OFFSET
1,1
COMMENTS
Sigma is the sum of divisors function, A000203, and sigma(k)/k is known as the abundancy ratio or abundancy index. If we specify this to be at least 2 instead of at least 4 we get the primitive nondeficient numbers (A006039).
Numbers whose proper multiples are all 4-abundant, and whose proper divisors are all 4-deficient. [A definition wording that encompasses the 4-perfect numbers, A027687, and is unaffected by whether the definition of "4-abundant" includes 4-perfect numbers or not.]
From David A. Corneth, Jan 28 2026: (Start)
If sigma(k)/k >= 4 then k is a term if for each prime p | k we have sigma(k/p)/(k/p) < 4.
Depending on an upper bound we can use branch and cutting to ease the search. An example is in the example section. (End)
The term with the greatest abundancy ratio is a(837) = 19399380 = A307111(4). - Peter Munn, Jun 05 2026
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000 (first 5000 terms from Michael de Vlieger)
Eric Weisstein's World of Mathematics, Abundancy.
FORMULA
{a(n)} = {k >= 2 : A000203(k) >= 4k and A000203(A395192(k)) < 4*A395192(k)}. - Peter Munn, Jun 05 2026
EXAMPLE
27720 is in the sequence since sigma(27720)/27720 = 4.0519... >= 4.
From David A. Corneth, Jan 28 2026 re calculation efficiency: (Start)
If a term k is of the form 6*m where m is a 5-rough number and we look for terms < 10^9 then we can use that there is no 5-rough number m < floor(10^9/6) such that sigma(6*m) / (6*m) = sigma(6)/6 * sigma(m)/m = 2 * sigma(m)/m >= 4 so sigma(m) >= 2.
If it exists then there must be such m that is a product of consecutive primes starting at 5. Also if two primes p, q such that p < q divide m then p has multiplicity of at least q. This is a similar idea to A025487 but we just start at 5 instead of 2. There are 143 such numbers <= 10^9 and all of them have sigma(m)/m < 2 so we can stop the search at 6 for this case. (End)
MATHEMATICA
fQ[x_] := DivisorSigma[1, x]/x >= 4; s = Select[Range[2^19], fQ]; Select[s, NoneTrue[Divisors[#][[2 ;; -2]], fQ] &] (* Michael De Vlieger, Jan 28 2026 *)
PROG
(PARI) is(n) = {my(f = factor(n)); if(sigma(f) < 4*n, return(0)); for(i = 1, #f~, c = n\f[i, 1]; if(sigma(c) >= 4*c, return(0))); 1} \\ David A. Corneth, Jan 28 2026
CROSSREFS
Subsequence of A023198.
KEYWORD
nonn
AUTHOR
Peter Munn, Jan 27 2026
STATUS
approved