A357606 - OEIS

%I #10 Oct 09 2022 04:22:49

%S 36,48,80,120,162,168,200,224,264,270,280,300,312,352,378,392,408,416,

%T 450,456,500,552,588,594,630,696,700,702,744,750,882,888,918,968,980,

%U 984,1026,1032,1050,1088,1100,1128,1216,1232,1242,1272,1300,1372,1416,1452

%N Primitive terms of A357605: numbers in A357605 with no proper divisor in A357605.

%C Numbers k such that A162296(k) > 2*k but for all the aliquot divisors d of k (i.e., d | k, d < k), A162296(d) <= 2*d.

%C If k is a term then all the positive multiples of k are terms of A357605.

%C The least odd term is a(144) = 4725.

%H Amiram Eldar, <a href="/A357606/b357606.txt">Table of n, a(n) for n = 1..10000</a>

%e 36 is a term since A162296(36) = 79 > 2*36, but for all the divisors d of 36, 1, 2, 3, 4, 6, 9, 12 and 18, A162296(d) <= 2*d. E.g., A162296(18) = 28 < 2*18.

%t q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) > 2*n]; q[1] = False; primQ[n_] := q[n] && AllTrue[Most @ Divisors[n], ! q[#] &]; Select[Range[1500], primQ]

%Y Cf. A162296.

%Y Subsequence of A005101, A013929 and A357605.

%Y Similar sequences: A091191, A302574.

%K nonn

%O 1,1

%A _Amiram Eldar_, Oct 06 2022