A071365 - OEIS

%I #42 Aug 04 2024 03:03:03

%S 18,50,54,75,90,98,108,126,147,150,162,198,234,242,245,250,270,294,

%T 300,306,324,338,342,350,363,375,378,414,450,486,490,500,507,522,525,

%U 540,550,558,578,588,594,600,605,630,648,650,666,686,702,722,726,735,738

%N Numbers k such that A071364(k) <> A046523(k).

%C A071364(k) and A046523(k) have the same prime factors, but not the same sequence of exponents in their prime factorizations.

%C A046523(k) <> k, as A046523(k) <= A071366(k) <= k.

%C 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

%C 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

%C 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

%H Alois P. Heinz, <a href="/A071365/b071365.txt">Table of n, a(n) for n = 1..10000</a>

%H Y. Choie, N. Lichiardopol, P. Moree, and P. Sole, <a href="https://doi.org/10.5802/jtnb.591">On Robin's criterion for the Riemann hypothesis</a>, J. Theor. Nombr. Bord. 19 (2) (2007), 357-372

%e For k = 50 = 2*5*5: A071364(50) = 2*3*3 = 18, A046523(50) = 2*2*3 = 12.

%e For k = 500 = 2*2*5*5*5: A071364(500) = 2*2*3*3*3 = 108, A046523(500) = 2*2*2*3*3 = 72.

%p a:= proc(n) option remember; local i, k, l;

%p       for k from 1 +`if`(n=1, 0, a(n-1))

%p       do l:= sort(ifactors(k)[2], (x, y)->x[1]<y[1]);

%p          for i to nops(l)-1 do

%p            if l[i][2]<l[i+1][2] then return k fi

%p          od

%p       od

%p     end:

%p seq(a(n), n=1..60);  # _Alois P. Heinz_, Aug 18 2014

%t Select[Range[750], (e = Last /@ FactorInteger[ # ]) != Sort[e, Greater] &] (* _Ray Chandler_, Sep 23 2005 *)

%t Select[Range[750],

%t OrderedQ[FactorInteger[#][[All, 2]], GreaterEqual] == False &] (* _Kenneth A Klinger_, Nov 22 2016 *)

%o (PARI) is(k) = {my(e = factor(k)[,2]); e != vecsort(e, , 4);} \\ _Amiram Eldar_, Aug 04 2024

%Y Cf. A046523, A071364, A071366.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, May 21 2002

%E Extended by _Ray Chandler_, Sep 23 2005