A355579 -id:A355579 - cp's OEIS frontend

A355578 Numbers whose sum of 3-smooth divisors sets a new record.

1, 2, 3, 4, 6, 8, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 108, 144, 192, 216, 288, 324, 384, 432, 576, 648, 768, 864, 972, 1152, 1296, 1536, 1728, 1944, 2304, 2592, 2916, 3072, 3456, 3888, 4608, 5184, 5832, 6912, 7776, 8748, 9216, 10368, 11664, 13824, 15552, 17496

Offset: 1

Amiram Eldar, Jul 08 2022

nonn

Numbers m such that A072079(m) > A072079(k) for all k < m.
All the terms are 3-smooth numbers (A003586).
Equivalently, 3-smooth numbers k such that A000203(k) sets a new record.
Analogous to highly abundant numbers (A002093) with 3-smooth numbers only.

			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.

Michael S. Branicky, Table of n, a(n) for n = 1..11233 (all < 10^60)

Subsequence of A003586.
A355579 is a subsequence.
Cf. A000203, A002093, A072079, A309015.

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

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)};

from sympy import multiplicity as v
from itertools import count, takewhile
def f(n): return (2**(v(2, n)+1)-1) * (3**(v(3, n)+1)-1)//2
def smooth3(lim):
    pows2 = list(takewhile(lambda x: x record: record = v; records.append(argv)
    return records
print(aupto(10**5)) # Michael S. Branicky, Jul 08 2022

cp's OEIS Frontend

A355578 Numbers whose sum of 3-smooth divisors sets a new record.

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