A355579 Numbers k such that A072079(k)/k sets a new record.

1, 2, 4, 6, 12, 24, 36, 48, 72, 144, 288, 432, 864, 1728, 2592, 3456, 5184, 10368, 20736, 31104, 41472, 62208, 124416, 248832, 373248, 746496, 1492992, 2239488, 2985984, 4478976, 8957952, 17915904, 26873856, 53747712, 107495424, 161243136, 214990848, 322486272

Offset

1,2

Comments

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

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

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

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

Links

Michael S. Branicky, Table of n, a(n) for n = 1..4370

Formula

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

Example

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.

Mathematica

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

Prog

(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)};
(Python)
from fractions import Fraction
from sympy import multiplicity as v
from itertools import count, takewhile
def f(n): return Fraction((2**(v(2, n)+1)-1) * (3**(v(3, n)+1)-1)//2, n)
def smooth3(lim):
    pows2 = list(takewhile(lambda x: x<lim, (2**i for i in count(0))))
    pows3 = list(takewhile(lambda x: x<lim, (3**i for i in count(0))))
    return sorted(c*d for c in pows2 for d in pows3 if c*d <= lim)
def aupto(lim):
    data, records, record = smooth3(lim), [], -1
    for argv, v in zip(data, map(f, data)):
        if v > record: record = v; records.append(argv)
    return records
print(aupto(10**9)) # Michael S. Branicky, Jul 08 2022

Crossrefs

Subsequence of A003586 and A355578.

Cf. A000203, A004394, A072079.

Sequence in context: A141320 A307122 A309015 * A357173 A349424 A134865

Adjacent sequences: A355576 A355577 A355578 * A355580 A355581 A355582

Keyword

nonn

Author

Amiram Eldar, Jul 08 2022

Status

approved