A323056 -id:A323056 - cp's OEIS frontend

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

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&]

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

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI