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).

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..2025

Index entries for sequences defined by congruent products between domains N and GF(2)[X].

Index entries for sequences related to polynomials in ring GF(2)[X].

Index entries for sequences related to sigma(n).

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

Adjacent sequences: A379118 A379119 A379120 * A379122 A379123 A379124

Keyword

nonn

Author

Antti Karttunen, Dec 18 2024

Status

approved