%I #18 Jul 08 2022 15:56:00

%S 1,2,3,4,6,8,12,16,18,24,32,36,48,64,72,96,108,144,192,216,288,324,

%T 384,432,576,648,768,864,972,1152,1296,1536,1728,1944,2304,2592,2916,

%U 3072,3456,3888,4608,5184,5832,6912,7776,8748,9216,10368,11664,13824,15552,17496

%N Numbers whose sum of 3-smooth divisors sets a new record.

%C Numbers m such that A072079(m) > A072079(k) for all k < m.

%C All the terms are 3-smooth numbers (A003586).

%C Equivalently, 3-smooth numbers k such that A000203(k) sets a new record.

%C Analogous to highly abundant numbers (A002093) with 3-smooth numbers only.

%H Michael S. Branicky, <a href="/A355578/b355578.txt">Table of n, a(n) for n = 1..11233</a> (all < 10^60)

%e The numbers of 3-smooth divisors of the first 6 positive integers are 1, 3, 4, 7, 1 and 12. The record values, 1, 3, 4 and 12, occur at 1, 2, 3, 4 and 6, the first 5 terms of this sequence.

%t s[n_] := Module[{e = IntegerExponent[n, {2, 3}], p}, p = {2, 3}^e; If[Times @@ p == n, (2^(e[[1]] + 1) - 1)*(3^(e[[2]] + 1) - 1)/2, 0]]; sm = 0; seq = {}; Do[sn = s[n]; If[sn > sm, sm = sn; AppendTo[seq, n]], {n, 1, 18000}]; seq

%o (PARI) lista(nmax) = {my(list = List(), smax = 0, e2, e3, s); for(n = 1, nmax, e2 = valuation(n, 2); e3 = valuation(n, 3); s = if(2^e2 * 3^e3 == n, (2^(e2 + 1) - 1)*(3^(e3 + 1) - 1)/2, 0); if(s > smax, smax = s; listput(list, n))); Vec(list)};

%o (Python)

%o from sympy import multiplicity as v

%o from itertools import count, takewhile

%o def f(n): return (2**(v(2, n)+1)-1) * (3**(v(3, n)+1)-1)//2

%o def smooth3(lim):

%o pows2 = list(takewhile(lambda x: x<lim, (2**i for i in count(0))))

%o pows3 = list(takewhile(lambda x: x<lim, (3**i for i in count(0))))

%o return sorted(c*d for c in pows2 for d in pows3 if c*d <= lim)

%o def aupto(lim):

%o data, records, record = smooth3(lim), [], -1

%o for argv, v in zip(data, map(f, data)):

%o if v > record: record = v; records.append(argv)

%o return records

%o print(aupto(10**5)) # _Michael S. Branicky_, Jul 08 2022

%Y Subsequence of A003586.

%Y A355579 is a subsequence.

%Y Cf. A000203, A002093, A072079, A309015.

%K nonn

%O 1,2

%A _Amiram Eldar_, Jul 08 2022