

A301975


Numbers whose abundance is divisible by its number of divisors.


1



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

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



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



KEYWORD

nonn


AUTHOR



STATUS

approved



