A335368 Harmonic numbers k with a record number of primes p not dividing k such that k*p is also a harmonic number.

1, 28, 1638, 30240, 2178540, 2457000, 32997888, 142990848, 1307124000, 71271827200, 547929930240, 2198278051200, 2567400675840, 54409216942080

Offset

1,2

Comments

The corresponding record values are 0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 17.

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, k*tau(k)/sigma(k) = k*A000005(k)/A000203(k).

Links

Table of n, a(n) for n=1..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 the first harmonic number, and it has 0 primes p such that 1*p = p is a harmonic number, since a prime number cannot be harmonic. The next harmonic number k with at least one prime p such that k*p is also a harmonic number is 28, since 28 * 5 = 140 is a harmonic number.

Mathematica

harmNums = Cases[Import["https://oeis.org/A001599/b001599.txt", "Table"], {_, _}][[;; , 2]]; harMean[n_] := n*DivisorSigma[0, n]/DivisorSigma[1, n]; primeCount[n_] := Module[{d = Divisors[harMean[n]]}, Length @ Select[2*d - 1, PrimeQ[#] && ! Divisible[n, #] &]]; primeCountMax = -1; seq = {}; Do[If[(pc = primeCount[harmNums[[k]]]) > primeCountMax, primeCountMax = pc; AppendTo[seq, harmNums[[k]]]], {k, 1, Length[harmNums]}]; seq

Crossrefs

Cf. A000005, A000203, A001599, A099377, A099378, A335369, A335370, A335371.

Sequence in context: A060335 A335371 A335370 * A339120 A194190 A190397

Adjacent sequences: A335365 A335366 A335367 * A335369 A335370 A335371

Keyword

nonn,more

Author

Amiram Eldar, Jun 03 2020

Status

approved