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

1,1

Comments

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.

Links

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

Maple

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A086527 A361445 A161554 * A353758 A245644 A371168

Adjacent sequences: A328941 A328942 A328943 * A328945 A328946 A328947

Keyword

nonn

Author

Jaroslav Krizek, Oct 31 2019

Status

approved