A301975 Numbers whose abundance is divisible by its number of divisors.

1, 3, 5, 6, 7, 11, 13, 14, 17, 19, 22, 23, 28, 29, 31, 37, 38, 41, 43, 45, 46, 47, 52, 53, 56, 59, 60, 61, 62, 67, 71, 73, 76, 79, 83, 86, 89, 94, 96, 97, 99, 101, 103, 107, 109, 113, 118, 124, 126, 127, 130, 131, 132, 134, 137, 139, 142, 147, 148, 149, 150, 151, 153, 157, 158, 163, 166, 167, 168, 170, 172, 173, 175, 176, 179

Offset

1,2

Comments

Numbers n such that f(n) = A033880(n)/A000005(n) is an integer.

Perfect numbers (A000396) and odd primes (A065091) are members, unified (along with 1) into a subsequence on which abs(f(n)) reaches record extrema. For perfect numbers, these are global minima, for the other terms, maxima.

Another notable subsequence is defined by f(n)=1: numbers whose abundance equals their number of divisors. They all belong to A056075. The first 3 terms are 56, 7192, 7232. There are 11 of them up to 10^9.

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000

Example

11 is a term as its abundance is -10 and its number of divisors is 2, the former number being divisible by the latter.

Mathematica

Select[Range[180], Divisible[DivisorSigma[1, #]-2#, DivisorSigma[0, #]]&]

Prog

(PARI) for(n=1, 180, ((sigma(n)-2*n)%numdiv(n)==0) && print1(n ", "))
(PARI) isok(n) = !((sigma(n)-2*n)%numdiv(n)); \\ Michel Marcus, Apr 09 2018

Crossrefs

Cf. A033880 (abundance), A000005 (number of divisors), A065091, A000396 (subsequences), A056075 (related sequence).

Sequence in context: A047582 A333217 A342438 * A015814 A227026 A361827

Adjacent sequences: A301972 A301973 A301974 * A301976 A301977 A301978

Keyword

nonn

Author

Waldemar Puszkarz, Mar 29 2018

Status

approved