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A365517
Numbers k such that the sum of the squarefree part of k and the squarefree kernel of k is a perfect square.
2
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
1,1
COMMENTS
In other words, numbers k such that A007913(k) + A007947(k) is in A000290 (the sum of the core and the kernel (rad) 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.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..3140 (all terms less than or equal to 2^28.)
EXAMPLE
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,...).
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved