cp's OEIS Frontend

A385722 a(n) = the n-th term of the sequence B_2 defined as the ordered union of the ranges of A_x, where the sequence A_x is defined in A384669, and x is a proper positive fraction with denominator 2, 3, 4, or 6.

%I A385722 #26 Jul 21 2025 13:16:01
%S A385722 1,2,4,6,8,12,16,24,30,32,36,48,60,64,72,96,120,128,144,180,192,210,
%T A385722 240,256,288,360,384,420,480,512,576,720,768,840,960,1024,1152,1260,
%U A385722 1440,1536,1680,1920,2048,2304,2310,2520,2880,3072,3360,3840,4096,4320,4608,4620,5040
%N A385722 a(n) = the n-th term of the sequence B_2 defined as the ordered union of the ranges of A_x, where the sequence A_x is defined in A384669, and x is a proper positive fraction with denominator 2, 3, 4, or 6.
%C A385722 The sequences A_x were defined in A384669; please see that sequence for more details.
%C A385722 Because of the continuity of the functions g_k(x) defined in A384669, if k is a term of A_y, then k is a term of A_x with x rational sufficiently close to y; so it suffices to study A_x for x rational, 0 < x < 1.
%C A385722 Let x = p/q, with p and q natural numbers, p < q. Then B_n is the ordered union of the ranges of A_x, where q has Euler totient <= n (that is, A000010(q) <= n). B_1 is just the sequence A_(1/2), that is, A384669. The present sequence B_2 is the ordered union of the ranges of A_x, where x = 1/6, 1/4, 1/3, 1/2, 2/3, 3/4, 5/6.
%C A385722 8 is the first term in B_2 (this sequence) that is not in B_1 = A384669.
%C A385722 15120 appears to be the first term in B_4 that is not in B_2 (this sequence).
%C A385722 More discussion about B_n and B_oo appears in a linked pdf.
%H A385722 Pontus von Brömssen, <a href="/A385722/b385722.txt">Table of n, a(n) for n = 1..2000</a> (terms 1..186 from Hal M. Switkay)
%H A385722 Hal M. Switkay, <a href="/A385722/a385722.pdf">More information about B_n and B_oo</a>
%e A385722 8 is a term of A_(2/3) and therefore of this sequence as well as A_x for x > 2/3, even though 8 is not a term of A384669, because 3^(2/3) (corresponding to 8) > 2 = 1^(2/3) + 1^(2/3) (corresponding to 6). Thus 8 qualifies to be a term in B_2, this sequence.
%o A385722 (PARI) s(n, q) = my(f=factor(n)); sum(k=1, #f~, f[k, 2]^q);
%o A385722 listaq(nn, q) = my(r=-oo, list=List()); for (n=1, nn, my(ss=s(n, q)); if (ss > r, r = ss; listput(list, n)); ); Vec(list);
%o A385722 putlist(list, elems) = for (i=1, #elems, listput(list, elems[i])); list;
%o A385722 lista(nn) = my(list=List(), vq=[1/6, 1/4, 1/3, 1/2, 2/3, 3/4, 5/6]); for (i=1, #vq, list = putlist(list, listaq(nn, vq[i]))); Set(Vec(list)); \\ _Michel Marcus_, Jul 08 2025
%Y A385722 Cf. A000010, A000079, A002110, A002182, A025487, A095848, A384669.
%K A385722 nonn
%O A385722 1,2
%A A385722 _Hal M. Switkay_, Jul 07 2025