A373947 -id:A373947 - cp's OEIS frontend

A373952 Number of integer compositions of n whose run-compression sums to 3.

0, 0, 0, 3, 2, 4, 5, 6, 6, 9, 8, 10, 11, 12, 12, 15, 14, 16, 17, 18, 18, 21, 20, 22, 23, 24, 24, 27, 26, 28, 29, 30, 30, 33, 32, 34, 35, 36, 36, 39, 38, 40, 41, 42, 42, 45, 44, 46, 47, 48, 48, 51, 50, 52, 53, 54, 54, 57, 56, 58, 59, 60, 60, 63, 62, 64, 65, 66

Offset: 0

Gus Wiseman, Jun 29 2024

nonn

We define the (run-) compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has compression (1,2,1).

			The a(3) = 3 through a(9) = 9 compositions:
  (3)   (112)  (122)   (33)     (1222)    (11222)    (333)
  (12)  (211)  (221)   (1122)   (2221)    (22211)    (12222)
  (21)         (1112)  (2211)   (11122)   (111122)   (22221)
               (2111)  (11112)  (22111)   (221111)   (111222)
                       (21111)  (111112)  (1111112)  (222111)
                                (211111)  (2111111)  (1111122)
                                                     (2211111)
                                                     (11111112)
                                                     (21111111)

John Tyler Rascoe, Table of n, a(n) for n = 0..10000

For partitions we appear to have A137719.
Column k = 3 of A373949, rows-reversed A373951.
The compression-sum statistic is represented by A373953, difference A373954.
A003242 counts compressed compositions (anti-runs).
A011782 counts compositions.
A114901 counts compositions with no isolated parts.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A240085 counts compositions with no unique parts.
A333755 counts compositions by compressed length.
A373948 represents the run-compression transformation.
Cf. A037201 (halved A373947), A106356, A124762, A238130, A238279, A238343, A285981, A333213, A333489, A373950.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#]]==3&]],{n,0,10}]

A_x(N)={my(x='x+O('x^N)); concat([0, 0, 0], Vec(x^3 *(3-x-x^2-x^3)/((1-x)*(1-x^2)*(1-x^3))))}
A_x(50) \\ John Tyler Rascoe, Jul 01 2024

G.f.: x^3 * (3-x-x^2-x^3)/((1-x)*(1-x^2)*(1-x^3)). - John Tyler Rascoe, Jul 01 2024

a(26) onwards from John Tyler Rascoe, Jul 01 2024

A376343 Positions of twos in the run-compressed (A037201) first differences (A001223) of the primes (A000040).

Original entry on oeis.org

2, 4, 6, 9, 12, 15, 18, 24, 26, 31, 33, 37, 39, 41, 44, 47, 50, 53, 57, 62, 73, 75, 81, 90, 95, 99, 102, 105, 108, 127, 129, 131, 135, 139, 156, 158, 161, 163, 167, 173, 182, 187, 190, 193, 196, 205, 210, 214, 216, 232, 235, 241, 244, 247, 254, 263, 265, 270

Offset: 1

Gus Wiseman, Sep 26 2024

nonn

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, we can remove all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

			The sequence of prime numbers (A000040) is:
  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, ...
with first differences (A001223):
  1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, ...
with run-compression (A037201):
  1, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, ...
with twos at (A376343):
  2, 4, 6, 9, 12, 15, 18, 24, 26, 31, 33, 37, 39, 41, 44, 47, 50, 53, 57, 62, 73, ...

Positions of 2's in A037201.
The repeats were at positions A064113 before being omitted.
A variation for squarefree numbers is A376342.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A005117 lists squarefree numbers, differences A076259 (ones A375927).
A013929 lists nonsquarefree numbers, differences A078147.
A333254 lists run-lengths of differences between consecutive primes.
Cf. A007921, A030173, A053289, A061398, A373817, A373822, A373947, A376305, A376308.

Join@@Position[First/@Split[Differences[Select[Range[100],PrimeQ]]],2]

For just the odd primes we have a(n) - 1.

A376521 Sorted positions of first appearances in the run-compression (A037201) of the first differences (A001223) of the prime numbers (A000040).

Original entry on oeis.org

1, 2, 3, 8, 22, 28, 32, 42, 91, 141, 172, 198, 242, 259, 341, 400, 556, 692, 1119, 1737, 1779, 2072, 2101, 2913, 3126, 3204, 3246, 3457, 3598, 4294, 4383, 7596, 7651, 8284, 11986, 13729, 14220, 15101, 16273, 18217, 22303, 29523, 30243, 32236, 32808, 32820

Offset: 1

Gus Wiseman, Sep 26 2024

nonn

			The sequence of prime numbers (A000040) is:
  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, ...
with first differences (A001223):
  1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, ...
with run-compression (A037201):
  1, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, ...
with first appearances at (A376521):
  1, 2, 3, 8, 22, 28, 32, 42, 91, 141, 172, 198, 242, 259, 341, 400, 556, 692, 1119, ...

These are the sorted positions of first appearances in A037201.
For positions of twos instead of first appearances we have A376343.
The unsorted version is A376520.
A000040 lists the prime numbers, differences A001223.
A003242 counts compressed compositions, ranks A333489.
A333254 lists run-lengths of differences between consecutive primes.
A373948 encodes compression using compositions in standard order.
Cf. A007921, A030173, A061398, A064113, A373817, A373822, A373947, A376305, A376308, A376312.

q=First/@Split[Differences[Select[Range[1000],PrimeQ]]];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A376520 Position of first appearance of 2n in the run-compression (A037201) of the first differences (A001223) of the prime numbers (A000040).

Original entry on oeis.org

2, 3, 8, 22, 32, 42, 28, 259, 91, 141, 172, 242, 341, 400, 556, 692, 198, 1119, 3126, 2072, 1779, 1737, 7596, 2913, 3246, 2101, 3598, 7651, 4383, 4294, 3457, 8284, 14220, 11986, 15101, 3204, 32808, 18217, 16273, 42990, 22303, 37037, 13729, 43117, 32820, 70501

Offset: 1

Gus Wiseman, Sep 26 2024

nonn

			The sequence of prime numbers (A000040) is:
  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, ...
with first differences (A001223):
  1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, ...
with run-compression (A037201):
  1, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, ...
with first appearance of 2n at (A376520):
  2, 3, 8, 22, 32, 42, 28, 259, 91, 141, 172, 242, 341, 400, 556, 692, 198, 1119, ...

This is the position of first appearance of 2n in A037201.
For positions of twos instead of first appearances we have A376343.
The sorted version is A376521.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259 (ones A375927).
A013929 lists nonsquarefree numbers, differences A078147.
A116861 counts partitions by compressed sum, compositions A373949.
A116608 counts partitions by compressed length, compositions A333755.
A274174 counts contiguous compositions, ranks A374249.
A333254 lists run-lengths of differences between consecutive primes.
A373948 encodes compression using compositions in standard order.
Cf. A007921, A030173, A064113, A373817, A373822, A373947, A376305, A376308, A376312.

mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=First/@Split[Differences[Select[Range[10000],PrimeQ]]];
Table[Position[q,2k][[1,1]],{k,mnrm[Rest[q]/2]}]

cp's OEIS Frontend

A373952 Number of integer compositions of n whose run-compression sums to 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A376343 Positions of twos in the run-compressed (A037201) first differences (A001223) of the primes (A000040).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A376521 Sorted positions of first appearances in the run-compression (A037201) of the first differences (A001223) of the prime numbers (A000040).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A376520 Position of first appearance of 2n in the run-compression (A037201) of the first differences (A001223) of the prime numbers (A000040).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica