A328945 - OEIS

%I #18 Dec 14 2023 19:31:50

%S 2,4,8,9,10,12,16,18,24,25,26,32,34,36,40,48,50,52,58,63,64,72,74,75,

%T 76,80,81,82,84,88,90,98,100,104,106,108,112,117,120,121,122,124,128,

%U 130,136,144,146,148,152,156,160,162,170,171,172,175,176,178,180

%N Numbers m that are neither arithmetic (A003601) nor harmonic (A001599).

%C Numbers m such that neither the arithmetic mean of the divisors of m nor the harmonic mean of the divisors of m is an integer.

%C Numbers m such that neither A(m) = A000203(m)/A000005(m) nor H(m) = m * A000005(m)/A000203(m) is an integer.

%C Corresponding values of A(m): 3/2, 7/3, 15/4, 13/3, 9/2, 14/3, 31/5, 13/2, 15/2, 31/3, 21/2, 21/2, 27/2, ...

%C Corresponding values of H(m): 4/3, 12/7, 32/15, 27/13, 20/9, 18/7, 80/31, 36/13, 16/5, 75/31, 52/21, 64/21, ...

%H Robert Israel, <a href="/A328945/b328945.txt">Table of n, a(n) for n = 1..10000</a>

%p filter:= proc(n) local D,d,t;

%p   D:=numtheory:-divisors(n);

%p   d:= nops(D);

%p   convert(D,`+`) mod d <> 0 and not ((d/add(1/t,t=D))::integer)

%p end proc:

%p select(filter, [$1..200]); # _Robert Israel_, Dec 14 2023

%t Select[Range[180], !Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] && !Divisible[# * DivisorSigma[0, #], DivisorSigma[1, #]] &] (* _Amiram Eldar_, Nov 01 2019 *)

%o (Magma) [m: m in [1..10^5] | not IsIntegral(m * NumberOfDivisors(m) / SumOfDivisors(m)) and not IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m))]

%o (PARI) isok(m) = my(f = factor(m), prd = sigma(f)/numdiv(f)); (denominator(prd) != 1) && (denominator(m/prd) != 1); \\ _Michel Marcus_, Nov 05 2019

%Y Cf. A000005, A000203, A001599, A003601, A007340, A328944.

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Oct 31 2019