A353507 Product of multiplicities of the prime exponents (signature) of n; a(1) = 0.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 2, 2, 2, 2

Offset: 1

Author

Gus Wiseman, May 19 2022

Keywords

• nonn

Comments

Warning: If the prime multiplicities of n are a multiset y, this sequence gives the product of multiplicities in y, not the product of y.

Differs from A351946 at A351946(1260) = 4, a(1260) = 2.

Differs from A327500 at A327500(450) = 3, a(450) = 2.

We set a(1) = 0 so that the positions of first appearances are the primorials A002110.

Also the product of the prime metasignature of n (row n of A238747).

Examples

			The prime signature of 13860 is (2,2,1,1,1), with multiplicities (2,3), so a(13860) = 6.
		

Links

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

Crossrefs

Positions of first appearances are A002110.

The prime indices themselves have product A003963, counted by A339095.

The prime signature itself has product A005361, counted by A266477.

A001222 counts prime factors with multiplicity, distinct A001221.

A056239 adds up prime indices, row sums of A112798 and A296150.

A071625 counts distinct prime exponents (third omega).

A124010 gives prime signature, sorted A118914.

A130091 lists numbers with distinct prime exponents, counted by A098859.

A181819 gives prime shadow, with an inverse A181821.

A238747 gives prime metasignature, sorted A353742.

A323022 gives fourth omega.

Cf. A085629, A097318, A182850, A304678, A353394, A353399, A353500, A353503.

Programs

Maple

f:= proc(n) local M,s;
  M:= ifactors(n)[2][..,2];
  mul(numboccur(s,M),s=convert(M,set));
end proc:
f(1):= 0:
map(f, [$1..100]); # Robert Israel, May 19 2023

Mathematica

Table[If[n==1,0,Times@@Length/@Split[Sort[Last/@FactorInteger[n]]]],{n,100}]
Join[{0},Table[Times@@(Length/@Split[FactorInteger[n][[;;,2]]]),{n,2,100}]] (* Harvey P. Dale, Oct 20 2024 *)

Python

from math import prod
from itertools import groupby
from sympy import factorint
def A353507(n): return 0 if n == 1 else prod(len(list(g)) for k, g in groupby(factorint(n).values())) # Chai Wah Wu, May 20 2022

Formula

a(n) = A005361(A181819(n)) = A003963(A181819(A181819(n))).