A392936 - OEIS

%I #31 Jun 05 2026 09:20:07

%S 27720,30240,32760,50400,75600,85680,95760,105840,115920,120120,

%T 128520,141120,143640,176400,180180,184800,205920,207900,214200,

%U 218400,235620,239400,264600,289800,292320,299880,308880,312480,314160,351120,371280,372960,414960,425040,438480,462000,468720,514800,517440,521640

%N Primitive terms of A023198: numbers k with the property sigma(k)/k >= 4 that are not divisible by any other number with that property.

%C 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).

%C 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.]

%C From _David A. Corneth_, Jan 28 2026: (Start)

%C If sigma(k)/k >= 4 then k is a term if for each prime p | k we have sigma(k/p)/(k/p) < 4.

%C Depending on an upper bound we can use branch and cutting to ease the search. An example is in the example section. (End)

%C The term with the greatest abundancy ratio is a(837) = 19399380 = A307111(4). - _Peter Munn_, Jun 05 2026

%H David A. Corneth, <a href="/A392936/b392936.txt">Table of n, a(n) for n = 1..10000</a> (first 5000 terms from Michael de Vlieger)

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Abundancy.html">Abundancy</a>.

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.

%F {a(n)} = {k >= 2 : A000203(k) >= 4k and A000203(A395192(k)) < 4*A395192(k)}. - _Peter Munn_, Jun 05 2026

%e 27720 is in the sequence since sigma(27720)/27720 = 4.0519... >= 4.

%e From _David A. Corneth_, Jan 28 2026 re calculation efficiency: (Start)

%e 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.

%e 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)

%t 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 *)

%o (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

%Y Subsequence of A023198.

%Y Subsequences: A027687, A358414.

%Y Cf. A006039 (for abundancy ratio 2), A388019 (for ratio 3, with a list for other ratios).

%Y Cf. A000203, A023199, A025487, A133812, A307111, A307115, A395192.

%K nonn

%O 1,1

%A _Peter Munn_, Jan 27 2026