A323025 -id:A323025 - cp's OEIS frontend

A071625 Number of distinct exponents when n is factorized as a product of primes.

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

Offset: 1

Labos Elemer, May 29 2002

nonn

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
Sanna (2020) proved that for each k>=1, the sequence of numbers n with A071625(n) = k has an asymptotic density A_k = (6/Pi^2) * Sum_{n>=1, n squarefree} rho_k(n)/psi(n), where psi is the Dedekind psi function (A001615), and rho_k(n) is defined by rho_1(n) = 1 if n = 1 and 0 otherwise, rho_{k+1}(n) = 0 if n = 1 and (1/(n-1)) * Sum_{d|n, dAmiram Eldar, Oct 18 2020

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

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, Proceedings - Mathematical Sciences, Indian Academy of Sciences, Vol. 130, No. 1 (2020), Article 27, alternative link, arXiv preprint, arXiv:1902.09224 [math.NT], 2019.

Cf. A051903, A051904, A006939, A124010.

Cf. A001221, A001222, A001615, A059404, A062770, A118914, A181819, A323014, A323022, A323023, A323024, A323025.

# Using function 'PrimeSignature' from A124010.
a := n -> nops(convert(PrimeSignature(n), set)):
seq(a(n), n = 1..105); # Peter Luschny, Jun 15 2025

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 *)

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

from sympy import factorint
def a(n): return len(set(factorint(n).values()))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Sep 01 2022

A323024 Numbers with exactly three distinct exponents in their prime factorization, or three distinct parts in their prime signature.

Original entry on oeis.org

360, 504, 540, 600, 720, 756, 792, 936, 1008, 1176, 1188, 1200, 1224, 1350, 1368, 1400, 1404, 1440, 1500, 1584, 1620, 1656, 1836, 1872, 1960, 2016, 2052, 2088, 2160, 2200, 2232, 2250, 2268, 2352, 2400, 2448, 2484, 2520, 2600, 2646, 2664, 2736, 2800, 2880, 2904

Offset: 1

Gus Wiseman, Jan 02 2019

nonn

Positions of 3's in A071625.
Numbers k such that A001221(A181819(k)) = 3.
The asymptotic density of this sequence is (6/Pi^2) * Sum_{n>=2, n squarefree} r(n)/((n-1)*psi(n)) = 0.030575..., where psi is the Dedekind psi function (A001615), and r(n) = Sum_{d|n, 1Amiram Eldar, Oct 18 2020

			1500 = 2^2 * 3^1 * 5^3 has three distinct exponents {1, 2, 3}, so belongs to the sequence.
52500 = 2^2 * 3^1 * 5^4 * 7^1 has three distinct exponents {1, 2, 4}, so belongs to the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carlo Sanna, On the number of distinct exponents in the prime factorization of an integer, Proceedings - Mathematical Sciences, Indian Academy of Sciences, Vol. 130, No. 1 (2020), Article 27, alternative link.

Cf. A001221, A001222, A001615, A006939, A033992, A059404, A062770, A071625, A118914, A181819, A323014, A323022, A323025.

tom[n_]:=Length[Union[Last/@If[n==1,{},FactorInteger[n]]]];
Select[Range[1000],tom[#]==3&]

is(n) = #Set(factor(n)[, 2]) == 3 \\ David A. Corneth, Jan 02 2019

A323055 Numbers with exactly two distinct exponents in their prime factorization, or two distinct parts in their prime signature.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200

Offset: 1

Gus Wiseman, Jan 03 2019

nonn

The first term is A006939(2) = 12.
First differs from A059404 in lacking 360, whose prime signature has three distinct parts.
Positions of 2's in A071625.
Numbers k such that A001221(A181819(k)) = 2.
The asymptotic density of this sequence is (6/Pi^2) * Sum_{n>=2, n squarefree} 1/((n-1)*psi(n)) = 0.3611398..., where psi is the Dedekind psi function (A001615) (Sanna, 2020). - Amiram Eldar, Oct 18 2020

			3000 = 2^3 * 3^1 * 5^3 has two distinct exponents {1, 3}, so belongs to the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carlo Sanna, On the number of distinct exponents in the prime factorization of an integer, Proceedings - Mathematical Sciences, Indian Academy of Sciences, Vol. 130, No. 1 (2020), Article 27, alternative link.

One distinct exponent: A062770 or A072774.
Two distinct exponents: this sequence.
Three distinct exponents: A323024.
Four distinct exponents: A323025.
Five distinct exponents: A323056.
Cf. A001221, A001222, A001615, A006939, A007774, A059404, A071625, A118914, A181819, A323014, A323022.

isA323055 := proc(n)
    local eset;
    eset := {};
    for pf in ifactors(n)[2] do
        eset := eset union {pf[2]} ;
    end do:
    simplify(nops(eset) = 2 ) ;
end proc:
for n from 12 to 1000 do
    if isA323055(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 09 2019

Select[Range[100],Length[Union[Last/@FactorInteger[#]]]==2&]

A323056 Numbers with exactly five distinct exponents in their prime factorization, or five distinct parts in their prime signature.

Original entry on oeis.org

174636000, 206388000, 244490400, 261954000, 269892000, 274428000, 288943200, 291060000, 301644000, 309582000, 343980000, 349272000, 365148000, 366735600, 377848800, 383292000, 404838000, 411642000, 412776000, 422301600, 433414800, 449820000, 452466000, 457380000

Offset: 1

Gus Wiseman, Jan 03 2019

nonn

The first term is A006939(5) = 174636000.
Positions of 5's in A071625.
Numbers k such that A001221(A181819(k)) = 5.

			174636000 = 2^5 * 3^4 * 5^3 * 7^2 * 11^1 has five distinct exponents so belongs to the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

One distinct exponent: A062770 or A072774.
Two distinct exponents: A323055.
Three distinct exponents: A323024.
Four distinct exponents: A323025.
Five distinct exponents: A323056.
Cf. A001221, A001222, A006939, A051270, A059404, A071625, A118914, A181819, A182855, A323014, A323022, A323024.

Select[Range[300000000],Length[Union[Last/@FactorInteger[#]]]==5&]

is(n) = #Set(factor(n)[, 2]) == 5 \\ David A. Corneth, Jan 12 2019

a(13)-a(24) from Daniel Suteu, Jan 12 2019

A324206 Numbers with exactly six distinct exponents in their prime factorization, or six distinct parts in their prime signature.

Original entry on oeis.org

5244319080000, 6197831640000, 6857955720000, 7342046712000, 7664774040000, 7866478620000, 8241072840000, 8676964296000, 8740531800000, 9278410680000, 9296747460000, 9578467080000, 9601138008000, 10286933580000, 10329719400000, 10488638160000, 10598658840000, 10705345560000

Offset: 1

David A. Corneth, Feb 17 2019

			6197831640000 = 2^6 * 3^5 * 5^4 * 7^3 * 11 * 13^2 is in the sequence as there are 6 distinct exponents; 1 through 6.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A006939, A051270, A059404, A071625, A118914, A181819, A182855, A323014, A323022, A323024, A323025, A323056.

PARI
```
is(n) = #Set(factor(n)[, 2]) == 6
```

A324207 Numbers with exactly seven distinct exponents in their prime factorization, or seven distinct parts in their prime signature.

Original entry on oeis.org

2677277333530800000, 2992251137475600000, 3164055030536400000, 3501054974617200000, 3536296798834800000, 3622198745365200000, 3748188266943120000, 4015916000296200000, 4189151592465840000, 4207150095548400000, 4280780335431600000, 4373290124002800000, 4429677042750960000

Offset: 1

David A. Corneth, Feb 17 2019

			2677277333530800000 = 2^7 * 3^6 * 5^5 * 7^4 * 11^3 * 13^2 * 17 is in the sequence. There are exactly 7 distinct exponents; 1 through 7 in it.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A006939, A051270, A059404, A071625, A118914, A181819, A182855, A323014, A323022, A323024, A323025, A323056.

PARI
```
is(n) = #Set(factor(n)[, 2]) == 7
```

A324208 Numbers with exactly eight distinct exponents in their prime factorization, or eight distinct parts in their prime signature.

Original entry on oeis.org

25968760179275365452000000, 29023908435660702564000000, 30690352939143613716000000, 31435867585438600284000000, 33959147926744708668000000, 34300982696689921212000000, 36356264250985511632800000, 37151479873700163972000000, 38953140268913048178000000, 39267640824717421116000000

Offset: 1

David A. Corneth, Feb 17 2019

nonn

			29023908435660702564000000 = 2^8 * 3^7 * 5^6 * 7^5 * 11^4 * 13^3 * 17 * 19^2 is in the sequence as there are exactly 8 distinct exponents; 1 through 8.

Cf. A001221, A001222, A006939, A051270, A059404, A071625, A118914, A181819, A182855, A323014, A323022, A323024, A323025, A323056.

PARI
```
is(n) = #Set(factor(n)[, 2]) == 8
```

cp's OEIS Frontend

A071625 Number of distinct exponents when n is factorized as a product of primes.

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A323024 Numbers with exactly three distinct exponents in their prime factorization, or three distinct parts in their prime signature.

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A323055 Numbers with exactly two distinct exponents in their prime factorization, or two distinct parts in their prime signature.

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A323056 Numbers with exactly five distinct exponents in their prime factorization, or five distinct parts in their prime signature.

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A324206 Numbers with exactly six distinct exponents in their prime factorization, or six distinct parts in their prime signature.

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A324207 Numbers with exactly seven distinct exponents in their prime factorization, or seven distinct parts in their prime signature.

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A324208 Numbers with exactly eight distinct exponents in their prime factorization, or eight distinct parts in their prime signature.

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