A197756 -id:A197756 - cp's OEIS frontend

A332072 The Blue Comma sequence: Differences of indices of even terms gives back the sequence; lexicographically earliest such permutation of positive integers.

1, 2, 4, 3, 6, 5, 7, 9, 8, 11, 13, 10, 15, 17, 19, 21, 23, 12, 25, 27, 29, 31, 14, 33, 35, 37, 39, 41, 43, 16, 45, 47, 49, 51, 53, 55, 57, 59, 18, 61, 63, 65, 67, 69, 71, 73, 20, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 22, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 24, 119, 121, 123, 125, 127

Offset: 1

Eric Angelini and M. F. Hasler, May 19 2020

nonn

			The even terms are at indices 2, 3, 5, 9, 12, 18, 23, 30, ...; first differences give 1, 2, 4, 3, 6, 5, 7, ... = the sequence itself.

Eric Angelini, Yellow commas, pink, red, blue..., personal blog "Cinquante signes" on blogspot.com, May 17 2020
Eric Angelini, Yellow commas, pink, red, blue..., personal blog "Cinquante signes" on blogspot.com, May 17 2020 [Cached copy]

Cf. A197756 (the Yellow Comma sequence: count digits between odd terms).

upto(N)={my(a=Vec([1,2],N),i=1,L=2); local(U=[2], nxt(e)=while(#U>1&&U[2]==U[1]+1, U=U[^1]); my(t=U[1]); until( t++%2!=e && !setsearch(U,t),); U=setunion(U,[t]);t ); for(n=3,N,a[n]=nxt(n-L==a[i] && i++ && L=n));a}

A334829 The sum a(n) + a(n+1) is visible around the comma that follows a(n+1). See the Comments and Example sections for details.

Original entry on oeis.org

1, 11, 23, 46, 91, 374, 6506, 8801, 53076, 18777, 18533, 73109, 16428, 95371, 117992, 133632, 516246, 4987805, 50405105, 539291005, 896961101, 4362521065, 2594821666, 9573427311, 21682489773, 12559170843, 42416606165, 49757770089, 21743762547, 15015326363, 67590889108, 26062154719, 36530438276, 25925929956

Offset: 1

Eric Angelini and Jean-Marc Falcoz, May 13 2020

The rule used here is that the rightmost digit of a(n+1) is the first digit of the sum a(n) + a(n+1), the other digits of the said sum being put after the comma in order to start a(n+2).
As no digit 0 (zero) can start a term, one will have to backtrack sometimes in order to extend the sequence - and pick another term for a(n+1), compatible with the above rule. This is always possible.
Note that the sequence is not monotonically increasing as shown by a(10) and a(11) for instance; still, the 1000th term is 406-digit long.
The sequence is always extended with the smallest available integer not yet present that does not lead to a contradiction.

			a(1) + a(2) is 1 + 11 = 12 and 12 can be seen here: 1(1,2)3,
a(2) + a(3) is 11 + 23 = 34 and 34 can be seen here: 2(3,4)6,
a(3) + a(4) is 23 + 46 = 69 and 69 can be seen here: 4(6,9)1,
a(4) + a(5) is 46 + 91 = 137 and 137 can be seen here: 9(1,37)4,
a(5) + a(6) is 91 + 374 = 465 and 465 can be seen here: 37(4,65)06, etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..1002
E. Angelini More and more commas on the SeqFan mailing list, May 12 2020.

Cf. A121805, A230450, A197756.

A332073 The Silver Comma sequence: Differences of indices of prime terms gives back the sequence; lexicographically earliest such permutation of positive integers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 11, 14, 15, 16, 18, 13, 20, 21, 22, 24, 25, 17, 26, 27, 28, 30, 32, 33, 34, 19, 35, 36, 38, 39, 40, 42, 23, 44, 45, 46, 48, 49, 50, 51, 52, 29, 54, 55, 56, 57, 58, 60, 62, 63, 64, 31, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 37, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 41

Offset: 1

Eric Angelini and M. F. Hasler, May 19 2020

nonn

			The prime terms are at indices 2, 3, 5, 8, 12, 17, 23, 31, 38, 47, 57, 69, 80, 94, ...; first differences give 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 11, 14, ... = the sequence itself.

Eric Angelini, Yellow commas, pink, red, blue..., personal blog "Cinquante signes" on blogspot.com, May 17 2020
Eric Angelini, Yellow commas, pink, red, blue..., personal blog "Cinquante signes" on blogspot.com, May 17 2020 [Cached copy]

Cf. A197756 (the Yellow Comma sequence: count digits between odd terms).

upto(N)={local(U=[2]); my(a=Vec([1,2],N), i=1, L=2, nxt(p)=while(#U>1 && U[2]==U[1]+1, U=U[^1]); my(t=U[1]); until(if(p, t=nextprime(t+1), !isprime(t+=1)) && !setsearch(U,t),); U=setunion(U,[t]);t); for(n=3,N,a[n]=nxt(n-L==a[i] && i++ && L=n));a}

cp's OEIS Frontend

A332072 The Blue Comma sequence: Differences of indices of even terms gives back the sequence; lexicographically earliest such permutation of positive integers.

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A334829 The sum a(n) + a(n+1) is visible around the comma that follows a(n+1). See the Comments and Example sections for details.

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Author

Keywords

Comments

Examples

Links

Crossrefs

A332073 The Silver Comma sequence: Differences of indices of prime terms gives back the sequence; lexicographically earliest such permutation of positive integers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI