A048108 Numbers with at least as many non-unitary divisors (A048105) as unitary divisors (A034444).

8, 16, 24, 27, 32, 36, 40, 48, 54, 56, 64, 72, 80, 81, 88, 96, 100, 104, 108, 112, 120, 125, 128, 135, 136, 144, 152, 160, 162, 168, 176, 180, 184, 189, 192, 196, 200, 208, 216, 224, 225, 232, 240, 243, 248, 250, 252, 256, 264, 270, 272, 280, 288, 296, 297

Offset

1,1

Comments

Numbers divisible by a prime cubed or two distinct primes squared. - Charles R Greathouse IV, Jun 07 2013

Equals A013929 \ A060687. The asymptotic density of this sequence is 1 - A059956 - A271971 = 0.1913171761... - Amiram Eldar, Nov 07 2020

Numbers k such that 2*A325973(k) = A034448(k)+A048250(k) < A000203(k), or equally, for which 2*A325974(k) > sigma(k), thus numbers k for which A325973(k) < A325974(k). See A048107 for a proof. - Antti Karttunen, Oct 05 2025

Numbers k such that A000005(k) > 2^(A001221(k)) + 2^(A001221(k) - 1). Proof: Let k = Product_{i = 1 .. A001221(k)} p_i^{e_i} and t(k) = (A001222(k) - A001221(k) + 2) * 2^(A001221(k) - 1). with e_1, e_2 >= 2. With A000005(k) = Product_{i = 1 .. A001221(k)} (e_i + 1) follows: If less than two e_i >= 2 and the rest are 1, then A000005(k) = t(k). If at least two e_i >= 2, compare to the previous pattern by transferring 1 from the big exponent, say m >= 3, to an exponent 1: then (m + 1)*2 < m*3, hence A000005(k) > t(k). - Felix Huber, May 29 2026

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Maple

with(numtheory): for n from 1 to 800 do if 2^nops(ifactors(n)[2])<=tau(n)-2^nops(ifactors(n)[2]) then printf(`%d, `, n) fi; od:
# Alternative:
with(NumberTheory):
A048108 := proc(n)
    option remember;
    local i, k;
    if n = 1 then return 8 end if;
    for k from A048108(n - 1) + 1 do
        i := Omega(k, 'distinct');
        if tau(k) > (2^i + 2^(i - 1)) then
            return k
        end if
    end do;
end proc:
seq(A048108(n), n = 1 .. 100); # Felix Huber, May 29 2026

Mathematica

Select[Range[300], Function[n, # <= DivisorSigma[0, n] - # &@ DivisorSum[n, 1 &, CoprimeQ[#, n/#] &]]] (* Michael De Vlieger, Aug 01 2017 *)
(* Alternative: *)
Select[Range[300], Or[Count[#, p_ /; Last@ p >= 2] >= 2, Count[#, p_ /; Last@ p >= 3] == 1] &@ FactorInteger@ # &] (* Michael De Vlieger, Aug 01 2017 *)

Prog

(PARI) is(n)=my(f=vecsort(factor(n)[, 2], , 4)); #f && (f[1]>2 || (#f>1 && f[2]>1)) \\ Charles R Greathouse IV, Jun 07 2013
(PARI) is(n)=factorback(factor(n)[, 2]) > 2 \\ Charles R Greathouse IV, Aug 25 2016

Crossrefs

Complement of A048107.

Subsequence of A013929.

Supersequence of A036785 and A338539.

Cf. A000203, A034448, A048250, A059956, A060687, A271971, A325973, A325974.

Sequence in context: A122612 A078130 A062171 * A228957 A137845 A046099

Adjacent sequences: A048105 A048106 A048107 * A048109 A048110 A048111

Keyword

nonn

Author

N. J. A. Sloane, Labos Elemer

Extensions

More terms from James Sellers, Jun 20 2000

Status

approved