|
|
|
|
18, 50, 54, 75, 90, 98, 108, 126, 147, 150, 162, 198, 234, 242, 245, 250, 270, 294, 300, 306, 324, 338, 342, 350, 363, 375, 378, 414, 450, 486, 490, 500, 507, 522, 525, 540, 550, 558, 578, 588, 594, 600, 605, 630, 648, 650, 666, 686, 702, 722, 726, 735, 738
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
A071364(k) and A046523(k) have the same prime factors, but not the same sequence of exponents in their prime factorizations.
Numbers with more than one prime factor and, in the ordered factorization, at least one exponent is greater than the previous exponent when read from left to right; contains A097319. - Ray Chandler, Sep 23 2005
Choie et al. call the complementary set of integers (n = p1^e1 * p2^e^2 * ... with exponents e1 >= e2 >= e3 >= ... in their ordered prime factorization) Hardy-Ramanujan integers. - R. J. Mathar, Dec 08 2011
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 0, 6, 71, 759, 7758, 77948, 780216, 7803437, 78033303, 780315757, ... . Apparently, the asymptotic density of this sequence exists and equals 0.07803... . - Amiram Eldar, Aug 04 2024
|
|
LINKS
|
|
|
EXAMPLE
|
For k = 50 = 2*5*5: A071364(50) = 2*3*3 = 18, A046523(50) = 2*2*3 = 12.
For k = 500 = 2*2*5*5*5: A071364(500) = 2*2*3*3*3 = 108, A046523(500) = 2*2*2*3*3 = 72.
|
|
MAPLE
|
a:= proc(n) option remember; local i, k, l;
for k from 1 +`if`(n=1, 0, a(n-1))
do l:= sort(ifactors(k)[2], (x, y)->x[1]<y[1]);
for i to nops(l)-1 do
if l[i][2]<l[i+1][2] then return k fi
od
od
end:
|
|
MATHEMATICA
|
Select[Range[750], (e = Last /@ FactorInteger[ # ]) != Sort[e, Greater] &] (* Ray Chandler, Sep 23 2005 *)
Select[Range[750],
OrderedQ[FactorInteger[#][[All, 2]], GreaterEqual] == False &] (* Kenneth A Klinger, Nov 22 2016 *)
|
|
PROG
|
(PARI) is(k) = {my(e = factor(k)[, 2]); e != vecsort(e, , 4); } \\ Amiram Eldar, Aug 04 2024
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|