A376309 Run-lengths of the sequence of first differences of prime-powers.

3, 1, 2, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2

Offset: 1

Author

Gus Wiseman, Sep 22 2024

Keywords

• nonn

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, ...
with runs:
  (1,1,1),(2),(1,1),(2,2),(3),(1),(2),(4),(2,2,2,2),(1),(5),(4),(2),(4), ...
with lengths A376309 (this sequence).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

For runs of prime-powers increasing by one we have A174965.

For primes instead of prime-powers we have A333254.

For squarefree numbers instead of prime-powers we have A376306.

For compression instead of run-lengths we have A376308.

For run-sums instead of run-lengths 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, first differences A057820.

A003242 counts compressed compositions, ranks A333489.

A005117 lists squarefree numbers, differences A076259.

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

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

A124767 counts runs in standard compositions, anti-runs A333381.

A238130, A238279, A333755 count compositions by number of runs.

A274174 counts contiguous compositions, ranks A374249.

A373948 encodes compression using compositions in standard order.

Cf. A001597, A006549, A025475, A037201, A053289, A110969, A120430, A251092, A373822, A375706.

Programs

Mathematica

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

PARI

up_to = 20000;
A376309list(up_to) = { my(v=vector(up_to), ppp=2, pd=1, d, rl=0, k=2, i=0); while(i<#v, k++; if(isprimepower(k), d = k-ppp; ppp = k; if(d == pd, rl++, i++; v[i] = rl; rl = 1; pd = d))); (v); };
v376309 = A376309list(up_to);
A376309(n) = v376309[n]; \\ Antti Karttunen, Jan 18 2025

Extensions

More terms from Antti Karttunen, Jan 18 2025