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A335369 Harmonic numbers k such that k*p is not a harmonic number for all the primes p that do not divide k. 4
1, 6, 140, 496, 672, 2970, 27846, 105664, 173600, 237510, 539400, 695520, 726180, 753480, 1421280, 1539720, 2229500, 2290260, 8872200, 11981970, 14303520, 15495480, 33550336, 50401728, 71253000, 80832960, 90409410, 144963000, 221557248, 233103780, 287425800, 318177800 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

If k is a harmonic number (A001599) and p is a prime that does not divide k, then k*p is a harmonic number if and only if (p+1)/2 is a divisor of the harmonic mean of the divisors of k, h(k) = k*tau(k)/sigma(k) = k*A000005(k)/A000203(k). The terms of this sequence are harmonic numbers k such that for all the divisors d of h(k), 2*d - 1 is either a nonprime or a prime divisor of k.

The even perfect numbers, 2^(p-1)*(2^p - 1) where p is a Mersenne exponent (A000043), have harmonic mean of divisors p. Therefore, they are in this sequence if p = 2 or if 2*p - 1 is composite (i.e., not in A172461). Of the first 47 Mersenne exponents there are 37 such primes (p = 2, 5, 13, 17, ...), with the corresponding even perfect numbers 6, 496, 33550336, 8589869056, ...

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..246 (terms below 10^14)

Mariano Garcia, On numbers with integral harmonic mean, The American Mathematical Monthly, Vol. 61, No. 2 (1954), pp. 89-96. See page 95.

EXAMPLE

1 is a term since it is a harmonic number, and there is no prime p such that 1*p = p is a harmonic number (if p is a prime, h(p) = 2*p/(p+1) cannot be an integer).

MATHEMATICA

harmNums = Cases[Import["https://oeis.org/A001599/b001599.txt", "Table"], {_, _}][[;; , 2]]; harMean[n_] := n * DivisorSigma[0, n]/DivisorSigma[1, n]; primeCountQ[n_] := Module[{d = Divisors[harMean[n]]}, Select[2*d - 1, PrimeQ[#] && ! Divisible[n, #] &] == {}]; Select[harmNums, primeCountQ]

CROSSREFS

Cf. A000005, A000043, A000203, A000396, A001599, A099377, A099378, A172461, A335368, A335370, A335371.

Sequence in context: A335318 A342358 A122483 * A335388 A123729 A193835

Adjacent sequences:  A335366 A335367 A335368 * A335370 A335371 A335372

KEYWORD

nonn

AUTHOR

Amiram Eldar, Jun 03 2020

STATUS

approved

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Last modified May 6 09:45 EDT 2021. Contains 343580 sequences. (Running on oeis4.)