A370820 - OEIS

%I #14 May 02 2024 09:47:15

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

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

%U 4,4,6,2,4,6,3,4,4,4,4,2,2,2,2,3,3,4,4

%N Number of positive integers that are a divisor of some prime index 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.

%C This sequence contains all nonnegative integers. In particular, a(prime(n)!) = n.

%e 2045 has prime indices {3,80} with combined divisors {1,2,3,4,5,8,10,16,20,40,80}, so a(2045) = 11. In fact, 2045 is the least number with this property.

%t Table[Length[Union@@Divisors/@PrimePi/@First/@If[n==1,{},FactorInteger[n]]],{n,100}]

%o (PARI) a(n) = my(list=List(), f=factor(n)); for (i=1, #f~, fordiv(primepi(f[i,1]), d, listput(list, d))); #Set(list); \\ _Michel Marcus_, May 02 2024

%Y a(prime(n)) = A000005(n).

%Y Positions of ones are A000079 except for 1.

%Y a(n!) = A000720(n).

%Y a(prime(n)!) = a(prime(A005179(n))) = n.

%Y Counting prime factors instead of divisors gives A303975.

%Y Positions of 2's are A371127.

%Y Position of first appearance of n is A371131(n), sorted version A371181.

%Y RHS of A370802, A371128, A371130, A371165-A371170, A371177, A371178.

%Y A001221 counts distinct prime factors.

%Y A003963 gives product of prime indices.

%Y A027746 lists prime factors, A112798 indices, length A001222.

%Y A355731 counts choices of a divisor of each prime index, firsts A355732.

%Y A355741 counts choices of a prime factor of each prime index.

%Y Cf. A000792, A007416, A048249, A319899, A355737, A355739, A370348, A370808, A370809.

%K nonn

%O 1,3

%A _Gus Wiseman_, Mar 15 2024