A388019 Primitive terms of A023197: numbers k such that sigma(k)/k >= 3, but for every proper divisor, d|k, d<k, sigma(d)/d < 3.

120, 180, 420, 504, 660, 672, 780, 1584, 1848, 1872, 1890, 2184, 2352, 2376, 2772, 2856, 3150, 3192, 3276, 4284, 4410, 4788, 4896, 5100, 5292, 5700, 5796, 6864, 6900, 6930, 7344, 7728, 8190, 8208, 8424, 9744, 10296, 10416, 10710, 10944, 11550, 11970, 12012, 12432, 12870, 13464, 13650, 13776, 14448, 14490, 14850

Offset

1,1

Comments

Analogous to primitive nondeficient numbers (A006039) but with abundancy index 3 instead of 2.

The term with the greatest abundancy is a(30) = 6930 = A307111(3). - Peter Munn, Jun 05 2026

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Abundancy.

Index entries for sequences related to sigma(n).

Formula

{a(n)} = {k >= 2 : A000203(k) >= 3k and A000203(A395192(k)) < 3*A395192(k)}. - Peter Munn, Jun 05 2026

Maple

N:= 50000: # for terms <= N
R:= NULL: V:= Vector(N):
for n from 1 to N do
  if andmap(p -> V[n/p] = 1, numtheory:-factorset(n)) then
    if numtheory:-sigma(n)/n < 3 then V[n]:= 1 else R:= R, n
fi fi od:
R; # Robert Israel, Sep 15 2025

Mathematica

A388019Q[k_] := DivisorSigma[1, k] >= 3*k && AllTrue[Most[Divisors[k]], DivisorSigma[1, #] < 3*# &];
Select[Range[15000], A388019Q] (* Paolo Xausa, Sep 15 2025 *)

Prog

(PARI) is_A388019(n) = if((sigma(n)/n)<3, 0, fordiv(n, d, if(d<n && (sigma(d)/d)>=3, return(0))); (1));

Crossrefs

Subsequence of A023197.

Disjoint union of A005820 and A307112.

Other subsequences: A358413, A388020 (terms of the form 4u+2).

Equivalent sets of primitives for abundancy >= r: A006039 (r=2), A392936 (r=4), A307115 (nearly equivalent for r=5), A393013 (r=6), A389635 (r=4/3), {2} U A334118 (r=3/2), A353543 (r=Pi^2/6), A388014 (r=5/3).

Cf. A000203, A307111, A395192.

Sequence in context: A279088 A388024 A337479 * A322377 A388037 A247851

Adjacent sequences: A388016 A388017 A388018 * A388020 A388021 A388022

Keyword

nonn

Author

Antti Karttunen, Sep 15 2025

Status

approved