A065310 Number of occurrences of n-th prime in A065308, where A065308(j) = prime(j - pi(j)).

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

Offset: 1

Author

Labos Elemer, Oct 29 2001

Keywords

• nonn

Comments

Seems identical to A054546. Each odd prime arises once or twice!?

First differences of A018252 (positive nonprime numbers). Including 0 gives A054546. Removing 1 gives A073783. - Gus Wiseman, Sep 15 2024

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A000720, A054576, A065308-A065309.

For twin 2's see A169643.

Positions of 1's are A375926, complement A014689 (except first term).

Other families of numbers and their first-differences:

For prime numbers (A000040) we have A001223.

For composite numbers (A002808) we have A073783.

For nonprime numbers (A018252) we have A065310 (this).

For perfect powers (A001597) we have A053289.

For non-perfect-powers (A007916) we have A375706.

For squarefree numbers (A005117) we have A076259.

For nonsquarefree numbers (A013929) we have A078147.

For prime-powers inclusive (A000961) we have A057820.

For prime-powers exclusive (A246655) we have A057820(>1).

For non-prime-powers inclusive (A024619) we have A375735.

For non-prime-powers exclusive (A361102) we have A375708.

Cf. A046933, A053707, A054546, A110969, A176246, A251092, A373403, A375929.

Programs

Mathematica

t=Table[Prime[w-PrimePi[w]], {w, a, b}] Table[Count[t, Prime[n]], {n, c, d}]
Differences[Select[Range[100],!PrimeQ[#]&]] (* Gus Wiseman, Sep 15 2024 *)

PARI

{ p=1; f=2; m=1; for (n=1, 1000, a=0; p=nextprime(p + 1); while (p==f, a++; m++; f=prime(m - primepi(m))); write("b065310.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 16 2009