A396319 Denominator of the largest abundancy of any proper divisor of n as a fraction of the abundancy of n.

3, 4, 7, 6, 4, 8, 15, 13, 6, 12, 7, 14, 8, 6, 31, 18, 13, 20, 7, 8, 12, 24, 15, 31, 14, 40, 8, 30, 6, 32, 63, 12, 18, 8, 13, 38, 20, 14, 15, 42, 8, 44, 12, 13, 24, 48, 31, 57, 31, 18, 14, 54, 40, 12, 15, 20, 30, 60, 7, 62, 32, 13, 127, 14, 12, 68, 18, 24, 8, 72, 15, 74, 38, 31

Offset

2,1

Comments

The denominator is that of the reduced fraction. The abundancy of k is sigma(k)/k, where sigma is the sum of divisors function, A000203.

Links

Table of n, a(n) for n=2..75.

Formula

For e > 0, a(prime(k)^e) = Sum_{j=0..e} prime(k)^j = A319076(e,k), where prime(k) = A000040(k) is the k-th prime.

For i, j > 1, a(lcm(i, j)) = max(a(i), a(j)).

a(n) = j/gcd(i, j), where i = sigma(A395192(n))*n, j = A395192(n)*sigma(n) and A395192(n) is the least proper divisor of n with the greatest abundancy.

a(n) = A396316(n) + 1.

For squarefree n, a(n) = A006530(n) + 1.

Example

For any number n, we need consider only its maximal proper divisors (row n of A355079), since the abundancy of any number is never greater than that of any of its multiples.
For n = 20, we consider 10 and 4. The abundancy of 10 is sigma(10)/10 = 18/10 = 1.8. The abundancy of 4 is sigma(4)/4 = 7/4 = 1.75. The larger of these is 1.8. As a fraction of the abundancy of 20, which is 42/20 = 2.1, this is 1.8/2.1 = 6/7, whose denominator is 7. So a(20) = 7.

Crossrefs

Cf. A000203, A355079, A396316 (numerators).

See the formula section for the relationships with A000040, A006530, A319076, A395192.

Range of terms: A108348\{1}.

Sequence in context: A322656 A105827 A393157 * A230289 A367502 A023888

Adjacent sequences: A396316 A396317 A396318 * A396320 A396321 A396322

Keyword

nonn,frac,new

Author

Peter Munn, Jun 12 2026

Status

approved