A316749 -id:A316749 - cp's OEIS frontend

A316750 A self-"read and extend" sequence built following the rules visible in the Comments section.

1, 2, 4, 8, 16, 32, 46, 92, 841, 1268, 2356, 1247, 9442, 14888, 97762, 955421, 1910842, 1234688, 2346679, 3345689, 9876631, 12235679, 12344578, 12456689, 98743321, 124466789, 233457889, 145667789, 123355789, 877654211, 1755308422, 8665443110, 88763322100, 1224456677, 1233444589, 2466889178

Offset: 1

Jean-Marc Falcoz and Eric Angelini, Jul 12 2018

Start with a(1) = 1 and a(2) = 2; read the sequence digit by digit starting from the left:
when the read digit is smaller than the next one, double the last integer of the sequence and extend the sequence with the result rearranged (smallest digits first and leading zeros erased);
when the read digit is bigger than the next one, double the last integer of the sequence and extend the sequence with the result rearranged (biggest digits first, zeros at the end);
when both digits are equal, double the last integer of the sequence and extend the sequence with the simple result (no rearrangement).

			As the only digit of a(1) = 1 is smaller than 2 (the next digit), we extend the sequence with 4 (that is 2 times 2);
as the only digit of a(2) = 2 is smaller than 4 (the next digit), we extend the sequence with 8 (that is 2 times 4);
as the only digit of a(3) = 4 is smaller than 8 (the next digit), we extend the sequence with 16 (that is 2 times 8 -- with 1 coming before 6);
as the only digit of a(4) = 8 is bigger than 1 (the next digit), we extend the sequence with 32 (that is 2 times 16 -- with 3 coming before 2);
as the first digit of a(5) = 1 is smaller than 6 (the next digit), we extend the sequence with 46 (that is 2 times 32 = 64 that is rearranged in 46);
as the last digit of a(5) = 6 is bigger than 3 (the next digit), we extend the sequence with 92 (that is 2 times 46 = 92 rearranged in 29);
. . .
as the last digit of a(9) = 1 is equal to 1 (the next digit), we extend the sequence with 1910842 (this is simply 2 times 955421, with no rearrangement);
etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..10002

Cf. A316749 (for another set of "read and extend" rules).

A316758 A self-"read and extend" sequence built following the rules visible in the Comments section.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 46, 92, 841, 1268, 2356, 1247, 9442, 14888, 97762, 955421, 554219, 113488, 226679, 334558, 966611, 1222339, 2444678, 3456889, 9877631, 12255679, 11234558, 11224669, 22334489, 98876644, 88766449, 988775321, 9776554210, 112345589, 112246789, 122467891, 224678911, 223445789

Offset: 1

Jean-Marc Falcoz and Eric Angelini, Jul 12 2018

Start with a(1) = 1 and a(2) = 2; read the sequence digit by digit starting from the left:
when the read digit is smaller than the next one, double the last integer of the sequence and extend the sequence with the result rearranged (smallest digits first and leading zeros erased);
when the read digit is bigger than the next one, double the last integer of the sequence and extend the sequence with the result rearranged (biggest digits first, zeros at the end);
when both digits are equal, do a circular permutation of the last integer of the sequence and extend the sequence with the result (this will erase a few zeros in some cases, as 100023850 becomes 238501).

			As the only digit of a(1) = 1 is smaller than 2 (the next digit), we extend the sequence with 4 (that is 2 times 2);
as the only digit of a(2) = 2 is smaller than 4 (the next digit), we extend the sequence with 8 (that is 2 times 4);
as the only digit of a(3) = 4 is smaller than 8 (the next digit), we extend the sequence with 16 (that is 2 times 8 -- with 1 coming before 6);
as the only digit of a(4) = 8 is bigger than 1 (the next digit), we extend the sequence with 32 (that is 2 times 16 -- with 3 coming before 2);
as the first digit of a(5) = 1 is smaller than 6 (the next digit), we extend the sequence with 46 (that is 2 times 32 = 64 that is rearranged in 46);
as the last digit of a(5) = 6 is bigger than 3 (the next digit), we extend the sequence with 92 (that is 2 times 46 = 92 rearranged in 29);
. . .
as the last digit of a(9) = 1 is equal to 1 (the next digit), we extend the sequence with 554219 (this is the circular permutation of the previous term, 955421);
etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..3002

Cf. A316749 and A316750 (for other sets of "read and extend" rules).

A316765 A self-"read and extend" sequence built following the three rules given in the Comments section.

Original entry on oeis.org

1, 0, 2, 6, 18, 9, 27, 13, 39, 19, 9, 4, 2, 1, 0, 0, 0, 0, 0, 0, 3, 5, 7, 8, 10, 11, 5, 2, 1, 3, 1, 12, 6, 3, 1, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 15, 16, 17, 20, 21, 22, 23, 24, 12, 36, 18, 9, 4, 12, 6, 3, 9, 25, 75, 37, 111, 333, 999, 499, 1497, 4491, 2245, 6735, 3367, 10101, 5050, 15150, 7575

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Jul 12 2018

Start with a(1) = 1 and read the sequence digit by digit starting from the left:
when the read digit is odd, we divide by 2 the last term of the sequence, then extend the sequence with the entire part of the result;
when the read digit is even (but not 0), we multiply by 3 the last term of the sequence, then extend the sequence with the result;
when the read digit is 0, we extend the sequence with the smallest integer not yet present in the sequence.

			The odd digit 1 divides 1 by two (which is 0,5), and |0,5| is 0;
the digit 0 extends the sequence with the smallest integer not present yet in the sequence, which is 2;
the digit 2 multiplies 2 by three, which is 6;
the digit 6 multiplies 6 by three, which is 18;
the odd digit 1 divides 18 by two, which is 9;
the digit 8 multiplies 9 by three, which is 27; etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..10001

Cf. (for more self-"read and extend" sequences) A316749, A316750, A316758, A316764.

A339853 Read one digit d at the time, starting from the first one; extend S with the smallest multiple of d not yet present in the sequence. Zeros are not read.

Original entry on oeis.org

1, 2, 4, 8, 16, 3, 6, 9, 12, 18, 5, 10, 7, 24, 15, 11, 14, 20, 28, 13, 25, 17, 19, 21, 32, 22, 26, 40, 23, 27, 30, 35, 29, 42, 31, 36, 34, 33, 39, 38, 44, 46, 48, 54, 52, 50, 45, 56, 49, 51, 57, 55, 58, 63, 60, 62, 66, 37, 69, 72, 75, 64, 78, 81, 84, 90, 87, 80, 68, 76, 88, 96, 92, 104, 65, 100, 70, 74, 85, 108

Offset: 1

Eric Angelini and Carole Dubois, Dec 19 2020

This is a permutation of the integers > 0 (as the prime numbers are multiples of 1).

			The first d is 1; as 1 is already in the sequence, we extend it with 2;
the next d is now 2; as 2 is already in the sequence, we extend it with 4 (smallest multiple of 2 not in the sequence);
the next d is 4; as 4 is already in the sequence, we extend it with 8 (smallest multiple);
the next d is 8; as 8 is already in the sequence, we extend it with 16 (smallest multiple);
the next d is 1; we extend the sequence with 3 as 3 is the smallest multiple of 1 not yet present in the sequence;
the next d is 6; as 6 is not yet present, we extend the sequence with 6;
the next d is 3; we extend the sequence with 9 as 9 is the smallest multiple of 3 not yet present;
the next d is 6; we extend the sequence with 12 as 12 is the smallest multiple of 6 not yet present; etc.
As the zero of 10 will not be read, we will extend the sequence at that point with the smallest multiple of 7 not yet present -- which is 14.

Cf. A316749.

def aupto(n):
  alst, astr, used, strind = [1], "1", {1}, 0
  for k in range(1, n):
    while astr[strind] == "0": strind += 1
    ak = digit = int(astr[strind])
    while ak in used: ak += digit
    alst.append(ak); astr += str(ak); used.add(ak); strind += 1
  return alst    # use alst[n-1] for a(n)
print(aupto(71)) # Michael S. Branicky, Dec 19 2020

A316909 A self-"read and extend" sequence built following the three rules visible in the Comments section (a variation of A316765).

Original entry on oeis.org

1, 0, 2, 14, 4, 28, 196, 1372, 9604, 3201, 1067, 7469, 2489, 829, 276, 1932, 644, 4508, 3, 21, 7, 49, 5, 1, 0, 6, 42, 14, 4, 28, 196, 65, 455, 3185, 22295, 7431, 52017, 364119, 121373, 849611, 283203, 1982421, 660807, 220269, 73423, 513961, 3597727, 25184089, 176288623, 1234020361, 411340120, 8, 56

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Jul 16 2018

Start with a(1) = 1 and read the sequence digit by digit starting from the left:
when the read digit is odd, we divide by 3 the last term of the sequence, then extend the sequence with the entire part of the result;
when the read digit is even (but not 0), we multiply by 7 the last term of the sequence, then extend the sequence with the result;
when the read digit is 0, we extend the sequence with the smallest integer not yet present in the sequence.
This is a possible variation among many others of the first 2 rules illustrated by A316765 (where an odd digit divides by 3 and an even digit -except 0— multiplies by 2) that shows the flexibility of the "read-and-extend" idea.

			Reading the sequence one digit after the other, starting from the left:
the odd digit 1 divides 1 by three (which is 0,333...), and |0,333...| is 0;
the digit 0 extends the sequence with the smallest integer not present yet in the sequence, which is 2;
the digit 2 multiplies 2 by seven, which is 14;
the odd digit 1 divides 14 by three, (which is 4,666...) and |4,666...| is 4;
the digit 4 multiplies 4 by seven, which is 28;
the digit 4 multiplies 28 by seven, which is 196;
etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..10001

Cf. (for more self-"read and extend" sequences) A316749, A316750, A316758, A316764 and A316765.

cp's OEIS Frontend

A316750 A self-"read and extend" sequence built following the rules visible in the Comments section.

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A316758 A self-"read and extend" sequence built following the rules visible in the Comments section.

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A316765 A self-"read and extend" sequence built following the three rules given in the Comments section.

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A339853 Read one digit d at the time, starting from the first one; extend S with the smallest multiple of d not yet present in the sequence. Zeros are not read.

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A316909 A self-"read and extend" sequence built following the three rules visible in the Comments section (a variation of A316765).

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Author

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Comments

Examples

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