A379306 - OEIS

%I #8 Dec 27 2024 18:07:54

%S 0,1,1,2,1,2,0,3,2,2,1,3,1,1,2,4,1,3,0,3,1,2,0,4,2,2,3,2,1,3,1,5,2,2,

%T 1,4,0,1,2,4,1,2,1,3,3,1,1,5,0,3,2,3,0,4,2,3,1,2,1,4,0,2,2,6,2,3,1,3,

%U 1,2,0,5,1,1,3,2,1,3,1,5,4,2,1,3,2,2,2

%N Number of squarefree prime indices of n.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%F Totally additive with a(prime(k)) = A008966(k).

%e The prime indices of 39 are {2,6}, so a(39) = 2.

%e The prime indices of 70 are {1,3,4}, so a(70) = 2.

%e The prime indices of 98 are {1,4,4}, so a(98) = 1.

%e The prime indices of 294 are {1,2,4,4}, a(294) = 2.

%e The prime indices of 1911 are {2,4,4,6}, so a(1911) = 2.

%e The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 3.

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Table[Length[Select[prix[n],SquareFreeQ]],{n,100}]

%Y Positions of first appearances are A000079.

%Y Positions of zero are A379307, counted by A114374 (strict A256012).

%Y Positions of one are A379316, counted by A379308 (strict A379309).

%Y A000040 lists the primes, differences A001223.

%Y A005117 lists the squarefree numbers, differences A076259.

%Y A008966 is the characteristic function for the squarefree numbers.

%Y A013929 lists the nonsquarefree numbers, differences A078147.

%Y A055396 gives least prime index, greatest A061395.

%Y A056239 adds up prime indices, row sums of A112798, counted by A001222.

%Y A061398 counts squarefree numbers between primes, zeros A068360.

%Y A377038 gives k-th differences of squarefree numbers.

%Y Other counts of prime indices:

%Y - A087436 postpositive, see A038550.

%Y - A257991 odd, see A000041, A000070, A066207, A349158.

%Y - A257992 even, see A000009, A038348, A066208, A379317.

%Y - A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.

%Y - A330944 nonprime, see A000586, A000607, A076610, A330945.

%Y - A379300 composite, see A023895, A034891, A036497, A302540, A379301.

%Y - A379310 nonsquarefree, see A302478.

%Y - A379311 old prime, see A204389, A320629, A379312-A379315.

%Y Cf. A000720, A013928, A057627, A068361, A070321, A071403, A072284, A112925, A112929, A120327, A377430, A378086.

%K nonn,new

%O 1,4

%A _Gus Wiseman_, Dec 25 2024