A297131 -id:A297131 - cp's OEIS frontend

A297146 Numbers having an up-first zigzag pattern in base 10; see Comments.

12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 56, 57, 58, 59, 67, 68, 69, 78, 79, 89, 120, 121, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 145

Offset: 1

Clark Kimberling, Jan 15 2018

A number n having base-b digits d(m), d(m-1),..., d(0) such that d(i) != d(i+1) for 0 <= i < m shows a zigzag pattern of one or more segments, in the following sense.  Writing U for up and D for down, there are two kinds of patterns:  U, UD, UDU, UDUD, ... and D, DU, DUD, DUDU, ... .  In the former case, we say n has an "up-first zigzag pattern in base b"; in the latter, a "down-first zigzag pattern in base b".  Example:  2,4,5,3,0,1,4,2 has segments 2,4,5; 5,3,0; 0,1,4; and 4,2, so that 24530142, with pattern UDUD, has an up-first zigzag pattern in base 10, whereas 4,2,5,3,0,1,4,2 has a down-first pattern.  The sequences A297146-A297148 partition the natural numbers. In the following guide, column four, "complement" means the sequence of natural numbers not in the corresponding sequences in columns 2 and 3.
***
Base      up-first    down-first  complement
         (none)     A000975     A107907
        A297124     A297125     A297126
        A297128     A297129     A297130
        A297131     A297132     A297133
        A297134     A297135     A297136
        A297137     A297138     A297139
        A297140     A297141     A297142
        A297143     A297144     A297145
       A297146     A297147     A297148

			Base-10 digits of 59898: 5,9,8,9,8, with pattern UDUD, so that 59898 is in the sequence.

Cf. A297147, A297148.

a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
b = 10; t = Table[a[n, b], {n, 1, 10*z}];
u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297146 *)
v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297147 *)
Complement[Range[z], Union[u, v]]  (* A297148 *)

A297132 Numbers having a down-first zigzag pattern in base 5; see Comments.

Original entry on oeis.org

5, 10, 11, 15, 16, 17, 20, 21, 22, 23, 26, 27, 28, 29, 51, 52, 53, 54, 55, 57, 58, 59, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 88, 89, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 113, 114, 115, 116, 117, 119, 130, 132, 133, 134, 135, 136, 138, 139, 140

Offset: 1

Clark Kimberling, Jan 14 2018

A number n having base-b digits d(m), d(m-1),..., d(0) such that d(i) != d(i+1) for 0 <= i < m shows a zigzag pattern of one or more segments, in the following sense. Writing U for up and D for down, there are two kinds of patterns: U, UD, UDU, UDUD, ... and D, DU, DUD, DUDU, ... . In the former case, we say n has an "up-first zigzag pattern in base b"; in the latter, a "down-first zigzag pattern in base b". Example: 2,4,5,3,0,1,4,2 has segments 2,4,5; 5,3,0; 0,1,4; and 4,2, so that 24530142, with pattern UDUD, has an up-first zigzag pattern in base 10, whereas 4,2,5,3,0,1,4,2 has a down-first pattern. The sequences A297131-A297133 partition the natural numbers. See the guide at A297146.

			Base-5 digits of 3723: 1,0,4,3,4,3, with pattern DUDUD, so that 3723 is in the sequence.

Cf. A297131, A297133.

a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
b = 5; t = Table[a[n, b], {n, 1, 10*z}];
u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297131 *)
v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297132 *)
Complement[Range[z], Union[u, v]]  (* A297133 *)

A297133 Numbers whose base-5 digits d(m), d(m-1),..., d(0) have m=0 or else d(i) = d(i+1) for some i in {0,1,...,m-1}.

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 18, 24, 25, 30, 31, 32, 33, 34, 37, 43, 49, 50, 56, 60, 61, 62, 63, 64, 68, 74, 75, 81, 87, 90, 91, 92, 93, 94, 99, 100, 106, 112, 118, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 131, 137, 143, 149, 150, 151, 152, 153, 154, 155, 156

Offset: 1

Clark Kimberling, Jan 14 2018

These numbers comprise the complement of the set of numbers in the union of A297131 and A297132.

			Base-5 digits of 5000: 1,3,0,0,0,0, so that 5000is in the sequence.

Cf. A297131, A297132.

a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
b = 5; t = Table[a[n, b], {n, 1, 10*z}];
u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297131 *)
v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297132 *)
Complement[Range[z], Union[u, v]]  (* A297133 *)

cp's OEIS Frontend

A297146 Numbers having an up-first zigzag pattern in base 10; see Comments.

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A297132 Numbers having a down-first zigzag pattern in base 5; see Comments.

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A297133 Numbers whose base-5 digits d(m), d(m-1),..., d(0) have m=0 or else d(i) = d(i+1) for some i in {0,1,...,m-1}.

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