A328945 Numbers m that are neither arithmetic (A003601) nor harmonic (A001599).

2, 4, 8, 9, 10, 12, 16, 18, 24, 25, 26, 32, 34, 36, 40, 48, 50, 52, 58, 63, 64, 72, 74, 75, 76, 80, 81, 82, 84, 88, 90, 98, 100, 104, 106, 108, 112, 117, 120, 121, 122, 124, 128, 130, 136, 144, 146, 148, 152, 156, 160, 162, 170, 171, 172, 175, 176, 178, 180

Offset

1,1

Comments

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.

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

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, ...

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, ...

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Maple

filter:= proc(n) local D, d, t;
  D:=numtheory:-divisors(n);
  d:= nops(D);
  convert(D, `+`) mod d <> 0 and not ((d/add(1/t, t=D))::integer)
end proc:
select(filter, [$1..200]); # Robert Israel, Dec 14 2023

Mathematica

Select[Range[180], !Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] && !Divisible[# * DivisorSigma[0, #], DivisorSigma[1, #]] &] (* Amiram Eldar, Nov 01 2019 *)

Prog

(Magma) [m: m in [1..10^5] | not IsIntegral(m * NumberOfDivisors(m) / SumOfDivisors(m)) and not IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m))]
(PARI) isok(m) = my(f = factor(m), prd = sigma(f)/numdiv(f)); (denominator(prd) != 1) && (denominator(m/prd) != 1); \\ Michel Marcus, Nov 05 2019

Crossrefs

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

Sequence in context: A374664 A049642 A326713 * A050907 A100675 A196730

Adjacent sequences: A328942 A328943 A328944 * A328946 A328947 A328948

Keyword

nonn

Author

Jaroslav Krizek, Oct 31 2019

Status

approved