A071365 Numbers k such that A071364(k) <> A046523(k).

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

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.

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

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

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Y. Choie, N. Lichiardopol, P. Moree, and P. Sole, On Robin's criterion for the Riemann hypothesis, J. Theor. Nombr. Bord. 19 (2) (2007), 357-372

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:
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 18 2014

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

Cf. A046523, A071364, A071366.

Sequence in context: A222740 A335377 A317258 * A360252 A097319 A258211

Adjacent sequences: A071362 A071363 A071364 * A071366 A071367 A071368

Keyword

nonn

Author

Reinhard Zumkeller, May 21 2002

Extensions

Extended by Ray Chandler, Sep 23 2005

Status

approved