A378768 -id:A378768 - cp's OEIS frontend

A378900 Squares of numbers divisible by the squares of two distinct primes.

1296, 5184, 10000, 11664, 20736, 32400, 38416, 40000, 46656, 50625, 63504, 82944, 90000, 104976, 129600, 153664, 156816, 160000, 186624, 194481, 202500, 219024, 234256, 250000, 254016, 291600, 331776, 345744, 360000, 374544, 419904, 455625, 456976, 467856, 490000

Offset: 1

Michael De Vlieger, Dec 12 2024

Also, the squares in A376936.
Proper subset of A378767, in turn a proper subset of A286708, the intersection of A001694 and A024619.
Numbers that have 3 kinds of coreful divisor pairs (d, k/d), d | k, i.e., rad(d) = rad(k/d) = rad(k) where rad = A007947. These kinds are described as follows:
Type A: d = k/d, which pertain to square k (in A000290).
Type B: d | k/d, d < k/d, which pertain to k in A320966, powerful numbers divisible by a cube.
Type C: neither d | k/d nor k/d | d, which pertain to k in A376936.
Since divisors d, k/d may either divide or not divide the other, there are no other cases.
In addition the following kinds of divisor pairs are also seen:
Type D: (d, k/d) such that d | k/d but there exists a factor Q | k/d that does not divide d. Then omega(d) < omega(k/d) = omega(k).
Type E: Nontrivial unitary divisor pairs (d, k/d) such that gcd(d, k/d) = 1, d > 1, k/d > 1. Let prime power factor p^m | k be such that m is maximized. Then set d = p^m and it is clear that for any k in A024619, there exists at least 1 nontrivial unitary divisor pair.

			Let b = A036785.
Table of the first 12 terms of this sequence, showing examples of types A, B, and C of coreful pairs of divisors.
   n    a(n)   Factors of a(n)    b(n)   Type B       Type C
  -------------------------------------------------------------
   1    1296   2^4  * 3^4          36    6 * 216      24 * 54
   2    5184   2^6  * 3^4          72    6 * 864      48 * 108
   3   10000   2^4  * 5^4         100   10 * 1000     40 * 250
   4   11664   2^4  * 3^6         108    6 * 1944     24 * 486
   5   20736   2^8  * 3^4         144    6 * 3456     54 * 384
   6   32400   2^4  * 3^4 * 5^2   180   30 * 1080    120 * 270
   7   38416   2^4  * 7^4         196   14 * 2744     56 * 686
   8   40000   2^6  * 5^4         200   10 * 4000     80 * 500
   9   46656   2^6  * 3^6         216    6 * 7776     48 * 972
  10   50625   3^4  * 5^4         225   15 * 3375    135 * 375
  11   63504   2^4  * 3^4 * 7^2   252   42 * 1512    168 * 378
  12   82944   2^10 * 3^4         288    6 * 13824    54 * 1536

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Listing of select divisor pairs of a(n), n = 1..12, showing divisor pairs of type A in light gray, type B in orange and purple, and type C in black.

Cf. A000290, A001694, A007947, A024619, A036785, A126706, A286708, A320966, A376936, A378768.

Cf. A013661, A082020.

s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}],   IntegerQ@ Sqrt[#] &] &[500000];
Union@ Select[s, Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]

a(n) = A036785(n)^2.

Sum_{n>=1} 1/a(n) = Pi^2/6 - (15/Pi^2) * (1 + Sum_{p prime} 1/(p^4-1)) = 0.0015294876575980711757... . - Amiram Eldar, Dec 21 2024

A380857 Squares of numbers that are neither squarefree nor prime powers.

Original entry on oeis.org

144, 324, 400, 576, 784, 1296, 1600, 1936, 2025, 2304, 2500, 2704, 2916, 3136, 3600, 3969, 4624, 5184, 5625, 5776, 6400, 7056, 7744, 8100, 8464, 9216, 9604, 9801, 10000, 10816, 11664, 12544, 13456, 13689, 14400, 15376, 15876, 17424, 18225, 18496, 19600, 20736

Offset: 1

Michael De Vlieger, Feb 06 2025

Proper subset of A359280 which is a proper subset of A286708 (powerful numbers that are not prime powers, a proper subset of A126706).

Does not intersect A362605.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A059404, A126706, A177492 (k^2 for k in A120944), A286708, A359280, A362605, A378768 (k^2 for k in A286708).

Cf. A013661, A082020, A085548, A154945.

Select[Range[150], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]^2

isok(k) = !issquarefree(k) && !isprimepower(k); \\ A126706
apply(sqr, select(isok, [1..200])) \\ Michel Marcus, Feb 07 2025

from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A380857(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 int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    return bisection(f,n,n)**2 # Chai Wah Wu, Feb 08 2025

a(n) = A126706(n)^2.

Sum_{n>=1} 1/a(n) = Pi^2/6 - 15/Pi^2 - Sum_{p prime} 1/(p^2*(p^2-1)) = A013661 - A082020 + A085548 - A154945 = 0.025670434597226178881... . - Amiram Eldar, Feb 08 2025

cp's OEIS Frontend

A378900 Squares of numbers divisible by the squares of two distinct primes.

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