A365517 Numbers k such that the sum of the squarefree part of k and the squarefree kernel of k is a perfect square.

2, 8, 9, 12, 32, 48, 81, 98, 108, 128, 150, 192, 225, 252, 363, 392, 432, 512, 578, 600, 729, 768, 972, 1008, 1100, 1225, 1350, 1568, 1728, 1805, 1922, 2025, 2028, 2048, 2268, 2312, 2366, 2400, 2940, 3072, 3174, 3267, 3750, 3888, 4032, 4400, 4802, 5400, 5625

Offset: 1

Author

David James Sycamore, Sep 07 2023

Keywords

• nonn

Comments

In other words, numbers k such that A007913(k) + A007947(k) is in A000290 (the sum of the core and the kernel (radical) of k is a square).

The corresponding sequence of squares starts: 4,4,4,9,4,9,4,16,9,4,36,9,... and it seems that 25 is the smallest square which is not the sum of the core and kernel of any integer.

Examples

			108 is a term because A007913(108) = 3, A007947(108) = 6 and 3 + 6 = 9.
Let c, r denote the core and the rad (kernel) respectively, of any number, then for m >= 0, 2^(2*m+1) is a term (c = r = 2)-->4 (2,8,32,128,...).
For m >= 1, h > 0, 2^(2*m)*3^(2*h+1) is a term (c = 6, r = 3)-->9 (12,48,108,...).
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..3140 (all terms less than or equal to 2^28.)

Crossrefs

Cf. A000290, A007913, A007947.

Programs

Mathematica

squareFreePart[n_Integer?Positive] := squareFreePart[n] = n/Times @@ (First[#]^(2*Floor[Last[#]/2]) & /@ FactorInteger[n]); squareFreeKernel[n_Integer?Positive] := squareFreeKernel[n] = Times @@ (First[#] & /@ FactorInteger[n]); a[max_Integer?Positive] := a[max] = Select[Range[max], IntegerQ@Sqrt[squareFreePart[#] + squareFreeKernel[#]] &]; a[5625] (* Robert P. P. McKone, Sep 07 2023 *)

PARI

isok(s) = issquare(core(s) + factorback(factorint(s)[, 1])); \\ Michel Marcus, Sep 08 2023

Python

from itertools import count, islice
from sympy.ntheory.primetest import is_square
from sympy import factorint
def A365517_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 is_square(a*(b+1)):
            yield k
A365517_list = list(islice(A365517_gen(),30)) # Chai Wah Wu, Sep 19 2023