A370820 Number of positive integers that are a divisor of some prime index of n.

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, 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, 4, 4, 6, 2, 4, 6, 3, 4, 4, 4, 4, 2, 2, 2, 2, 3, 3, 4, 4

Offset: 1

Author

Gus Wiseman, Mar 15 2024

Keywords

• nonn

Comments

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.

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

Examples

			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.
		

Crossrefs

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

Positions of ones are A000079 except for 1.

a(n!) = A000720(n).

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

Counting prime factors instead of divisors gives A303975.

Positions of 2's are A371127.

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

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

A001221 counts distinct prime factors.

A003963 gives product of prime indices.

A027746 lists prime factors, A112798 indices, length A001222.

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

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

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

Programs

Mathematica

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

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