A363084 Numbers k such that sqrt(A007947(k) - A007913(k)) is an integer m > 0.

4, 16, 18, 25, 64, 72, 100, 162, 180, 256, 288, 289, 294, 400, 507, 625, 648, 676, 720, 722, 1024, 1152, 1176, 1210, 1369, 1458, 1600, 1620, 2178, 2205, 2500, 2548, 2592, 2646, 2704, 2880, 2888, 3150, 4096, 4225, 4500, 4563, 4608, 4704, 4840, 5202, 5832, 5887

Offset: 1

Author

Michael De Vlieger, Sep 05 2023

Keywords

• nonn

Comments

Let core(k) = A007913(k) and rad(k) = A007947(k).

Squarefree numbers k imply rad(k) - core(k) = k - k = 0.

Perfect squares k^2 such that rad(k) = m^2+1 and k > 1 imply rad(k^2) - core(k^2) = (m^2+1) - 1 = m^2, with integers k, m.

Generally, if there exists a minimal d such that d | k, k/d = m^2, and rad(k) - d = m^2, then k is in the sequence.

Subsets of this sequence include the sets of squares k^2 such that k is in A002496, A003592, and A089653, since A089653 contains both A002496 and A003592.

Examples

			a(1) = 4 since rad(4) = 1+1; rad(4) - core(4) = 2 - 1 = 1, a nonzero square.
a(2) = 18 since 18/2 = 9, and rad(18) - core(18) = 6 - 2 = 4, a nonzero square, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..3170 (all terms less than 2^28.)

Crossrefs

Cf. A002496, A003592, A007913, A007947, A089653, A365517.

Programs

Mathematica

Select[Range[6000], And[IntegerQ[#], # > 0] &[Sqrt[Times @@ FactorInteger[#][[All, 1]] - (Sqrt[#] /. (c_ : 1)*a_^(b_ : 0) :> (c*a^b)^2)] ] &]

PARI

isok(k) = my(s=factorback(factorint(k)[, 1])-core(k)); (s>0) && issquare(s); \\ Michel Marcus, Sep 18 2023

Python

from itertools import count, islice
from sympy.ntheory.primetest import is_square
from sympy import factorint
def A363084_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue,1)):
        a, b = 1, 1
        for p, e in factorint(k).items():
            if e&1:
                a *= p
            else:
                b *= p
        if b>1 and is_square(a*(b-1)):
            yield k
A363084_list = list(islice(A363084_gen(),30)) # Chai Wah Wu, Sep 19 2023