OFFSET
1,1
COMMENTS
Of the first 2025 terms, only two, a(520) and a(1087) have multiple solutions. See the examples.
See also comments in A379123.
LINKS
FORMULA
EXAMPLE
k = 225 = 15^2 is included, because x = A379113(k) = 9, y = A379119(k) = 225/9 = 25, and A048720(A065621(sigma(9)), sigma(25)) = A048720(A065621(13), 31) = A048720(21, 31) = 403 = sigma(225).
a(8) = k = 216225 = 465^2 = (3*5*31)^2 is included, because x = A379113(k) = 9, y = A379119(k) = k/9 = 24025, sigma(9) = 13, A065621(13) = 21, sigma(24025) = 30783 and A048720(21, 30783) = 400179 = sigma(k). Note that pair x = 31^2 = 961, y = k / 961 = 225 is not among the solutions (we have A379129(k) = 1, not 2), because A048720(A065621(sigma(961)), sigma(k/961)) = 425971 > 400179.
a(520) = k = 383942431613601 = 19594449^2 is included, because x = A379113(k) = 16129, y = A379119(k) = 23804478369, and A048720(A065621(sigma(x)),sigma(y)) = 703777973774337 = sigma(k). This is the first term that has more than one such solution (A379129(k) = 2), the other solution pair being x=961 and y=399523862241.
a(1087) = k = 19012955210325729 = 137887473^2 is included, because x = A379113(k) = 8649, y = k/8649 = 2198283640921, and A048720(A065621(sigma(x)),sigma(y)) = A048720(22197, 2198285123583) = sigma(x)*sigma(y) = 28377662660332947 = A379125(1087). Note that 8649 = 9*961 and here also x=961 and x=9 satisfy the condition, so there are three solutions in total.
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Dec 18 2024
STATUS
approved