A376308 Run-compression of the sequence of first differences of prime-powers.

1, 2, 1, 2, 3, 1, 2, 4, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, 4, 2, 6, 2, 6, 8, 4, 2, 4, 2, 4, 8, 4, 2, 1, 3, 6, 2, 10, 2, 6, 4, 2, 4, 6, 2, 10, 2, 4, 2, 12, 4, 2, 4, 6, 2, 8, 5, 1, 6, 2, 6, 4, 2, 6, 4, 14, 4, 2, 4, 14, 6, 4, 2, 4, 6, 2, 6, 4, 6, 8, 4, 8, 10, 2, 10

Offset

1,2

Comments

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, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

Links

Table of n, a(n) for n=1..85.

Example

The sequence of prime-powers (A246655) is:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, ...
The sequence of first differences (A057820) of prime-powers is:
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, ...
The run-compression is A376308 (this sequence).

Mathematica

First/@Split[Differences[Select[Range[100], PrimePowerQ]]]

Crossrefs

For primes instead of prime-powers we have A037201, halved A373947.

For squarefree numbers instead of prime-powers we have A376305.

For run-lengths instead of compression we have A376309.

For run-sums instead of compression we have A376310.

For positions of first appearances we have A376341, sorted A376340.

A000040 lists the prime numbers, differences A001223.

A000961 and A246655 list prime-powers, differences A057820.

A003242 counts compressed compositions, ranks A333489.

A024619 and A361102 list non-prime-powers, differences A375708.

A116861 counts partitions by compressed sum, by compressed length A116608.

A373948 encodes compression using compositions in standard order.

Cf. A001597, A006549, A007916, A025475, A034296, A053289, A076259, A110969, A120430, A124767, A174965, A374251.

Sequence in context: A214614 A265692 A370263 * A194976 A195082 A210535

Adjacent sequences: A376305 A376306 A376307 * A376309 A376310 A376311

Keyword

nonn

Author

Gus Wiseman, Sep 20 2024

Status

approved