A395192 - OEIS

%I #27 Apr 21 2026 18:01:11

%S 1,1,2,1,2,1,4,3,2,1,6,1,2,3,8,1,6,1,10,3,2,1,12,5,2,9,4,1,6,1,16,3,2,

%T 5,12,1,2,3,20,1,6,1,4,15,2,1,24,7,10,3,4,1,18,5,28,3,2,1,30,1,2,21,

%U 32,5,6,1,4,3,10,1,36,1,2,15,4,7,6,1,40,27,2,1

%N a(n) is the least k with the largest abundancy where k is a proper divisor of n.

%C The abundancy of k is sigma(k)/k, where sigma is the sum of divisors function, A000203.

%C a(n) is of the form n/p where p is a prime divisor of n. Proof: sigma(k)/k is multiplicative with sigma(p^m)/p^m = (p^(m+1)-1)/((p-1)*p^m) which increases as m increases. To ensure sigma(k)/k is maximal, divide by as few primes as possible. To ensure k is a proper divisor of n, divide by one prime.

%C When we have an abundancy-defined set, S, such as A005101, A005231, A023196 or A023197, we can determine whether a member, k, is a primitive member of S using only the abundancy of a(k).

%C For all positive integer m, a(p*m) = m for all except finitely many primes p. Thus all positive integers appear infinitely often. These values p*m are all values k such that a(k) = m.

%C The integers m for which a(p*m) = m for all primes p are given in A137825.

%C For all k in A362619 we have a(k) = k/A006530(k) but this property is not exclusive to A362619. For example 200 is not in A362619 but a(200) = 200/A006530(200) = 200/5 = 40.

%H Paolo Xausa, <a href="/A395192/b395192.txt">Table of n, a(n) for n = 2..10000</a>

%F a(n) = n/A137826(m), where m = min({k : n|A137825(k)}).

%F a(p) = 1 for prime p.

%F a(p*q) = p for primes p <= q.

%e Usually a number n has a unique proper divisor with the largest abundancy. The first exception is 400, where this property is shared by its divisors 80 and 200, whose abundancy is 93/40, so a(400) = 80.

%t A395192[n_] := Min[MaximalBy[n/FactorInteger[n][[All, 1]], DivisorSigma[1, #]/# &]];

%t Array[A395192, 100, 2] (* _Paolo Xausa_, Apr 21 2026 *)

%o (PARI) a(n) = {my(f = factor(n), r = 0, res = 0, c); forstep(i = #f~, 1, -1, c = sigma(n/f[i,1])/(n/f[i,1]); if(c > r, r = c; res = n/f[i,1])); res}

%Y Cf. A004394.

%Y See the comments for the relationships with A000203, A005101, A005231, A006530, A023196, A023197, A362619.

%Y See the formula section for the relationships with A137825, A137826.

%K nonn

%O 2,3

%A _David A. Corneth_ and _Peter Munn_, Apr 15 2026