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 A335268 Numbers that are not powers of primes (A024619) whose harmonic mean of their unitary divisors that are larger than 1 is an integer. 3
 6, 15, 20, 24, 28, 30, 45, 60, 72, 90, 91, 96, 100, 112, 153, 216, 220, 240, 264, 272, 325, 352, 360, 364, 378, 496, 703, 765, 780, 816, 832, 1056, 1125, 1170, 1225, 1360, 1431, 1512, 1656, 1760, 1891, 1900, 1984, 2275, 2448, 2520, 2701, 2912, 3024, 3168, 3321 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. The corresponding harmonic means are 3, 5, 6, 6, 7, 5, 9, 7, 12, 7, 13, 8, 10, 14, 17, ... 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. The squarefree terms of A335267 are also terms of this sequence. 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). The unitary perfect numbers (A002827) are terms of this sequence: if m is a unitary perfect number then usigma(m)-m = m. LINKS Amiram Eldar, Table of n, a(n) for n = 1..1000 EXAMPLE 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. MATHEMATICA usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[3000], (omega = PrimeNu[#]) > 1 && Divisible[# * (2^omega-1), usigma[#] - #] &] CROSSREFS The unitary version of A335267. A002827 is subsequence. Cf. A006086, A000961, A024619, A034444, A034448, A063919, A077610, A309307, A335269, A335270. Sequence in context: A282173 A328226 A045848 * A294651 A368309 A044439 Adjacent sequences: A335265 A335266 A335267 * A335269 A335270 A335271 KEYWORD nonn AUTHOR Amiram Eldar, May 29 2020 STATUS approved

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Last modified May 18 18:37 EDT 2024. Contains 372664 sequences. (Running on oeis4.)