A065310 - OEIS

%I #25 Sep 15 2024 07:13:13

%S 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,

%T 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,

%U 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

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

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

%C First differences of A018252 (positive nonprime numbers). Including 0 gives A054546. Removing 1 gives A073783. - _Gus Wiseman_, Sep 15 2024

%H Harry J. Smith, <a href="/A065310/b065310.txt">Table of n, a(n) for n = 1..1000</a>

%t t=Table[Prime[w-PrimePi[w]], {w, a, b}] Table[Count[t, Prime[n]], {n, c, d}]

%t Differences[Select[Range[100],!PrimeQ[#]&]] (* _Gus Wiseman_, Sep 15 2024 *)

%o (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

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

%Y For twin 2's see A169643.

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

%Y Other families of numbers and their first-differences:

%Y For prime numbers (A000040) we have A001223.

%Y For composite numbers (A002808) we have A073783.

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

%Y For perfect powers (A001597) we have A053289.

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

%Y For squarefree numbers (A005117) we have A076259.

%Y For nonsquarefree numbers (A013929) we have A078147.

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

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

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

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

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

%K nonn

%O 1,1

%A _Labos Elemer_, Oct 29 2001