A379311 - OEIS

%I #6 Dec 28 2024 16:27:36

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

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

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

%N Number of prime indices of n that are 1 or prime.

%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)) = A080339(k).

%e The prime indices of 39 are {2,6}, so a(39) = 1.

%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) = 1.

%e The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 2.

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

%t Table[Length[Select[prix[n],#==1||PrimeQ[#]&]],{n,100}]

%Y Positions of first appearances are A000079.

%Y These "old" primes are listed by A008578.

%Y Positions of zero are A320629, counted by A023895 (strict A204389).

%Y Positions of one are A379312, counted by A379314 (strict A379315).

%Y Positions of nonzero terms are A379313.

%Y A000040 lists the prime numbers, differences A001223.

%Y A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.

%Y A055396 gives least prime index, greatest A061395.

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

%Y A080339 is the characteristic function for the old prime numbers.

%Y A376682 gives k-th differences of old prime numbers, see A030016, A075526, A173390, A376683, A376855.

%Y Other counts of prime indices:

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

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

%Y - A257994 prime, see A000586, A000607, A076610, A331386, A331915, A379304, A379305.

%Y - A330944 nonprime, see A002095, A096258, A320628, A330945.

%Y - A379306 squarefree, see A302478, A379308, A379309, A379316.

%Y - A379310 nonsquarefree, see A114374, A256012, A379307.

%Y Cf. A034891, A036234, A036497, A038550, A087436, A302540, A379300, A379301.

%K nonn

%O 1,4

%A _Gus Wiseman_, Dec 27 2024