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A379121
Odd squares k for which A379113(k) > 1, i.e., k that have a proper unitary divisor d > 1 such that A048720(A065621(sigma(d)),sigma(k/d)) is equal to sigma(k).
7
225, 3025, 3249, 12321, 29241, 38025, 91809, 216225, 247009, 354025, 408321, 751689, 772641, 855625, 919681, 1366561, 1595169, 3814209, 9828225, 11189025, 12173121, 12709225, 29430625, 47927929, 52403121, 66471409, 67486225, 77457601, 80263681, 94148209, 100661089, 110397049, 126540001, 204232681, 264875625, 328878225
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.
FORMULA
{k such that k is an odd square and A379113(k) > 1 (or equally, A379129(k) > 0)}.
a(n) = A379122(n)^2.
a(n) = A379123(n)*A379124(n).
For all n, A379125(n) = sigma(a(n)) = A277320(sigma(A379123(n)), sigma(A379124(n))).
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.
PROG
(PARI) is_A379121(n) = (n%2 && issquare(n) && A379113(n)>1);
CROSSREFS
Intersection of A016754 and A379114.
Cf. A000203, A048720, A065621, A277320, A379113, A379122 (square roots).
Cf. A379123 [= A379113(a(n))], A379124 [= A379119(a(n))], A379125 [= sigma(a(n))], A379129.
Sequence in context: A014771 A184878 A134741 * A110204 A180697 A230065
KEYWORD
nonn
AUTHOR
Antti Karttunen, Dec 18 2024
STATUS
approved