A387114 Number of divisors in common to all prime indices of n.

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

Offset: 1

Author

Gus Wiseman, Aug 19 2025

Keywords

• nonn
• new

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.

Also the number of divisors of the greatest common divisor of the prime indices of n.

Examples

			The prime indices of 703 are {8,12}, with divisors {{1,2,4,8},{1,2,3,4,6,12}}, with {1,2,4} in common, so a(703) = 3.
		

Crossrefs

For initial interval instead of divisors we have A055396.

Positions of 1 are A289509, complement A318978.

Positions of 2 are A387119.

For prime factors or indices instead of divisors we have A387135, see A010055 or A069513.

A000005 counts divisors.

A001414 adds up distinct prime divisors, counted by A001221.

A003963 multiplies together the prime indices of n.

A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.

A120383 lists numbers divisible by all of their prime indices.

A289508 gives greatest common divisor of prime indices.

Cf. A000720, A061395, A326841, A335433, A335448, A355731, A355733, A355734, A355737, A355739, A355740, A370820.

Programs

Mathematica

Table[If[n==1,0,Length[Divisors[GCD@@PrimePi/@First/@FactorInteger[n]]]],{n,100}]

Formula

a(1) = 0; a(n) = A000005(A289508(n)) for n > 1.