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A071625 Number of distinct exponents when n is factorized as a product of primes. 75
0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,12

COMMENTS

First term greater than 2 is a(360) = 3.

From Michel Marcus, Apr 24 2016: (Start)

A006939(n) gives the least m such that a(m) = n.

A062770 is the sequence of integers m such that a(m) = 1. (End)

We define the k-th omega of n to be Omega(red^{k-1}(n)) where Omega = A001222 and red^{k} is the k-th functional iteration of A181819. The first two omegas are A001222 and A001221, while this sequence is the third, and A323022 is the fourth. The zeroth omega is not uniquely determined from prime signature, but one possible choice is A056239 (sum of prime indices). - Gus Wiseman, Jan 02 2019

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..10000

E. T. Bell, Functions of coprime divisors of integers, Bull. Amer. Math. Soc. 43 (1937), 818-822.

Carlo Sanna, On the number of distinct exponents in the prime factorization of an integer, arXiv:1902.09224 [math.NT], 2019.

EXAMPLE

n = 5040 = 2^4*(3*5)^2*7, three different exponents arise:4,2 and 1; so a(5040)=3.

MATHEMATICA

ffi[x_] := Flatten[FactorInteger[x]];

lf[x_] := Length[FactorInteger[x]];

ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}];

Table[Length[Union[ep[w]]], {w, 1, 256}]

(* Second program: *)

{0}~Join~Array[Length@ Union@ FactorInteger[#][[All, -1]] &, 104, 2] (* Michael De Vlieger, Apr 10 2019 *)

PROG

(PARI) a(n) = #Set(factor(n)[, 2]); \\ Michel Marcus, Mar 12 2015

CROSSREFS

Cf. A051903, A051904, A006939.

Cf. A001221, A001222, A059404, A062770, A118914, A181819, A323014, A323022, A323023, A323024, A323025.

Sequence in context: A058060 A088323 A003652 * A309004 A304779 A049100

Adjacent sequences:  A071622 A071623 A071624 * A071626 A071627 A071628

KEYWORD

nonn

AUTHOR

Labos Elemer, May 29 2002

STATUS

approved

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Last modified October 17 14:39 EDT 2019. Contains 328114 sequences. (Running on oeis4.)