A328945 -id:A328945 - cp's OEIS frontend

A328944 Arithmetic numbers (A003601) that are not harmonic (A001599).

3, 5, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 77, 78, 79, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 99, 101

Offset: 1

Jaroslav Krizek, Oct 31 2019

nonn

Numbers m such that the arithmetic mean of the divisors of m is an integer but the harmonic mean of the divisors of m is not an integer.
Numbers m such that A(m) = A000203(m)/A000005(m) is an integer but H(m) = m * A000005(m)/A000203(m) is not an integer.
Corresponding values of A(m): 2, 3, 4, 6, 7, 6, 6, 9, 10, 7, 8, 9, 12, 10, 15, 9, 16, 12, 12, 19, 15, 14, 21, 12, 22, ...
Corresponding values of H(m): 3/2, 5/3, 7/4, 11/6, 13/7, 7/3, 5/2, 17/9, 19/10, 20/7, 21/8, 22/9, ...
Complement of A007340 with respect to A003601.

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

Cf. A000005, A000203, A003601, A007340, A046999, A328945.

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

harm:= proc(S) local s; nops(S)/add(1/s,s=S) end proc:
filter:= proc(n) local S;
  S:= numtheory:-divisors(n);
  (convert(S,`+`)/nops(S))::integer and not harm(S)::integer
end proc:
select(filter, [$1..200]); # Robert Israel, May 04 2025

Select[Range[100], Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] && !Divisible[# * DivisorSigma[0, #], DivisorSigma[1, #]] &] (* Amiram Eldar, Nov 01 2019 *)

cp's OEIS Frontend

A328944 Arithmetic numbers (A003601) that are not harmonic (A001599).

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