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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). 2
 4, 9, 25, 18, 49, 80, 121, 169, 112, 135, 289, 361, 441, 352, 529, 416, 841, 360, 961, 891, 1088, 875, 1369, 1216, 1053, 1681, 672, 1849, 1472, 2209, 2601, 2809, 3025, 3249, 1856, 3481, 3721, 1984, 4225, 1584, 4489, 4761, 1960, 5041, 5329, 4736, 5929, 2496, 6241 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Equivalently, subsequence of terms of A339744 excluding terms whose prime factor set has already been encountered. 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 From Bernard Schott, Jan 19 2021: (Start) Indeed, a(n) satisfies the double inequality A005117(n+1) < a(n) <= A005117(n+1)^2. 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) LINKS David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Michel Marcus) FORMULA a(n) <= A005117(n+1)^2. - David A. Corneth, Dec 19 2020 EXAMPLE n a(n) prime factor set 1 4 [2] A000079 2 9 [3] A000244 3 25 [5] A000351 4 18 [2, 3] A033845 5 49 [7] A000420 6 80 [2, 5] A033846 7 121 [11] A001020 8 169 [13] A001022 9 112 [2, 7] A033847 10 135 [3, 5] A033849 11 289 [17] A001026 12 361 [19] A001029 13 441 [3, 7] A033850 14 352 [2, 11] A033848 15 529 [23] A009967 16 416 [2, 13] A288162 17 841 [29] A009973 18 360 [2, 3, 5] A143207 PROG (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)); ); ); } lista(nn) = {for (n=2, nn, if (issquarefree(n), print1(u(n), ", "); ); ); } CROSSREFS Cf. A000203 (sigma), A007947 (rad). Cf. A005117 (squarefree numbers), A027748, A265668, A339744. Subsequence: A001248 (squares of primes). Sequence in context: A131826 A366786 A051961 * A251544 A175119 A093867 Adjacent sequences: A339791 A339792 A339793 * A339795 A339796 A339797 KEYWORD nonn AUTHOR Michel Marcus, Dec 17 2020 STATUS approved

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Last modified April 22 08:12 EDT 2024. Contains 371893 sequences. (Running on oeis4.)