A162142 -id:A162142 - cp's OEIS frontend

A377845 Numbers that have more than one odd exponent larger than 1 in their prime factorization.

216, 864, 1000, 1080, 1512, 1944, 2376, 2744, 2808, 3000, 3375, 3456, 3672, 4000, 4104, 4320, 4968, 5400, 6048, 6264, 6696, 6750, 7000, 7560, 7776, 7992, 8232, 8856, 9000, 9261, 9288, 9504, 9720, 10152, 10584, 10648, 10976, 11000, 11232, 11448, 11880, 12000, 12744, 13000

Offset: 1

Amiram Eldar, Nov 09 2024

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p^2*(p+1))) * (1 + Sum_{p prime} (1/(p^3+p^2-1))) = 0.0035024748296318122535... .

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

Complement of the union of A335275 and A377844.
Subsequence of A295661.
Subsequences: A162142, A179671, A190011.
Cf. A065465.

q[n_] := Count[FactorInteger[n][[;; , 2]], _?(# > 1 && OddQ[#] &)] > 1; Select[Range[13000], q]

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

A177493 Products of cubes of 2 or more distinct primes.

Original entry on oeis.org

216, 1000, 2744, 3375, 9261, 10648, 17576, 27000, 35937, 39304, 42875, 54872, 59319, 74088, 97336, 132651, 166375, 185193, 195112, 238328, 274625, 287496, 328509, 343000, 405224, 456533, 474552, 551368, 614125, 636056, 658503, 753571, 804357, 830584, 857375

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 10 2010

nonn

			216 = 2^3 * 3^3.
9261 = 3^3 * 7^3.
27000 = 2^3 * 3^3 * 5^3.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000469, A085986, A162142, A177492.

q:= n-> not isprime(n) and numtheory[issqrfree](n):
map(x-> x^3, select(q, [$4..120]))[];  # Alois P. Heinz, Aug 02 2024

f1[n_]:=Length[Last/@FactorInteger[n]]; f2[n_]:=Union[Last/@FactorInteger[n]]; lst={};Do[If[f1[n]>1&&f2[n]=={3},AppendTo[lst,n]],{n,0,9!}];lst
Reap[Do[{p, e}=Transpose[FactorInteger[n]]; If[Length[p]>1 && Union[e]=={3}, Sow[n]], {n, 343000}]][[2, 1]]

[k^3 | k<-[1..100], k>1 && !isprime(k) && issquarefree(k)] \\ Andrew Howroyd, Jan 14 2020

from math import isqrt
from sympy import primepi, mobius
def A177493(n):
    def f(x): return n+1+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n+1, f(n+1)
    while m != k:
        m, k = k, f(k)
    return m**3 # Chai Wah Wu, Aug 02 2024

a(n) = A120944(n)^3. - R. J. Mathar, Dec 06 2010

Definition corrected by R. J. Mathar, Dec 06 2010

Terms a(25) and beyond from Andrew Howroyd, Jan 14 2020

A355462 Powerful numbers divisible by exactly 2 distinct primes.

Original entry on oeis.org

36, 72, 100, 108, 144, 196, 200, 216, 225, 288, 324, 392, 400, 432, 441, 484, 500, 576, 648, 675, 676, 784, 800, 864, 968, 972, 1000, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1352, 1372, 1444, 1521, 1568, 1600, 1728, 1936, 1944, 2000, 2025, 2116, 2304, 2312, 2500

Offset: 1

Amiram Eldar, Jul 03 2022

nonn

First differs from A286708 at n = 25.
Number of the form p^i * q^j, where p != q are primes and i,j > 1.
Numbers k such that A001221(k) = 2 and A051904(k) >= 2.
The possible values of the number of the divisors (A000005) of terms in this sequence is any composite number that is not 8 or twice a prime (A264828 \ {1, 8}).
675 = 3^3*5^2 and 676 = 2^2*13^2 are 2 consecutive integers in this sequence. There are no other such pairs below 10^22 (the lesser members of such pairs are terms of A060355).

			36 is a term since 36 = 2^2 * 3^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index to sequences related to prime signature.

Intersection of A001694 and A007774.
Subsequence of A286708.
Subsequences: A085986, A143610, A162142, A179646, A179666, A179671, A179689, A179694, A179699, A179702, A179705, A189988, A189990, A189991, A190464, A190465, A303661.
Cf. A000005, A001221, A051904, A060355, A136141, A264828.

Select[Range[2500], Length[(e = FactorInteger[#][[;; , 2]])] == 2 && Min[e] > 1 &]

is(n) = {my(f=factor(n)); #f~ == 2 && vecmin(f[,2]) > 1};

Sum_{n>=1} 1/a(n) = ((Sum_{p prime} (1/(p*(p-1))))^2 - Sum_{p prime} (1/(p^2*(p-1)^2)))/2 = 0.1583860791... .

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A377845 Numbers that have more than one odd exponent larger than 1 in their prime factorization.

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A177493 Products of cubes of 2 or more distinct primes.

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A355462 Powerful numbers divisible by exactly 2 distinct primes.

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