A355579 - OEIS

%I #10 Jul 09 2022 15:29:03

%S 1,2,4,6,12,24,36,48,72,144,288,432,864,1728,2592,3456,5184,10368,

%T 20736,31104,41472,62208,124416,248832,373248,746496,1492992,2239488,

%U 2985984,4478976,8957952,17915904,26873856,53747712,107495424,161243136,214990848,322486272

%N Numbers k such that A072079(k)/k sets a new record.

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

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

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

%C Analogous to superabundant numbers (A004394) with 3-smooth numbers only.

%H Michael S. Branicky, <a href="/A355579/b355579.txt">Table of n, a(n) for n = 1..4370</a>

%F Limit_{n->oo} A072079(a(n))/a(n) = lim_{n->oo} A000203(a(n))/a(n) = 3.

%e The numbers of 3-smooth divisors of the first 6 positive integers are 1, 3, 4, 7, 1 and 12. The corresponding values of A072079(k)/k are 1, 3/2, 4/3, 7/4, 1/5 and 2. The record values, 1, 3/2, 7/4 and 2, occur at 1, 2, 4 and 6, the first 4 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*n), 0]]; sm = 0; seq = {}; Do[sn = s[n]; If[sn > sm, sm = sn; AppendTo[seq, n]], {n, 1, 10^6}]; seq

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

%o (Python)

%o from fractions import Fraction

%o from sympy import multiplicity as v

%o from itertools import count, takewhile

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

%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**9)) # _Michael S. Branicky_, Jul 08 2022

%Y Subsequence of A003586 and A355578.

%Y Cf. A000203, A004394, A072079.

%K nonn

%O 1,2

%A _Amiram Eldar_, Jul 08 2022