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A355578
Numbers whose sum of 3-smooth divisors sets a new record.
2
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
1,2
COMMENTS
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.
LINKS
Michael S. Branicky, Table of n, a(n) for n = 1..11233 (all < 10^60)
EXAMPLE
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.
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, 0]]; sm = 0; seq = {}; Do[sn = s[n]; If[sn > sm, sm = sn; AppendTo[seq, n]], {n, 1, 18000}]; seq
PROG
(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)};
(Python)
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<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**5)) # Michael S. Branicky, Jul 08 2022
CROSSREFS
Subsequence of A003586.
A355579 is a subsequence.
Sequence in context: A362668 A048874 A092824 * A084094 A217689 A018718
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jul 08 2022
STATUS
approved