A387135 Number of distinct prime factors in common to all prime indices of n.

0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1

Offset

1,13

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.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = A001221(A289508(n)).

a(prime(n)) = A001221(n).

Example

The prime indices of 2257 are {12,18}, with common prime factors {2,3}, so a(2257) = 2.

Maple

f:= proc(n) local g;
  g:= igcd(op(map(NumberTheory:-pi, NumberTheory:-PrimeFactors(n))));
  nops(NumberTheory:-PrimeFactors(g))
end proc: f(1):= 0:
map(f, [$1..100]); # Robert Israel, Jan 13 2026

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[If[n==1, 0, PrimeNu[GCD@@prix[n]]], {n, 100}]

Crossrefs

For initial interval instead of prime factors we have A055396.

Positions of first appearances appear to be A062447, see also A033844.

Positions of 0 are A289509, complement A318978.

For LCM (or product) instead of GCD we have A303975.

For all divisors instead of just prime factors we have A387114.

For bigomega instead of omega we have A387579, for LCM A387879.

A000005 counts divisors.

A001222 counts prime factors, distinct A001221.

A003963 multiplies together prime indices.

A112798 lists prime indices, row sums A056239 or A066328.

A120383 lists numbers divisible by all of their prime indices.

A289508 gives greatest common divisor of prime indices.

Cf. A000720, A001414, A061395, A335433, A335448, A355731, A355733, A355737, A355740, A370820, A387119.

Sequence in context: A085491 A321013 A284258 * A322389 A380656 A336388

Adjacent sequences: A387132 A387133 A387134 * A387136 A387137 A387138

Keyword

nonn

Author

Gus Wiseman, Sep 11 2025

Status

approved