this.
a(4) = 4 is the least positive integer > a(2) = 2 such that a(4)*a(3) (= 24) has a digit 2: The smaller choice 3 yields 3*6 = 18 and does not satisfy this.
Now, the least available positive integer a(5) = 3 is such that 3*4 = 12, which has again a digit 2. And so on.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A299403, A298974, ..., A298979 (analog with digit 3, ..., 9).

Cf. A299957, A299969, ..., A299988 (analog with addition instead of multiplication, and different digits).

N:= 100: # to get a(1)..a(n) where a(n+1) > N
S:= [$2..N]: nS:= N-1:
R:= 1: x:= 1; found:= true;
while found do
  found:= false;
  for i from 1 to nS do
    if member(2, convert(S[i]*x,base,10)) then
       found:= true;
       x:= S[i];
       R:= R,x;
       S:= subsop(i=NULL,S);
       nS:= nS-1;
       break
    fi
  od
od:
R; # Robert Israel, Feb 12 2023

a(n,f=1,d=2,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}

A299970 Lexicographic first sequence of nonnegative integers such that a(n) + a(n+1) has a digit 0, and no term occurs twice.

Original entry on oeis.org

0, 10, 20, 30, 40, 50, 51, 9, 1, 19, 11, 29, 21, 39, 31, 49, 41, 59, 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

Offset: 0

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

It happens that from a(18) = 42 on, the sequence coincides with the "strictly positive variant" A299971. Indeed, n = 18 is the first index for which the same value occurs, and {a(n), 0 <= n < 18} = {0} U {A299971(n), 1 <= n < 18}. - M. F. Hasler, Feb 28 2018

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

Cf. A299980, A299981, A299402, A299403, A298974, A298975, A299996, A299997, A298978, A298979 for an analog using multiplication: a(n)*a(n+1) has a digit 0, resp. 1, ..., resp. 9.

Nest[Append[#, Block[{k = 1}, While[Nand[FreeQ[#, k], DigitCount[#[[-1]] + k, 10, 0] > 0], k++]; k]] &, {0}, 67] (* Michael De Vlieger, Mar 01 2018 *)

a(n,f=1,d=0,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}

A298975 Lexicographic first sequence of positive integers such that a(n)*a(n+1) has a digit 5, and no term occurs twice.

Original entry on oeis.org

1, 5, 3, 15, 7, 8, 19, 24, 21, 12, 13, 4, 14, 11, 23, 22, 16, 32, 17, 9, 6, 25, 2, 26, 20, 27, 28, 18, 29, 33, 35, 10, 45, 30, 50, 31, 34, 37, 41, 38, 40, 39, 55, 43, 36, 42, 49, 44, 57, 62, 46, 56, 51, 52, 53, 48, 47, 54, 64, 68, 67, 59, 60, 75, 58, 61, 65, 63, 66, 69, 73, 70, 72, 71, 74, 76, 77, 85, 79, 83, 91, 94, 80, 82, 86, 88, 87, 95, 81

Offset: 1

M. F. Hasler, Feb 22 2018

A permutation of the positive integers.

			a(1) = 1 is the least positive integer, and a(1) has no other constraint to satisfy.
a(2) = 5 is the least positive integer > a(1) = 1 such that a(2)*a(1) = 5 has a digit 5.
a(3) = 3 is the least positive integer not in {1, 5} such that a(3)*a(2) (= 15) has a digit 5: The smaller choice 2 does not satisfy this.
a(4) = 15 is the least positive integer not in {1, 3, 5} such that a(4)*a(3) (= 75) has a digit 5: All available smaller choices do not satisfy this.

Cf. A299402, A299403, A298974, ..., A298979: analog with digit 2, 3; ..., 9.

Cf. A299957, A299969, ..., A299988 (analog with addition instead of multiplication, and different digits).

A298975(n,f=1,d=5,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}

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

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

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

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

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.

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 *)

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

M. F. Hasler, Feb 22 2018

A permutation of the positive integers.

			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.

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).

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

M. F. Hasler, Feb 22 2018

A permutation of the positive integers.

			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.

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).

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}

cp's OEIS Frontend

A299957 The sum a(n) + a(n+1) always has at least one digit "1". Lexicographically first such sequence of nonnegative integers without duplicate term.

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A299969 Lexicographic first sequence of nonnegative integers such that a(n) + a(n+1) has a digit 9, and no term occurs twice.

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A298974 Lexicographic first sequence of positive integers such that a(n)*a(n+1) has a digit 4, and no term occurs twice.

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A298979 Lexicographic first sequence of positive integers such that a(n)*a(n+1) has a digit 9, and no term occurs twice.

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A299402 Lexicographic first sequence of positive integers such that a(n)*a(n+1) has a digit 2, and no term occurs twice.

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A299970 Lexicographic first sequence of nonnegative integers such that a(n) + a(n+1) has a digit 0, and no term occurs twice.

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A298975 Lexicographic first sequence of positive integers such that a(n)*a(n+1) has a digit 5, and no term occurs twice.

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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.

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