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

Author

Gus Wiseman, Sep 20 2024

Keywords

• nonn

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).

Examples

			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).
		

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.

Programs

Mathematica

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