A065036 -id:A065036 - cp's OEIS frontend

A267891 Numbers with 8 odd divisors.

105, 135, 165, 189, 195, 210, 231, 255, 270, 273, 285, 297, 330, 345, 351, 357, 375, 378, 385, 390, 399, 420, 429, 435, 455, 459, 462, 465, 483, 510, 513, 540, 546, 555, 561, 570, 594, 595, 609, 615, 621, 627, 645, 651, 660, 663, 665, 690, 702, 705, 714, 715, 741, 750, 756, 759, 770, 777, 780, 783, 795, 798, 805, 837

Offset: 1

Omar E. Pol, Apr 03 2016

nonn

Positive integers that have exactly eight odd divisors.
Numbers n such that the symmetric representation of sigma(n) has 8 subparts. - Omar E. Pol, Dec 29 2016
Numbers n such that A000265(n) has prime signature {7} or {3,1} or {1,1,1}, i.e., is in A092759 or A065036 or A007304. - Robert Israel, Mar 15 2018
Numbers that can be formed in exactly 7 ways by summing sequences of 2 or more consecutive positive integers. - Julie Jones, Aug 13 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Column 8 of A266531.
Cf. A000265, A001227, A007304, A038547, A065036, A092759, A237593, A279387.
Numbers with exactly k odd divisors (k = 1..10): A000079, A038550, A072502, apparently A131651, A267696, A230577, A267697, this sequence, A267892, A267893.

[n: n in [1..1000] | #[d: d in Divisors(n) | IsOdd(d)] eq 8]; // Bruno Berselli, Apr 04 2016

filter:= proc(n) local r;
  r:= n/2^padic:-ordp(n,2);
  numtheory:-tau(r)=8
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 15 2018

Select[Range@ 840, Length@ Select[Divisors@ #, OddQ] == 8 &] (* Michael De Vlieger, Dec 30 2016 *)

isok(n) = sumdiv(n, d, (d%2)) == 8; \\ after Michel Marcus

A001227(a(n)) = 8.

A179694 Numbers of the form p^6*q^3 where p and q are distinct primes.

Original entry on oeis.org

1728, 5832, 8000, 21952, 85184, 91125, 125000, 140608, 250047, 314432, 421875, 438976, 778688, 941192, 970299, 1560896, 1601613, 1906624, 3176523, 3241792, 3581577, 4410944, 5000211, 5088448, 5359375, 6644672

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures

Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688, A179689, A179690, A179691, A179692, A179693.

Cf. A085541, A085966, A085969.

f[n_]:=Sort[Last/@FactorInteger[n]]=={3,6}; Select[Range[10^6], f]

list(lim)=my(v=List(),t);forprime(p=2, (lim\8)^(1/6), t=p^6;forprime(q=2, (lim\t)^(1/3), if(p==q, next);listput(v,t*q^3))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

from sympy import primepi, integer_nthroot, primerange
def A179694(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(integer_nthroot(x//p**6,3)[0]) for p in primerange(integer_nthroot(x,6)[0]+1))+primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Sum_{n>=1} 1/a(n) = P(3)*P(6) - P(9) = A085541 * A085966 - A085969 = 0.000978..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

a(n) = A054753(n)^3. - R. J. Mathar, May 05 2023

A179702 Numbers of the form p^4*q^5 where p and q are two distinct primes.

Original entry on oeis.org

2592, 3888, 20000, 50000, 76832, 151875, 253125, 268912, 468512, 583443, 913952, 1361367, 2576816, 2672672, 3557763, 4170272, 5940688, 6940323, 7503125, 8954912, 10504375, 13045131, 20295603, 22632992, 22717712, 29552672, 30074733

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

Subsequence of A046312 and of A137493. - R. J. Mathar, Jul 27 2010

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, List of Prime Signatures

Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688, A179689, A179690, A179691, A179692, A179693, A179694, A179695, A179696, A179698, A179699, A179700.

Cf. A085964, A085965, A085969.

fQ[n_] := Sort[Last /@ FactorInteger @n] == {4, 5}; Select[ Range@ 31668000, fQ] (* fixed by Robert G. Wilson v, Aug 26 2010 *)
lst = {}; Do[ If[p != q, AppendTo[lst, Prime@p^4*Prime@q^5]], {p, 12}, {q, 10}]; Take[ Sort@ Flatten@ lst, 27] (* Robert G. Wilson v, Aug 26 2010 *)
Take[Union[First[#]^4 Last[#]^5&/@Flatten[Permutations/@Subsets[ Prime[ Range[30]],{2}],1]],30] (* Harvey P. Dale, Jan 01 2012 *)

list(lim)=my(v=List(),t);forprime(p=2, (lim\16)^(1/5), t=p^5;forprime(q=2, (lim\t)^(1/4), if(p==q, next);listput(v,t*q^4))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from sympy import primepi, integer_nthroot, primerange
def A179702(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(integer_nthroot(x//p**5,4)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Sum_{n>=1} 1/a(n) = P(4)*P(5) - P(9) = A085964 * A085965 - A085969 = 0.000748..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

Edited and extended by Ray Chandler and R. J. Mathar, Jul 26 2010

A275387 Numbers of ordered pairs of divisors d < e of n such that gcd(d, e) > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 8, 0, 2, 2, 6, 0, 8, 0, 8, 2, 2, 0, 18, 1, 2, 3, 8, 0, 15, 0, 10, 2, 2, 2, 24, 0, 2, 2, 18, 0, 15, 0, 8, 8, 2, 0, 32, 1, 8, 2, 8, 0, 18, 2, 18, 2, 2, 0, 44, 0, 2, 8, 15, 2, 15, 0, 8, 2, 15, 0, 49, 0, 2, 8, 8, 2, 15, 0, 32, 6, 2

Offset: 1

Michel Lagneau, Aug 03 2016

nonn

Number of elements in the set {(x, y): x|n, y|n, x < y, gcd(x, y) > 1}.
Every element of the sequence is repeated indefinitely, for instance:
a(n)=0 if n prime;
a(n)=1 if n = p^2 for p prime (A001248);
a(n)=2 if n is a squarefree semiprime (A006881);
a(n)=3 if n = p^3 for p prime (A030078);
a(n)=6 if n = p^4 for p prime (A030514);
a(n)=8 if n is a number which is the product of a prime and the square of a different prime (A054753);
a(n)=10 if n = p^5 for p prime (A050997);
a(n)=15 if n is in the set {A007304} union {64} = {30, 42, 64, 66, 70,...} = {Sphenic numbers} union {64};
a(n)=18 if n is the product of the cube of a prime (A030078) and a different prime (see A065036);
a(n)=21 if n = p^7 for p prime (A092759);
a(n)=24 if n is square of a squarefree semiprime (A085986);
a(n)=32 if n is the product of the 4th power of a prime (A030514) and a different prime (see A178739);
a(n)=36 if n = p^9 for p prime (A179665);
a(n)=44 if n is the product of exactly four primes, three of which are distinct (A085987);
a(n)=45 if n is a number with 11 divisors (A030629);
a(n)=49 if n is of the form p^2*q^3, where p,q are distinct primes (A143610);
a(n)=50 if n is the product of the 5th power of a prime (A050997) and a different prime (see A178740);
a(n)=55 if n if n = p^11 for p prime(A079395);
a(n)=72 if n is a number with 14 divisors (A030632);
a(n)=80 if n is the product of four distinct primes (A046386);
a(n)=83 if n is a number with 15 divisors (A030633);
a(n)=89 if n is a number with prime factorization pqr^3 (A189975);
a(n)=96 if n is a number that are the cube of a product of two distinct primes (A162142);
a(n)=98 if n is the product of the 7th power of a prime and a distinct prime (p^7*q) (A179664);
a(n)=116 if n is the product of exactly 2 distinct squares of primes and a different prime (p^2*q^2*r) (A179643);
a(n)=126 if n is the product of the 5th power of a prime and different distinct prime of the 2nd power (p^5*q^2) (A179646);
a(n)=128 if n is the product of the 8th power of a prime and a distinct prime (p^8*q) (A179668);
a(n)=150 if n is the product of the 4th power of a prime and 2 different distinct primes (p^4*q*r) (A179644);
a(n)=159 if n is the product of the 4th power of a prime and a distinct prime of power 3 (p^4*q^3) (A179666).
It is possible to continue with a(n) = 162, 178, 209, 224, 227, 238, 239, 260, 289, 309, 320, 333,...

			a(12) = 8 because the divisors of 12 are {1, 2, 3, 4, 6, 12} and GCD(d_i, d_j)>1 for the 8 following pairs of divisors: (2,4), (2,6), (2,12), (3,6), (3,12), (4,6), (4,12) and (6,12).

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

Cf. A001248, A006881, A007304, A030078, A030514, A030632, A046386, A050997, A054753, A063647, A065036, A066446, A079395, A085986, A085987, A092759, A143610, A162142, A178739, A178740, A179644, A179646, A179664, A189975.

Cf. A333976 (same with d <= e).

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

Table[Sum[Sum[(1 - KroneckerDelta[GCD[i, k], 1]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}], {n, 100}] (* Wesley Ivan Hurt, Jan 01 2021 *)

a(n)=my(d=divisors(n)); sum(i=2,#d, sum(j=1,i-1, gcd(d[i],d[j])>1)) \\ Charles R Greathouse IV, Aug 03 2016

a(n)=my(f=factor(n)[,2],t=prod(i=1,#f,f[i]+1)); t*(t-1)/2 - (prod(i=1,#f,2*f[i]+1)+1)/2 \\ Charles R Greathouse IV, Aug 03 2016

a(n) = A066446(n) - A063647(n).

a(n) = Sum_{d1|n, d2|n, d1Wesley Ivan Hurt, Jan 01 2021

A215173 Numbers k such that k and k+1 are both of the form p*q^3 where p and q are distinct primes.

Original entry on oeis.org

135, 296, 375, 1431, 1592, 3992, 4023, 6183, 7624, 8936, 9368, 10071, 10232, 10375, 10984, 13256, 16551, 16712, 19143, 20871, 22328, 22375, 23031, 24488, 28375, 28376, 28647, 33271, 34856, 35127, 40311, 40472, 41336, 43767, 46791, 49624, 50408, 52375, 53271

Offset: 1

Michel Lagneau, Aug 05 2012

nonn

Intersection of A065036 and A065036 - 1. - Robert Israel, Jun 15 2014

			135 is a member as 135 = 5*3^3 and 136 = 17*2^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2048 from Robert Israel)

Cf. A074172, A065036.

with(numtheory):for n from 1 to 55000 do:x:=factorset(n):y:=factorset(n+1):x2:=sqrt(n):y2:=sqrt(n+1):n1:=nops(x):n2:=nops(y):if n1=2 and n2=2 and bigomega(n) = 4 and bigomega(n+1) = 4 and x2<>floor(x2) and y2<>floor(y2) then printf("%a, ", n):else fi:od:
# Alternative:
N:= 10^5: # to get all terms < N
P1:= select(isprime,{2,seq(2*i+1,i=1..floor(N/16))}):
P2:= select(t -> t^3 <= N/2,P1):
B:= {seq(seq(p^3*q,q=select(`<`,P1,floor(N/p^3)) minus {p}),p=P2)}:
B intersect map(`-`,B,1); # Robert Israel, Jun 15 2014

lst={}; Do[f1=FactorInteger[n]; If[Sort[Transpose[f1][[2]]]=={1, 3}, f2=FactorInteger[n+1]; If[Sort[Transpose[f2][[2]]]=={1, 3}, AppendTo[lst, n]]], {n, 3, 55000}]; lst

A331593 Numbers k that have the same number of distinct prime factors as A225546(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 28, 29, 31, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 121, 124, 127, 131, 135, 136, 137, 139, 144, 147, 148, 149

Offset: 1

Antti Karttunen and Peter Munn, Jan 21 2020

nonn

Numbers k for which A001221(k) = A331591(k).
Numbers k that have the same number of terms in their factorization into powers of distinct primes as in their factorization into powers of squarefree numbers with distinct exponents that are powers of 2. See A329332 for a description of the relationship between the two factorizations and A225546.
If k is included, then all such x that A046523(x) = k are also included, i.e., all numbers with the same prime signature as k. Notably, primes (A000040) are included, but squarefree semiprimes (A006881) are not.
k^2 is included if and only if k is included, for example A001248 is included, but A085986 is not.

			There are 2 terms in the factorization of 36 into powers of distinct primes, which is 36 = 2^2 * 3^2 = 4 * 9; but only 1 term in its factorization into powers of squarefree numbers with distinct exponents that are powers of 2, which is 36 = 6^(2^1). So 36 is not included.
There are 2 terms in the factorization of 40 into powers of distinct primes, which is 40 = 2^3 * 5^1 = 8 * 5; and also 2 terms in its factorization into powers of squarefree numbers with distinct exponents that are powers of 2, which is 40 = 10^(2^0) * 2^(2^1) = 10 * 4. So 40 is included.

Cf. A000120, A046523, A225546, A267116, A329332.
Sequences with related definitions: A001221, A331591, A331592.
Subsequences: A000040, A001248, A050376, A054753, A065036, A162142, A225547.
Subsequences of complement: A006881, A056824, A085986, A120944, A177492.

Select[Range@ 150, Equal @@ PrimeNu@ {#, If[# == 1, 1, Apply[Times, Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]]} &] (* Michael De Vlieger, Jan 26 2020 *)

A331591(n) = if(1==n,0,my(f=factor(n),u=#binary(vecmax(f[, 2])),xs=vector(u),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),xs[i]++)); m<<=1); #select(x -> (x>0),xs));
k=0; n=0; while(k<105, n++; if(omega(n)==A331591(n), k++; print1(n,", ")));

{a(n)} = {k : A001221(k) = A000120(A267116(k))}.

A381312 Numbers whose powerful part (A057521) is a power of a prime with an odd exponent >= 3 (A056824).

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520, 536, 544

Offset: 1

Amiram Eldar, Feb 19 2025

Subsequence of A301517 and A374459 and first differs from them at n = 21. A301517(21) = A374459(21) = 216 is not a term of this sequence.
Numbers having exactly one non-unitary prime factor and its multiplicity is odd.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m+1} with m >= 1, i.e., any number (including zero) of 1's and then a single odd number > 1.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/((p-1)*(p+1)^2) = 0.093382464285953613312...

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

Complement of A381311 within A190641.
Cf. A057521, A118914.
Subsequence of A268335, A301517 and A374459.
Subsequences: A030078, A050997, A056824, A065036, A079395, A092759, A138031, A178740, A179664, A179665, A179667, A179670, A179692, A179696, A179704, A189975, A189984, A190378, A190383, A190473.

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, e[[1]] > 1 && OddQ[e[[1]]] && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000], q]

isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); e[1] % 2 && e[1] > 1 && (#e == 1 || e[2] == 1));

A317534 Numbers k such that the poset of factorizations of k, ordered by refinement, is not a lattice.

Original entry on oeis.org

24, 32, 40, 48, 54, 56, 60, 64, 72, 80, 84, 88, 90, 96, 104, 108, 112, 120, 126, 128, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 198, 200, 204, 208, 216, 220, 224, 228, 232, 234, 240, 243, 248, 250, 252, 256, 260, 264, 270

Offset: 1

Gus Wiseman, Jul 30 2018

nonn

Includes 2^k for all k > 4.

Conjecture: Let S be the set of all numbers whose prime signature is either {1,3}, {5}, or {1,1,2}. Then the sequence consists of all multiples of elements of S. - David A. Corneth, Jul 31 2018.

			In the poset of factorizations of 24, the factorizations (2*2*6) and (2*3*4) have two least-upper bounds, namely (2*12) and (4*6), so this poset is not a lattice.

R. P Stanley, Enumerative Combinatorics Vol. 1, Sec. 3.3.

Wikipedia, Lattice (order)

Cf. A001055, A007716, A025487, A045778, A065036, A162247, A265947, A281113, A317142, A317144, A317145, A317146.

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];}

cp's OEIS Frontend

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