Alexandre Wajnberg - cp's OEIS frontend

A306807 An irregular fractal sequence: underline a(n) iff the absolute difference |a(n-1) - a(n)| is prime; all underlined terms rebuild the starting sequence.

1, 2, 3, 1, 5, 2, 6, 3, 1, 7, 5, 2, 8, 6, 3, 1, 9, 7, 5, 2, 10, 8, 6, 3, 1, 11, 9, 7, 5, 2, 12, 10, 8, 6, 3, 1, 13, 11, 9, 7, 5, 2, 14, 12, 10, 8, 6, 3, 1, 15, 13, 11, 9, 7, 5, 2, 16, 14, 12, 10, 8, 6, 3, 1, 17, 15, 13, 11, 9, 7, 5, 2, 18, 16, 14, 12, 10, 8, 6, 3, 1, 19, 17, 15, 13, 11, 9, 7, 5, 2, 20, 18, 16, 14, 12, 10, 8, 6, 3, 1

Offset: 1

Alexandre Wajnberg and Eric Angelini, Mar 11 2019

The sequence S starts with a(1) = 1 and a(2) = 2. S is extended by duplicating the first term A among the not yet duplicated terms, under the condition that the absolute difference |a(n-1) - a(n)| is prime. If this is not the case, we then extend S with the smallest integer X not yet present in S such that the absolute difference |a(n-1) - a(n)| is not prime. S is the lexicographically earliest sequence with this property.

			S starts with a(1) = 1 and a(2) = 2
Can we duplicate a(1) to form a(3)? No, as |a(2) - a(3)| would be 1 and 1 is not prime. We thus extend S with the smallest integer X not yet in S such that |a(2) - X| is not prime. We get a(3) = 3.
Can we duplicate a(1) to form a(4)? Yes, as |a(3) - a(4)| = 2, which is prime. We get a(4) = 1.
Can we duplicate a(2) to form a(5)? No, as |a(4) - a(5)| would be 1 and 1 is not prime. We thus extend S with the smallest integer X not yet in S such that |a(4) - X| is not prime; we get a(5) = 5.
Can we duplicate a(2) to form a(6)? Yes, as |a(6) - a(5)| = 3, which is prime; we get a(6) = 2.
Etc.

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

Cf. A306803 (obtained by replacing the absolute difference by the sum in the definition).

A306803 An irregular fractal sequence: underline a(n) iff [a(n-1) + a(n)] is prime; all underlined terms rebuild the starting sequence.

Original entry on oeis.org

0, 1, 3, 0, 4, 1, 5, 7, 2, 3, 0, 6, 8, 10, 11, 9, 4, 1, 13, 12, 5, 15, 17, 16, 7, 14, 18, 20, 19, 21, 2, 3, 0, 22, 23, 6, 24, 25, 26, 28, 27, 29, 8, 30, 32, 31, 10, 34, 35, 33, 36, 11, 37, 38, 9, 4, 1, 39, 41, 40, 13, 42, 43, 44, 46, 45, 47, 12, 5, 49, 50, 48, 51, 53, 52, 15, 54, 17, 55, 16, 7, 56, 58, 57, 14, 60, 59, 61, 18, 62

Offset: 1

Alexandre Wajnberg and Eric Angelini, Mar 11 2019

The sequence S starts with a(1) = 0 and a(2) = 1. S is extended by duplicating the first term A among the not yet duplicated terms, under the condition that [A + the last term Z of the sequence] is prime. If this is not the case, we then extend the S with the smallest integer X not yet present in S such that [X + the last term Z of the sequence] is not a prime. This is the lexicographically first sequence with this property.

			S starts with a(1) = 0 and a(2) = 1
Can we duplicate a(1) to form a(3)? No, as a(2) + a(3) would be 1 and 1 is not prime; we thus extend S with the smallest integer X not yet in S such that [X + a(2)]  is not prime. We get X = 3 and thus a(3) = 3.
Can we duplicate a(1) to form a(4)? Yes, as now [a(1) + a(3)] is prime; we get thus a(4) = 0.
Can we duplicate a(2) to form a(5)? No, as a(4) + a(2) would be 1 and 1 is not prime; we thus extend S with the smallest integer X not yet in S such that [X + a(4)]  is not prime. We get X = 4 and thus a(5) = 4.
Can we duplicate a(2) to form a(6)? Yes, as now [a(2) + a(5)] is prime; we get thus a(6) = 1
Can we duplicate a(3) to form a(7)? No, as a(6) + a(3) would be 4 and 4 is not prime; we thus extend S with the smallest integer X not yet in S such that [X + a(6)]  is not prime. We get X = 5 and thus a(7) = 5.
Can we duplicate a(3) to form a(8)? No, as a(7) + a(3) would be 8 and 8 is not prime; we thus extend S with the smallest integer X not yet in S such that [X + a(7)]  is not prime. We get X = 7 and thus a(8) = 7.
Can we duplicate a(3) to form a(9)? No, as a(8) + a(3) would be 10 and 10 is not prime; we thus extend S with the smallest integer X not yet in S such that [X + a(8)]  is not prime. We get X = 2 and thus a(9) = 2.
Can we duplicate a(3) to form a(10)? Yes, as now [a(3) + a(9)] is prime; we get thus a(10) = 3.
Can we duplicate a(4) to form a(11)? Yes, as [a(4) + a(10)] is prime; we get thus a(11) = 0.
Etc.

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

Cf. A306808 (which is obtained by replacing prime by palindrome in the definition).

A306801 An irregular fractal sequence: underline a(n) iff [a(n-1) + a(n)] is divisible by 3; all underlined terms rebuild the starting sequence.

Original entry on oeis.org

1, 3, 2, 1, 4, 6, 3, 5, 8, 9, 7, 2, 1, 10, 12, 11, 4, 13, 15, 6, 3, 14, 17, 18, 16, 5, 20, 21, 19, 8, 23, 24, 9, 22, 25, 27, 26, 7, 2, 1, 28, 30, 29, 10, 31, 33, 12, 32, 35, 36, 34, 11, 4, 37, 39, 38, 13, 40, 42, 15, 6, 3, 41, 44, 45, 43, 14, 47, 48, 46, 17, 50, 51, 18, 49, 52, 54, 53, 16, 5, 56, 57, 55, 20, 59, 60, 21, 58, 61, 63

Offset: 1

Alexandre Wajnberg and Eric Angelini, Mar 11 2019

The sequence S starts with a(1) = 1 and a(2) = 3. S is extended by duplicating the first term A among the not yet duplicated terms, under the condition that [A + the last term Z of the sequence] is divisible by 3. If this is not the case, we then extend S with the smallest integer X not yet present in S such that [X + the last term Z of the sequence] is not divisible by 3. This is the lexicographically first sequence with this property.

			S starts with a(1) = 1 and a(2) = 3.
Can we duplicate a(1) to form a(3)? No, as a(2) + a(3) would be 4 and 4 is not divisible by 3; we thus extend S with the smallest integer X not yet in S such that [X + a(2)]  is not divisible by 3. We get X = 2 and thus a(3) = 2.
Can we duplicate a(1) to form a(4)? Yes, as now [a(1) + a(3)] is divisible by 3; we get thus a(4) = 1.
Can we duplicate a(2) to form a(5)? No, as a(4) + a(2) would be 4 and 4 is not divisible by 3; we thus extend S with the smallest integer X not yet in S such that [X + a(4)]  is not divisible by 3. We get X = 4 and thus a(5) = 4.
Can we duplicate a(2) to form a(6)? No, as a(5) + a(2) would be 7 and 7 is not divisible by 3; we thus extend S with the smallest integer X not yet in S such that [X + a(5)]  is not divisible by 3. We get X = 6 and thus a(6) = 6.
Can we duplicate a(2) to form a(7)? Yes, as now [a(2) + a(6)] is divisible by 3; we get thus a(7) = 3.
Can we duplicate a(3) to form a(8)? No, as a(7) + a(3) would be 5 and 5 is not divisible by 3; we thus extend S with the smallest integer X not yet in S such that [X + a(6)]  is not divisible by 3. We get X = 6 and thus a(8) = 5.
Etc.

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

Cf. A122196 (which is obtained by replacing 3 by 2 in the definition of this sequence).

A113969 The first illegal prime number (the 1401 digits of its decimal expansion).

Original entry on oeis.org

4, 8, 5, 6, 5, 0, 7, 8, 9, 6, 5, 7, 3, 9, 7, 8, 2, 9, 3, 0, 9, 8, 4, 1, 8, 9, 4, 6, 9, 4, 2, 8, 6, 1, 3, 7, 7, 0, 7, 4, 4, 2, 0, 8, 7, 3, 5, 1, 3, 5, 7, 9, 2, 4, 0, 1, 9, 6, 5, 2, 0, 7, 3, 6, 6, 8, 6, 9, 8, 5, 1, 3, 4, 0, 1, 0, 4, 7, 2, 3, 7, 4, 4, 6, 9, 6, 8, 7, 9, 7, 4, 3, 9, 9, 2, 6, 1, 1, 7, 5, 1, 0, 9, 7, 3

Offset: 1

Alexandre Wajnberg, Jan 31 2006

The first illegal prime number was generated on March 2001 by Phil Carmody. Its binary representation corresponds to a compressed version of the C source code of a computer program implementing the DeCSS decryption scheme, making any DVD copy readable with any DVD player. Interpreted in this particular way, this number describes a computer program which bypasses copyright protection schemes on some DVDs. Such programs are illegal to possess or distribute under the Digital Millennium Copyright Act. Of course, any prime number is not illegal, although such an interpretation of it could be. It's fully displayed in the Wiki link below. Phil Carmody generated also other illegal primes; one of them (1811 digits) represents a non-compressed executable that performs the same task as this compressed program (cf. A113970).

David Wells, Prime numbers, John Wiley and Sons, Inc. (2005), p. 127.

Nathaniel Johnston, Table of n, a(n) for n = 1..1401 (full sequence)
Wikipedia, DeCSS.
Wikipedia, Illegal prime.

Cf. A113970.

A113970 The first illegal executable prime number (the 1811 digits of its decimal expansion).

Original entry on oeis.org

4, 9, 3, 1, 0, 8, 3, 5, 9, 7, 0, 2, 8, 5, 0, 1, 9, 0, 0, 2, 7, 5, 7, 7, 7, 6, 7, 2, 3, 9, 0, 7, 6, 4, 9, 5, 7, 2, 8, 4, 9, 0, 7, 7, 7, 2, 1, 5, 0, 2, 0, 8, 6, 3, 2, 0, 8, 0, 7, 5, 0, 1, 8, 4, 0, 9, 7, 9, 2, 6, 2, 7, 8, 8, 5, 0, 9, 7, 6, 5, 8, 8, 6, 4, 5, 5, 7, 8, 0, 2, 0, 1, 3, 6, 6, 0, 0, 7, 3, 2, 8, 6, 7, 9, 5

Offset: 1

Alexandre Wajnberg, Jan 31 2006

The first illegal prime number (1401 digits, cf. A113969) was generated on March 2001 by Phil Carmody. Its binary representation corresponds to the compressed version of the C source code of a computer program implementing the DeCSS decryption scheme, making any DVD copy readable with any DVD player. This prime number here (1811 digits) represents a *non-compressed executable* that performs the same task as the compressed program. Interpreted in this particular way, this number describes a computer program which bypasses copyright protection schemes on some DVDs. Such programs are illegal to possess or distribute under the Digital Millennium Copyright Act. Of course, any prime number is not illegal, although such an interpretation of it could be. It's fully displayed in the Wiki link below.

David Wells, Prime numbers, John Wiley and Sons, Inc. (2005), p. 127.

Nathaniel Johnston, Table of n, a(n) for n = 1..1811 (full sequence)
Wikipedia, DeCSS.
Wikipedia, Illegal prime.

Cf. A113969.

A114925 "Walking base" sequence: the number becomes the least base in which it could be read, once; written in base 10.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 4, 6, 8, 10, 3, 5, 7, 9, 11, 2, 5, 8, 11, 3, 6, 9, 12, 3, 7, 10, 4, 7, 11, 4, 8, 12, 4, 9, 13, 4, 10, 5, 9, 14, 5, 10, 6, 10, 7, 12, 5, 11, 5, 12, 6, 11, 6, 12, 7, 13, 5, 13, 6, 13, 7, 14, 6, 14, 7, 15, 6, 15, 7, 16, 7, 17, 8, 13, 8, 14, 8, 15, 8, 16, 8, 17, 9, 15

Offset: 0

Alexandre Wajnberg, Feb 20 2006

A "base" sequence visiting all the bases but nevertheless written here in base 10.
The number a(n) do not become its value in the new base, but becomes the base itself. So each term has a double status according to its preceding or following neighbor: regarding a(n-1), a(n) is a *base* (the least one not used so far) in which it is possible to read a(n-1); and regarding a(n+1), a(n) is a *number* to be read in the base expressed by a(n+1).
The first break, specific of this sequence written in base 10, occurs after a(9)=10. If, following the same principle, one build another sequence written, say in base 8, the beginning would be: 0,2,3,4,5,6,7,10,2,4... the first break occurring after a(7) instead of a(9). The inclusion of the unary base would lead to a different sequence since after the first occurrence of 11 would come 1 and not 2.
The word "walking base" refers to the "walking bass", a certain style of accompaniment in baroque music or jazz bass playing, in which the player, using a bass line composed of nonsyncopated notes of equal value, moves in stepwise motion to successive chord roots or notes, sometimes using passing notes.

			Examples: The beginning is 0,2,3 but could also be 1,2,3.
a(0)=0. Now the least base in which 0 has a meaning is the binary base, so next term, a(1)=2.
The least base in which 2 makes sense is 3, so next term, a(2)=3.
The least base in which "10" makes sense is not base 11 but base 2, so next term, a(10)=2 (although 2 was used to read 0, it has not yet been used to read "10").
The least base in which this second 2 makes sense now is not 3 (because 3 has already been used to read a(1)=2), but 4, so next term a(11)=4.
a(101)=10: the least base not used so far to read "10" is base 10, so a(102)=10; then a(103)=11 (and although the value a(102)="10" in base 11 should be written "A", which is impossible in the OEIS, this does not affect the next term a(103); anyway, this walking base is written all along in base 10, so a(102)=10).

The number base calculator,

A114148 Self-describing sequence: 2 odd integers between two even, then 3 odd integers between two even, then 5 odd integers, then 6, then 7... The length of each run of odd integers is given by the sequence itself (the smallest even available integer is used when needed).

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 19, 21, 22, 23, 25, 27, 29, 31, 33, 34, 35, 37, 39, 41, 43, 45, 47, 48, 49, 51, 53, 55, 57, 59, 61, 63, 65, 66, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 88, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 112, 113, 115, 117, 119

Offset: 2

Eric Angelini and Alexandre Wajnberg, Feb 03 2006

			Runs of odd integers are between brackets:
2,(3,5),6,(7,9,11),12,(13,15,17,19,21),22,(23,25,27,29,31,33),34...
2 odd 3 odd 5 odd 6 odd ... etc. = the sequence itself

A114147 Self-describing sequence : 1 prime between two nonprimes, then 2 primes between two nonprimes, then 4 primes, then 5, then 7, etc. The quantity of primes in each run is given by the sequence itself. (Sequence is strictly increasing and the smallest next available nonprime is used when needed).

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 11, 13, 17, 19, 20, 23, 29, 31, 37, 41, 42, 43, 47, 53, 59, 61, 67, 71, 72, 73, 79, 83, 89, 97, 101, 103, 107, 108, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 168, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 240, 241

Offset: 1

Eric Angelini and Alexandre Wajnberg

			Runs of primes are between brackets:
1,(2),4,(5,7),8,(11,13,17,19),20,(23,29,31,37,41),42,(43,47,53,59,61,67,71)
.^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
1 prime 2 pr. 4 primes 5 primes 7 primes (etc.) = the sequence itself

{m=10;k=1;v=[k];for(j=1,m,for(count=1,v[j],k=nextprime(k+1);v=concat(v,k)); while(isprime(k),k++);v=concat(v,k));for(n=1,#v,print1(v[n],","))} \\ Klaus Brockhaus

A114149 Self-describing sequence: 1 even integer between two odd, then 2 even integers between two odd, then 3 even integers, then 4, then 6... The length of each run of even integers is given by the sequence itself (the smallest odd available integer is used when needed).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 12, 13, 14, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 33, 34, 36, 38, 40, 42, 44, 46, 47, 48, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 83, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 107, 108, 110, 112, 114, 116, 118

Offset: 1

Eric Angelini and Alexandre Wajnberg, Feb 03 2006

			Runs of even integers are between brackets:
1,(2),3,(4,6),7,(8,10,12),13,(14,16,18,20),21,(22,24,26,28,30,32),33...
1 even 2 even 3 even 4 even 6 even... etc. = the sequence itself

A112433 Next term is the sum of the last 10 digits in the sequence, beginning with a(10) = 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 4, 8, 16, 23, 28, 38, 45, 42, 41, 41, 36, 34, 32, 31, 30, 28, 29, 33, 34, 37, 44, 42, 37, 41, 39, 41, 38, 43, 40, 39, 39, 46, 45, 47, 54, 51, 45, 44, 43, 39, 42, 42, 39, 43, 43, 38, 43, 44, 40, 37, 40, 33, 32, 29, 36, 35, 39, 45, 49, 51, 48, 52, 47

Offset: 1

Alexandre Wajnberg, Dec 11 2005

Variation on Angelini's A112395. The sequence cycles at a(17)=38 and the loop has 312 terms. It is exactly the same loop as in A112435 "Next term is the sum of the last 10 digits in the sequence, beginning with a(10) = 4". Computed by Gilles Sadowski.

			a(17)=38 because 2+2+4+8+1+6+2+3+2+8=38

Cf. A112395.

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User: Alexandre Wajnberg

A306807 An irregular fractal sequence: underline a(n) iff the absolute difference |a(n-1) - a(n)| is prime; all underlined terms rebuild the starting sequence.

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A306803 An irregular fractal sequence: underline a(n) iff [a(n-1) + a(n)] is prime; all underlined terms rebuild the starting sequence.

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A306801 An irregular fractal sequence: underline a(n) iff [a(n-1) + a(n)] is divisible by 3; all underlined terms rebuild the starting sequence.

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A113969 The first illegal prime number (the 1401 digits of its decimal expansion).

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A113970 The first illegal executable prime number (the 1811 digits of its decimal expansion).

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A114925 "Walking base" sequence: the number becomes the least base in which it could be read, once; written in base 10.

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A114148 Self-describing sequence: 2 odd integers between two even, then 3 odd integers between two even, then 5 odd integers, then 6, then 7... The length of each run of odd integers is given by the sequence itself (the smallest even available integer is used when needed).

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A114149 Self-describing sequence: 1 even integer between two odd, then 2 even integers between two odd, then 3 even integers, then 4, then 6... The length of each run of even integers is given by the sequence itself (the smallest odd available integer is used when needed).

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A112433 Next term is the sum of the last 10 digits in the sequence, beginning with a(10) = 2.

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