A323024 -id:A323024 - 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

A014311 Numbers with exactly 3 ones in binary expansion.

Original entry on oeis.org

7, 11, 13, 14, 19, 21, 22, 25, 26, 28, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 67, 69, 70, 73, 74, 76, 81, 82, 84, 88, 97, 98, 100, 104, 112, 131, 133, 134, 137, 138, 140, 145, 146, 148, 152, 161, 162, 164, 168, 176, 193, 194, 196, 200, 208, 224, 259, 261, 262, 265, 266, 268, 273, 274, 276, 280, 289, 290, 292, 296, 304

Offset: 1

Al Black (gblack(AT)nol.net)

Equivalently, sums of three distinct powers of 2.
Appears to give all n such that 64 is the highest power of 2 dividing A005148(n). - Benoit Cloitre, Jun 22 2002
From Gus Wiseman, Oct 05 2020: (Start)
These are numbers k such that the k-th composition in standard order has length 3. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. The sequence together with the corresponding standard compositions begins:
(1,1,1)     44: (2,1,3)     97: (1,5,1)
(2,1,1)     49: (1,4,1)     98: (1,4,2)
(1,2,1)     50: (1,3,2)    100: (1,3,3)
(1,1,2)     52: (1,2,3)    104: (1,2,4)
(3,1,1)     56: (1,1,4)    112: (1,1,5)
(2,2,1)     67: (5,1,1)    131: (6,1,1)
(2,1,2)     69: (4,2,1)    133: (5,2,1)
(1,3,1)     70: (4,1,2)    134: (5,1,2)
(1,2,2)     73: (3,3,1)    137: (4,3,1)
(1,1,3)     74: (3,2,2)    138: (4,2,2)
(4,1,1)     76: (3,1,3)    140: (4,1,3)
(3,2,1)     81: (2,4,1)    145: (3,4,1)
(3,1,2)     82: (2,3,2)    146: (3,3,2)
(2,3,1)     84: (2,2,3)    148: (3,2,3)
(2,2,2)     88: (2,1,4)    152: (3,1,4)
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. See p. 18 for Mathematica code irwinSums.m.
Stephen Morley, HAKMEM Item 175 (Gosper).
Tilman Piesk, First 56 elements in a tetrahedral array.

Cf. A038465 (base 3), A038471 (base 4), A038475 (base 5).
Cf. A081091 (primes), A212190 (squares), A212192 (triangular numbers), A173589 (Fibbinary).
Cf. A057168.
Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691 (Hammingweight = 1, 2, ..., 9).
Cf. A056558, A194848, A194847.
A000217(n-2) counts compositions into three parts.
A001399(n-3) = A069905(n) = A211540(n+2) counts the unordered case.
A001399(n-6) = A069905(n-3) = A211540(n-1) counts the unordered strict case.
A001399(n-6)*6 = A069905(n-3)*6 = A211540(n-1)*6 counts the strict case.
A014612 is an unordered version, with strict case A007304.
A337453 is the strict case.
A337461 counts the coprime case.
A033992 lists numbers divisible by exactly three different primes.
A323024 lists numbers with exactly three different prime multiplicities.
Cf. A220377, A307534, A337459, A337460, A337561, A337603, A337604.

unsigned hakmem175(unsigned x) {
    unsigned s, o, r;
    s = x & -x;  r = x + s;
    o = r ^ x;  o = (o >> 2) / s;
    return r | o;
}
unsigned A014311(int n) {
    if (n == 1) return 7;
    return hakmem175(A014311(n - 1));
}  // Peter Luschny, Jan 01 2014

a014311 n = a014311_list !! (n-1)
a014311_list = [2^x + 2^y + 2^z |
                x <- [2..], y <- [1..x-1], z <- [0..y-1]]
-- Reinhard Zumkeller, May 03 2012

Select[Range[200], (Count[IntegerDigits[#, 2], 1] == 3)&]
nn = 8; Flatten[Table[2^i + 2^j + 2^k, {i, 2, nn}, {j, 1, i - 1}, {k, 0, j - 1}]] (* T. D. Noe, Nov 05 2013 *)

for(n=0,10^3,if(hammingweight(n)==3,print1(n,", "))); \\ Joerg Arndt, Mar 04 2014

print1(t=7);for(i=2,50,print1(","t=A057168(t))) \\ M. F. Hasler, Aug 27 2014

A014311_list = [2**a+2**b+2**c for a in range(2,6) for b in range(1,a) for c in range(b)] # Chai Wah Wu, Jan 24 2021

from itertools import islice
def A014311_gen(): # generator of terms
    yield (n:=7)
    while True: yield (n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A014311_list = list(islice(A014311_gen(),20)) # Chai Wah Wu, Mar 10 2025

from math import isqrt, comb
from sympy import integer_nthroot
def A014311(n): return (1<<(r:=n-1-comb((m:=integer_nthroot(6*n,3)[0])+(t:=(n>comb(m+2,3)))+1,3))-comb((k:=isqrt(b:=r+1<<1))+(b>k*(k+1)),2))+(1<<(a:=isqrt(s:=n-comb(m-(t^1)+2,3)<<1))+((s<<2)>(a<<2)*(a+1)+1))+(1<Chai Wah Wu, Mar 10 2025

A000120(a(n)) = 3. - Reinhard Zumkeller, May 03 2012
Start with A084468. If n is in sequence, then 2n is too. - Ralf Stephan, Aug 16 2013
a(n+1) = A057168(a(n)). - M. F. Hasler, Aug 27 2014
a(n) = 2^A056558(n-1) + 2^A194848(n-1) + 2^A194847(n-1). - Ridouane Oudra, Sep 06 2020
Sum_{n>=1} 1/a(n) = A367110 = 1.428591545852638123996854844400537952781688750906133068397189529775365950039... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

Extension and program by Olivier Gérard

A323025 Numbers with exactly four distinct exponents in their prime factorization, or four distinct parts in their prime signature.

Original entry on oeis.org

75600, 105840, 113400, 118800, 126000, 140400, 151200, 158760, 178200, 183600, 198000, 205200, 210600, 211680, 232848, 234000, 237600, 246960, 248400, 252000, 261360, 275184, 275400, 280800, 283500, 294000, 302400, 306000, 307800, 313200, 315000, 334800

Offset: 1

Gus Wiseman, Jan 02 2019

nonn

Positions of 4's in A071625.
Numbers k such that A001221(A181819(k)) = 4.
Is a(n) ~ c * n for some c? - David A. Corneth, Jan 09 2019
The asymptotic density of this sequence is (6/Pi^2) * Sum_{n>=2, n squarefree} r(n)/((n-1)*psi(n)) = 0.00035750... (corresponding to c = 2797.1... in the question above, whose answer is affirmative), where psi is the Dedekind psi function (A001615), and r(n) = Sum_{d_1|n, 1Amiram Eldar, Oct 18 2020

			126000 = 2^4 * 3^2 * 5^3 * 7^1 has four distinct exponents {1, 2, 3, 4}, so belongs to the sequence.
831600 = 2^4 * 3^3 * 5^2 * 7^1 * 11^1 has four distinct exponents {1, 2, 3, 4}, so belongs to the sequence.

David A. Corneth, 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, A033993, A059404, A062770, A071625, A118914, A181819, A323014, A323022, A323024.

tom[n_]:=Length[Union[Last/@If[n==1,{},FactorInteger[n]]]];
Select[Range[100000],tom[#]==4&]

is(n) = #Set(factor(n)[, 2]) == 4 \\ David A. Corneth, Jan 09 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
```

cp's OEIS Frontend

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

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A014311 Numbers with exactly 3 ones in binary expansion.

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A323025 Numbers with exactly four distinct exponents in their prime factorization, or four 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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