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

Author

Waldemar Puszkarz, Mar 29 2018

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

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

Programs

Mathematica

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

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