cp's OEIS Frontend

A335268 Numbers that are not powers of primes (A024619) whose harmonic mean of their unitary divisors that are larger than 1 is an integer.

%I A335268 #9 Jun 06 2020 03:19:04
%S A335268 6,15,20,24,28,30,45,60,72,90,91,96,100,112,153,216,220,240,264,272,
%T A335268 325,352,360,364,378,496,703,765,780,816,832,1056,1125,1170,1225,1360,
%U A335268 1431,1512,1656,1760,1891,1900,1984,2275,2448,2520,2701,2912,3024,3168,3321
%N A335268 Numbers that are not powers of primes (A024619) whose harmonic mean of their unitary divisors that are larger than 1 is an integer.
%C A335268 Since the unitary divisors of a power of prime (A000961), p^e, are {1, p^e}, they are trivial terms and hence they are excluded from this sequence.
%C A335268 The corresponding harmonic means are 3, 5, 6, 6, 7, 5, 9, 7, 12, 7, 13, 8, 10, 14, 17, ...
%C A335268 Equivalently, numbers m such that omega(m) > 1 and (usigma(m)-m) | m * (2^omega(m)-1), or A063919(m) | (m * A309307(m)), where usigma is the sum of unitary divisors (A034448), and 2^omega(m) = A034444(m) is the number of the unitary divisors of m.
%C A335268 The squarefree terms of A335267 are also terms of this sequence.
%C A335268 The terms with 2 distinct prime divisors are of the form p^e * (2*p^e - 1), when the second factor is also a prime power. The least term which both of its 2 prime divisors are nonunitary (with multiplicity larger than 1) is 1225 = 5^2 * 7^2 = 5^2 * (2 * 5^2 - 1).
%C A335268 The unitary perfect numbers (A002827) are terms of this sequence: if m is a unitary perfect number then usigma(m)-m = m.
%H A335268 Amiram Eldar, <a href="/A335268/b335268.txt">Table of n, a(n) for n = 1..1000</a>
%e A335268 6 is a term since its unitary divisors other than 1 are 2, 3 and 6, and their harmonic mean, 3/(1/2 + 1/3 + 1/6) = 3, is an integer.
%t A335268 usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[3000], (omega = PrimeNu[#]) > 1 && Divisible[# * (2^omega-1), usigma[#] - #] &]
%Y A335268 The unitary version of A335267.
%Y A335268 A002827 is subsequence.
%Y A335268 Cf. A006086, A000961, A024619, A034444, A034448, A063919, A077610, A309307, A335269, A335270.
%K A335268 nonn
%O A335268 1,1
%A A335268 _Amiram Eldar_, May 29 2020