A143610 -id:A143610 - cp's OEIS frontend

A375073 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no other exponents.

72, 108, 200, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 1800, 2312, 2700, 2888, 3087, 3267, 3528, 4232, 4500, 4563, 5292, 5324, 5400, 6125, 6728, 7688, 7803, 8575, 8712, 8788, 9000, 9747, 9800, 10584, 10952, 11979, 12168, 12348, 13068, 13448, 13500, 14283, 14792

Offset: 1

Amiram Eldar, Jul 29 2024

Numbers k such that the set of distinct prime factorization exponents of k (row k of A136568) is {2, 3}.

Number k such that A051904(k) = 2 and A051903(k) = 3.

Equals A338325 \ (A062503 UNION A062838).
Subsequence of A001694 and A046100.
A143610 is a subsequence.
Cf. A051903, A051904, A136568.
Cf. A082020, A157289, A330595.

Select[Range[15000], Union[FactorInteger[#][[;; , 2]]] == {2, 3} &]

PARI
```
is(k) = Set(factor(k)[,2]) == [2, 3];
```

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/p^2 + 1/p^3) - 15/Pi^2 - zeta(3)/zeta(6) + 1 = A330595 - A082020 - A157289 + 1 = 0.047550294197921818806... .

A336530 Number of triples of divisors d_i < d_j < d_k of n such that gcd(d_i, d_j, d_k) > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 4, 0, 5, 0, 5, 0, 0, 0, 23, 0, 0, 1, 5, 0, 12, 0, 10, 0, 0, 0, 36, 0, 0, 0, 23, 0, 12, 0, 5, 5, 0, 0, 62, 0, 5, 0, 5, 0, 23, 0, 23, 0, 0, 0, 87, 0, 0, 5, 20, 0, 12, 0, 5, 0, 12, 0, 120, 0, 0, 5, 5, 0, 12, 0, 62, 4

Offset: 1

Michel Lagneau, Oct 04 2020

nonn

Number of elements in the set {(x, y, z): x|n, y|n, z|n, x < y < z, GCD(x, y, z) > 1}.
Every element of the sequence is repeated indefinitely, for instance:
a(n) = 0 for n = 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, ... (Numbers with at most 2 prime factors (counted with multiplicity). See A037143);
a(n) = 1 for n = 8, 27, 125, 343, 1331, 2197, 4913,... (cubes of primes. See A030078);
a(n) = 4 for n = 16, 81, 625, 2401, 14641, 28561, ... (prime(n)^4. See A030514);
a(n) = 5 for n = 12, 18, 20, 28, 44, 45, ... (Numbers which are the product of a prime and the square of a different prime (p^2 * q). See A054753);
a(n) = 12 for n = 30, 42, 66, 70, 78, 102, 105, 110,... (Sphenic numbers: products of 3 distinct primes. See A007304);
a(n) = 20 for n = 64, 729, 15625, 117649, ... (Numbers with 7 divisors. 6th powers of primes. See A030516);
a(n) = 23 for n = 24, 40, 54, 56, 88, 104, 135, 136, ... (Product of the cube of a prime (A030078) and a different prime. See A065036);
a(n) = 36 for n = 36, 100, 196, 225, 441, 484, 676,... (Squares of the squarefree semiprimes (p^2*q^2). See A085986);
a(n) = 62 for n = 48, 80, 112, 162, 176, 208, 272, ... (Product of the 4th power of a prime (A030514) and a different prime (p^4*q). See A178739);
a(n) = 87 for n = 60, 84, 90, 126, 132, 140, 150, 156, ... (Product of exactly four primes, three of which are distinct (p^2*q*r). See A085987);
a(n) = 120 for n = 72, 108, 200, 392, 500, 675, 968, ... (Numbers of the form p^2*q^3, where p,q are distinct primes. See A143610);
It is possible to continue with a(n) = 130, 235, 284, 289, 356, ...

			a(12) = 5 because the divisors of 12 are {1, 2, 3, 4, 6, 12} and GCD(d_i, d_j, d_k) > 1 for the 5 following triples of divisors: (2,4,6), (2,4,12), (2,6,12), (3,6,12) and (4,6,12).

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

Cf. A275387.

with(numtheory):nn:=100:
for n from 1 to nn do:
it:=0:d:=divisors(n):n0:=nops(d):
  for i from 1 to n0-2 do:
   for j from i+1 to n0-1 do:
     for k from j+1 to n0 do:
    if igcd(d[i],d[j],d[k])> 1
       then
       it:=it+1:
       else
      fi:
     od:
     od:
     od:
    printf(`%d, `,it):
   od:

Array[Count[GCD @@ # & /@ Subsets[Divisors[#], {3}], ?(# > 1 &)] &, 81] (* _Michael De Vlieger, Oct 05 2020 *)

a(n) = my(d=divisors(n)); sum(i=1, #d-2, sum (j=i+1, #d-1, sum (k=j+1, #d, gcd([d[i], d[j], d[k]]) > 1))); \\ Michel Marcus, Oct 31 2020

a(n) = {my(f = factor(n), vp = vecprod(f[,1]), d = divisors(vp), res = 0);
for(i = 2, #d, res-=binomial(numdiv(n/d[i]), 3)*(-1)^omega(d[i])); res} \\ David A. Corneth, Nov 01 2020

Name clarified by editors, Oct 31 2020

A381315 Numbers whose prime factorization exponents include exactly one 3 and no exponent greater than 3.

Original entry on oeis.org

8, 24, 27, 40, 54, 56, 72, 88, 104, 108, 120, 125, 135, 136, 152, 168, 184, 189, 200, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 360, 375, 376, 378, 392, 408, 424, 440, 456, 459, 472, 488, 500, 504, 513, 520, 536, 540, 552, 568, 584, 594

Offset: 1

Amiram Eldar, Feb 19 2025

Subsequence of A176297 and A375072, and first differs from them at n = 20: A176297(20) = A375072(20) = 216 = 2^3 * 3^3 is not a term of this sequence.

The asymptotic density of this sequence is (1/zeta(3)) * Sum_{p prime} 1/(p+p^2+p^3) = 0.089602607198058453295... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A046100, A176297, A375072 and A375145.
Subsequences: A030078, A048109, A065036, A143610, A163569, A179670, A179695, A179700, A189975, A189984, A190109, A190378, A190382.
Cf. A002117.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, MemberQ[e, 3] && Count[e, _?(# < 3 &)] == Length[e] - 1]; Select[Range[600], q]

isok(k) = {my(e = factor(k)[, 2]~); select(x -> x > 2, e) == [3];}

A382292 Numbers k such that A382290(k) = 1.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 64, 72, 88, 96, 104, 108, 120, 125, 135, 136, 152, 160, 168, 184, 189, 192, 200, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 320, 328, 343, 344, 351, 352, 360, 375, 376, 378, 392, 408, 416, 424, 432, 440, 448, 456, 459, 472, 480, 486, 488, 500

Offset: 1

Amiram Eldar, Mar 21 2025

First differs from A374590 and A375432 at n = 25: A374590(25) = A375432(25) = 216 is not a term of this sequence.
Numbers k such that A382291(k) = 2, i.e., numbers whose number of infinitary divisors is twice the number of their unitary divisors.
Numbers whose prime factorization has a single exponent that is a sum of two distinct powers of 2 (A018900) and all the other exponents, if they exist, are powers of 2. Equivalently, numbers of the form p^e * m, where p is a prime, e is a term in A018900, and m is a term in A138302 that is coprime to p.
If k is a term then k^2 is also a term. If m is a term in A138302 that is coprime to k then k * m is also a term. The primitive terms, i.e., the terms that cannot be generated from smaller terms using these rules, are the numbers of the form p^(2^i+1), where p is prime and i >= 1.
Analogous to A060687, which is the sequence of numbers k with prime excess A046660(k) = 2.
The asymptotic density of this sequence is A271727 * Sum_{p prime} (((1 - 1/p)/f(1/p)) * Sum_{k>=1} 1/p^A018900(k)) = 0.11919967112489084407..., where f(x) = 1 - x^3 + Sum_{k>=2} (x^(2^k)-x^(2^k+1)).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A018900, A046660, A060687, A271727, A374590, A375432, A382290, A382291.

Subsequences (numbers of the form): A030078 (p^3), A050997 (p^5), A030516 (p^6), A179665 (p^9), A030629 (p^10), A030631 (p^12), A065036 (p^3*q), A178740 (p^5*q), A189987 (p^6*q), A179692 (p^9*q), A143610 (p^2*q^3), A179646 (p^5*q^2), A189990 (p^2*q^6), A179702 (p^4*q^5), A179666 (p^4*q^3), A190464 (p^4*q^6), A163569 (p^3*q^2*r), A189975 (p*q*r^3), A190115 (p^2*q^3*r^4), A381315, A048109.

f[p_, e_] := DigitCount[e, 2, 1] - 1; q[1] = False; q[n_] := Plus @@ f @@@ FactorInteger[n] == 1; Select[Range[500], q]

isok(k) = vecsum(apply(x -> hammingweight(x) - 1, factor(k)[, 2])) == 1;

A203662 Achilles number whose largest proper divisor is also an Achilles number.

Original entry on oeis.org

864, 1944, 3888, 4000, 5400, 6912, 9000, 10584, 10800, 10976, 17496, 18000, 21168, 21600, 24696, 25000, 26136, 30375, 31104, 32000, 34992, 36000, 36504, 42336, 42592, 43200, 48600, 49000, 49392, 50000, 52272, 55296, 62208, 62424, 68600, 69984

Offset: 1

Antonio Roldán, Jan 04 2012

nonn

Exponent of smallest prime divisor of n is greater than or equal to 3.

Both N and the largest proper divisor of N share the same prime factors with different exponents.

			17496 is in the sequence because 17496=2^3*3^7 (Achilles number) and the largest proper divisor 8748=2^2*3^7 is also an Achilles number.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)

Cf. A052486, A143610.

(* First run the program for A052486 to define achillesQ *) Select[Range[50000], achillesQ[#] && achillesQ[Divisors[#][[-2]]] &] (* Alonso del Arte, Jan 05 2012 *)

# uses program in A052486
from itertools import count, islice
from math import gcd
from sympy import factorint
def A203662_gen(): # generator of terms
    def g(x):
        (f:=factorint(x))[min(f)]-=1
        return (x,f.values())
    return map(lambda x:x[0],filter(lambda x:all(d>1 for d in x[1]) and gcd(*x[1])==1,map(g,(A052486(i) for i in count(1)))))
A203662_list = list(islice(A204662_gen(),20)) # Chai Wah Wu, Sep 10 2024

A216426 Numbers of the form a^2*b^3, where a != b and a, b > 1.

Original entry on oeis.org

72, 108, 128, 200, 256, 288, 392, 432, 500, 512, 576, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1600, 1728, 1800, 1944, 2000, 2048, 2187, 2304, 2312, 2592, 2700, 2888, 2916, 3087, 3136, 3200, 3267, 3456, 3528, 3872, 3888, 4000

Offset: 1

V. Raman, Sep 07 2012

nonn

Terms of A216427 that are not 5th powers of squarefree numbers (A113850) and not 10th powers of primes (A030629). - Amiram Eldar, Feb 07 2023

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A143610.
Subsequence of A216427. - Zak Seidov, Jan 03 2014
Cf. A002117, A013661, A013663, A013664, A013668, A030629, A085966, A113850.

With[{upto=4000},Select[Union[Flatten[{#[[1]]^2 #[[2]]^3,#[[2]]^2 #[[1]]^3}& /@ Subsets[Range[2,Surd[upto,2]],{2}]]],#<=upto&]](* Harvey P. Dale, Jan 04 2014 *)
pMx = 25; mx = 2^3 pMx^2; t = Flatten[Table[If[a != b, a^2 b^3, 0], {a, 2, mx^(1/2)}, {b, 2, mx^(1/3)}]]; Union[Select[t, 0 < # <= mx &]] (* T. D. Noe, Jan 02 2014 *)

list(lim)=my(v=List()); for(b=2, sqrtnint(lim\4,3), for(a=2, sqrtint(lim\b^3), if(a!=b, listput(v, a^2*b^3)))); Set(v) \\ Charles R Greathouse IV, Jan 02 2014

from math import isqrt
from sympy import integer_nthroot, mobius, primepi
def A216426(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        j, b, a, d = isqrt(x), integer_nthroot(x,6)[0], integer_nthroot(x,5)[0], integer_nthroot(x,10)[0]
        l, c = 0, n+x-2+primepi(b)+sum(mobius(k)*(j//k**3) for k in range(d+1, b+1))+primepi(d)+sum(mobius(k)*(a//k**2+j//k**3) for k in range(1, d+1))
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = sum(mobius(k)*((k2-1)//k**2) for k in range(1, isqrt(k2-1)+1))
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    return bisection(f,n,n) # Chai Wah Wu, Sep 13 2024

Sum_{n>=1} 1/a(n) = 2 + ((zeta(2)-1)*(zeta(3)-1)-1)/zeta(6) - zeta(5)/zeta(10) - P(6) - P(10) = 0.09117811499514578262..., where P(s) is the prime zeta function. - Amiram Eldar, Feb 07 2023

Name corrected by Charles R Greathouse IV, Jan 02 2014

A272191 Either 8th power of a prime, or product of a square and a cube of two different primes.

Original entry on oeis.org

72, 108, 200, 256, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 2312, 2888, 3087, 3267, 4232, 4563, 5324, 6125, 6561, 6728, 7688, 7803, 8575, 8788, 9747, 10952, 11979, 13448, 14283, 14792, 15125, 17672, 19652, 19773, 21125, 22472, 22707, 25947, 27436, 27848, 29768

Offset: 1

Paolo P. Lava, Apr 22 2016

Numbers such that the sum of the number of divisors of their aliquot parts is four times the number of their divisors.

			72 = 2^3 * 3^2;  256 = 2^8.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A030516, A143610, A179645, A272190.

Cf. A085548, A085541, A085965, A085968.

with(numtheory): P:=proc(q) local a,k,n;  for n from 2 to q do a:=sort([op(divisors(n))]);
if 4*tau(n)= add(tau(a[k]),k=1..nops(a)-1) then print(n); fi; od; end: P(10^7);

Select[Range[30000], MemberQ[{{8}, {2, 3}}, Sort[FactorInteger[#][[;; , 2]]]] &] (* Amiram Eldar, Oct 03 2023 *)

isok(n) = 4*numdiv(n) == sumdiv(n, d, (n!=d)*numdiv(d)); \\ Michel Marcus, Apr 22 2016

is(n) = {my(e = vecsort(factor(n)[, 2])~); e == [8] || e == [2, 3];} \\ Amiram Eldar, Oct 03 2023

Sum_{n>=1} 1/a(n) = P(2)*P(3) - P(5) + P(8) = A085548 * A085541 - A085965 + A085968 = 0.047342..., where P is the prime zeta function. - Amiram Eldar, Oct 03 2023

A275345 Characteristic polynomials of a square matrix based on A051731 where A051731(1,N)=1 and A051731(N,N)=0 and where N=size of matrix, analogous to the Redheffer matrix.

Original entry on oeis.org

1, 1, -1, -1, -1, 1, -1, 0, 2, -1, 0, 0, 2, -3, 1, -1, 2, 1, -5, 4, -1, 1, -3, 5, -8, 9, -5, 1, -1, 4, -4, -5, 15, -14, 6, -1, 0, -1, 6, -17, 29, -31, 20, -7, 1, 0, 0, 2, -13, 36, -55, 50, -27, 8, -1, 1, -7, 23, -50, 84, -112, 112, -78, 35, -9, 1

Offset: 0

Mats Granvik, Jul 24 2016

From Mats Granvik, Sep 30 2017: (Start)
Conjecture: The largest absolute value of the eigenvalues of these characteristic polynomials appear to have the same prime signature in the factorization of the matrix sizes N.
In other words: Let b(N) equal the sequence of the largest absolute values of the eigenvalues of the characteristic polynomials of the matrices of size N. b(N) is then a sequence of truncated eigenvalues starting:
b(N=1..infinity)
= 1.00000, 1.61803, 1.61803, 2.00000, 1.61803, 2.20557, 1.61803, 2.32472, 2.00000, 2.20557, 1.61803, 2.67170, 1.61803, 2.20557, 2.20557, 2.61803, 1.61803, 2.67170, 1.61803, 2.67170, 2.20557, 2.20557, 1.61803, 3.08032, 2.00000, 2.20557, 2.32472, 2.67170, 1.61803, 2.93796, 1.61803, 2.89055, 2.20557, 2.20557, 2.20557, 3.21878, 1.61803, 2.20557, 2.20557, 3.08032, 1.61803, 2.93796, 1.61803, 2.67170, 2.67170, 2.20557, 1.61803, 3.45341, 2.00000, 2.67170, 2.20557, 2.67170, 1.61803, 3.08032, 2.20557, 3.08032, 2.20557, 2.20557, 1.61803, 3.53392, 1.61803, 2.20557, 2.67170, ...
It then appears that for n = 1,2,3,4,5,...,infinity we have the table:
Prime signature: b(Axxxxxx(n)) = Largest abs(eigenvalue):
p^0      : b(1)          = 1.0000000000000000000000000000...
p        : b(A000040(n)) = 1.6180339887498949025257388711...
p^2      : b(A001248(n)) = 2.0000000000000000000000000000...
p*q      : b(A006881(n)) = 2.2055694304005917238953315973...
p^3      : b(A030078(n)) = 2.3247179572447480566665944934...
p^2*q    : b(A054753(n)) = 2.6716998816571604358216518448...
p^4      : b(A030514(n)) = 2.6180339887498917939012699207...
p^3*q    : b(A065036(n)) = 3.0803227214906021558249449299...
p*q*r    : b(A007304(n)) = 2.9379558827528557962693867011...
p^5      : b(A050997(n)) = 2.8905508875432590620846440288...
p^2*q^2  : b(A085986(n)) = 3.2187765853016649941764626419...
p^4*q    : b(A178739(n)) = 3.4534111136673804054453285061...
p^2*q*r  : b(A085987(n)) = 3.5339198574905377192578725953...
p^6      : b(A030516(n)) = 3.1478990357047909043330946587...
p^3*q^2  : b(A143610(n)) = 3.7022736187975437971431347250...
p^5*q    : b(A178740(n)) = 3.8016448153137023524550386355...
p^3*q*r  : b(A189975(n)) = 4.0600260453688532535920785448...
p^7      : b(A092759(n)) = 3.3935083220984414431597997463...
p^4*q^2  : b(A189988(n)) = 4.1453038440113498808159420150...
p^2*q^2*r: b(A179643(n)) = 4.2413382309993874486053755390...
p^6*q    : b(A189987(n)) = 4.1311805192254587026923218218...
p*q*r*s  : b(A046386(n)) = 3.8825338629275134572083061357...
...
b(Axxxxxx(1)) in the sequences above, is given by A025487.
(End)
First column in the coefficients of the characteristic polynomials is the Möbius function A008683.
Row sums of coefficients start: 0, -1, 0, 0, 0, 0, 0, 0, 0, ...
Third diagonal is a signed version of A000096.
Most of the eigenvalues are equal to 1. The number of eigenvalues equal to 1 are given by A075795 for n>1.
The first three of the eigenvalues above can be calculated as nested radicals. The fourth eigenvalue 2.205569430400590... minus 1 = 1.205569430400590... is also a nested radical.

			{
{ 1},
{ 1, -1},
{-1, -1,  1},
{-1,  0,  2,  -1},
{ 0,  0,  2,  -3,  1},
{-1,  2,  1,  -5,  4,   -1},
{ 1, -3,  5,  -8,  9,   -5,   1},
{-1,  4, -4,  -5, 15,  -14,   6,  -1},
{ 0, -1,  6, -17, 29,  -31,  20,  -7,  1},
{ 0,  0,  2, -13, 36,  -55,  50, -27,  8, -1},
{ 1, -7, 23, -50, 84, -112, 112, -78, 35, -9, 1}
}

Mats Granvik, Mathematica program to verify that eigenvalues determine prime signature
OEIS Wiki, Prime signatures
Eric Weisstein, Prime signature
Index to sequences related to prime signature

Cf. A051731, A008683, A000040, A001248, A006881, A030078, A030514, A054753, A000096, A001622, A272874, A075795.

Cf. A025487. - Mats Granvik, Sep 30 2017

Clear[x, AA, nn, s]; Monitor[AA = Flatten[Table[A = Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, nn}], {n, 1, nn}]; MatrixForm[A]; a = A[[1, nn]]; A[[1, nn]] = A[[nn, nn]]; A[[nn, nn]] = a; CoefficientList[CharacteristicPolynomial[A, x], x], {nn, 1, 10}]], nn]

A376249 Numbers that are not prime powers and have a unique largest prime exponent that is larger than the second-largest prime exponent by 1.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 108, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 171, 172, 175, 188, 198, 200, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 260, 261, 268, 275, 276, 279, 284, 292, 294

Offset: 1

Amiram Eldar, Sep 16 2024

First differs from A325241 at n = 36: A325241(36) = 2^2 * 3^2 * 5 is not a term of this sequence. Also, a(71) = 360 = 2^3 * 3^2 * 5 is the least term that is not a term of A325241.
Numbers whose unordered prime signature (i.e., sorted, see A118914) ends with two consecutive integers: {..., k, k+1} for some k >= 1.
The asymptotic density of this sequence is Sum_{k >= 1, p prime} (d(k+1, p) - d(k, p))/p^(k+1) = 0.21831645263800520483..., where d(k, p) = 0 for k = 1, and (1-1/p)/((1-1/p^k)*zeta(k)) for k > 1, is the density of terms that have in their prime factorization a prime p with the largest exponent that is > k.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A356862.
Subsequences: A054753, A072357, A095990, A096156, A143610, A163569, A179666, A179702.
Cf. A118914, A246655, A325241.

q[k_] := Module[{e = Sort[FactorInteger[k][[;; , 2]]]}, Length[e] > 1 && e[[-1]] == e[[-2]] + 1]; Select[Range[300], q]

is(k) = {my(e = vecsort(factor(k)[, 2])); #e > 1 && e[#e] == e[#e-1] + 1;}

A377844 Numbers that have a single odd exponent larger than 1 in their prime factorization.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 432, 440, 456, 459, 472, 480, 486, 488, 500

Offset: 1

Amiram Eldar, Nov 09 2024

First differs from A295661, A325990 and A376142 at n = 24: A295661(24) = A325990(24) = A376142(24) = 216 = 2^3 * 3^3 is not a term of this sequence.
Differs from A060476 by having the terms 432, 648, 1728, ..., and not having the terms 1, 216, 256, 768, 864, ... .
The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^2*(p+1))) * Sum_{p prime} (1/(p^3+p^2-1)) = 0.11498368544519741081... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A295661.
Subsequences: A065036, A143610, A163569.
Cf. A060476, A065465, A325990, A376142, A377845.

q[n_] := Count[FactorInteger[n][[;; , 2]], _?(# > 1 && OddQ[#] &)] == 1; Select[Range[500], q]

is(k) = #select(x -> x>1 && x%2, factor(k)[, 2]) == 1;

cp's OEIS Frontend

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A382292 Numbers k such that A382290(k) = 1.

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A203662 Achilles number whose largest proper divisor is also an Achilles number.

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A216426 Numbers of the form a^2*b^3, where a != b and a, b > 1.

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A272191 Either 8th power of a prime, or product of a square and a cube of two different primes.

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