This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379121 #30 Dec 20 2024 13:15:06
%S A379121 225,3025,3249,12321,29241,38025,91809,216225,247009,354025,408321,
%T A379121 751689,772641,855625,919681,1366561,1595169,3814209,9828225,11189025,
%U A379121 12173121,12709225,29430625,47927929,52403121,66471409,67486225,77457601,80263681,94148209,100661089,110397049,126540001,204232681,264875625,328878225
%N 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).
%C A379121 Of the first 2025 terms, only two, a(520) and a(1087) have multiple solutions. See the examples.
%C A379121 See also comments in A379123.
%H A379121 Antti Karttunen, <a href="/A379121/b379121.txt">Table of n, a(n) for n = 1..2025</a>
%H A379121 <a href="/index/Con#CongruCrossDomain">Index entries for sequences defined by congruent products between domains N and GF(2)[X]</a>.
%H A379121 <a href="/index/Ge#GF2X">Index entries for sequences related to polynomials in ring GF(2)[X]</a>.
%H A379121 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A379121 {k such that k is an odd square and A379113(k) > 1 (or equally, A379129(k) > 0)}.
%F A379121 a(n) = A379122(n)^2.
%F A379121 a(n) = A379123(n)*A379124(n).
%F A379121 For all n, A379125(n) = sigma(a(n)) = A277320(sigma(A379123(n)), sigma(A379124(n))).
%e A379121 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).
%e A379121 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.
%e A379121 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.
%e A379121 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.
%o A379121 (PARI) is_A379121(n) = (n%2 && issquare(n) && A379113(n)>1);
%Y A379121 Intersection of A016754 and A379114.
%Y A379121 Cf. A000203, A048720, A065621, A277320, A379113, A379122 (square roots).
%Y A379121 Cf. A379123 [= A379113(a(n))], A379124 [= A379119(a(n))], A379125 [= sigma(a(n))], A379129.
%K A379121 nonn
%O A379121 1,1
%A A379121 _Antti Karttunen_, Dec 18 2024