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A339794 a(n) is the least integer k satisfying rad(k)^2 < sigma(k) and whose prime factors set is the same as the prime factors set of A005117(n+1).

%I A339794 #39 Jan 21 2021 13:57:17
%S A339794 4,9,25,18,49,80,121,169,112,135,289,361,441,352,529,416,841,360,961,
%T A339794 891,1088,875,1369,1216,1053,1681,672,1849,1472,2209,2601,2809,3025,
%U A339794 3249,1856,3481,3721,1984,4225,1584,4489,4761,1960,5041,5329,4736,5929,2496,6241
%N A339794 a(n) is the least integer k satisfying rad(k)^2 < sigma(k) and whose prime factors set is the same as the prime factors set of A005117(n+1).
%C A339794 Equivalently, subsequence of terms of A339744 excluding terms whose prime factor set has already been encountered.
%C A339794 a(n) = A005117(n + 1)^2 when A005117(n + 1) is prime. Proof: if A005117(n + 1) is a prime p then rad(A005117(n + 1))^2 = rad(p)^2 = p^2 and so integers whose prime factors set is the same as the prime factors set of A005117(n + 1) = p are p^m where m >= 1. p^2 > sigma(p^1) = p + 1 but p^2 < sigma(p^2) = p^2 + p + 1. Q.E.D. - _David A. Corneth_, Dec 19 2020
%C A339794 From _Bernard Schott_, Jan 19 2021: (Start)
%C A339794 Indeed, a(n) satisfies the double inequality A005117(n+1) < a(n) <= A005117(n+1)^2.
%C A339794 It is also possible that a(n) = A005117(n+1)^2, even when A005117(n+1) is not prime; the smallest such example is for a(13) = 441 = 21^2 = A005117(14)^2. (End)
%H A339794 David A. Corneth, <a href="/A339794/b339794.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Michel Marcus)
%F A339794 a(n) <= A005117(n+1)^2. - _David A. Corneth_, Dec 19 2020
%e A339794    n  a(n) prime factor set
%e A339794    1    4  [2]           A000079
%e A339794    2    9  [3]           A000244
%e A339794    3   25  [5]           A000351
%e A339794    4   18  [2, 3]        A033845
%e A339794    5   49  [7]           A000420
%e A339794    6   80  [2, 5]        A033846
%e A339794    7  121  [11]          A001020
%e A339794    8  169  [13]          A001022
%e A339794    9  112  [2, 7]        A033847
%e A339794   10  135  [3, 5]        A033849
%e A339794   11  289  [17]          A001026
%e A339794   12  361  [19]          A001029
%e A339794   13  441  [3, 7]        A033850
%e A339794   14  352  [2, 11]       A033848
%e A339794   15  529  [23]          A009967
%e A339794   16  416  [2, 13]       A288162
%e A339794   17  841  [29]          A009973
%e A339794   18  360  [2, 3, 5]     A143207
%o A339794 (PARI) u(n) = {my(fn=factor(n)[,1]); for (k = n, n^2, my(fk = factor(k)); if (fk[,1] == fn, if (factorback(fk[,1])^2 < sigma(fk), return (k));););}
%o A339794 lista(nn) = {for (n=2, nn, if (issquarefree(n), print1(u(n), ", ");););}
%Y A339794 Cf. A000203 (sigma), A007947 (rad).
%Y A339794 Cf. A005117 (squarefree numbers), A027748, A265668, A339744.
%Y A339794 Subsequence: A001248 (squares of primes).
%K A339794 nonn
%O A339794 1,1
%A A339794 _Michel Marcus_, Dec 17 2020