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%I #21 Nov 06 2019 03:46:34
%S 1,3,5,6,7,11,13,14,17,19,22,23,28,29,31,37,38,41,43,45,46,47,52,53,
%T 56,59,60,61,62,67,71,73,76,79,83,86,89,94,96,97,99,101,103,107,109,
%U 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
%N Numbers whose abundance is divisible by its number of divisors.
%C Numbers n such that f(n) = A033880(n)/A000005(n) is an integer.
%C 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.
%C 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.
%H Giovanni Resta, <a href="/A301975/b301975.txt">Table of n, a(n) for n = 1..10000</a>
%e 11 is a term as its abundance is -10 and its number of divisors is 2, the former number being divisible by the latter.
%t Select[Range[180], Divisible[DivisorSigma[1,#]-2#, DivisorSigma[0,#]]&]
%o (PARI) for(n=1, 180, ((sigma(n)-2*n)%numdiv(n)==0) && print1(n ", "))
%o (PARI) isok(n) = !((sigma(n)-2*n)%numdiv(n)); \\ _Michel Marcus_, Apr 09 2018
%Y Cf. A033880 (abundance), A000005 (number of divisors), A065091, A000396 (subsequences), A056075 (related sequence).
%K nonn
%O 1,2
%A _Waldemar Puszkarz_, Mar 29 2018