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A055081 Number of positive integers whose harmonic mean with n is a positive integer. 4
1, 2, 3, 3, 3, 7, 3, 4, 5, 6, 3, 10, 3, 6, 10, 5, 3, 11, 3, 10, 9, 6, 3, 13, 5, 6, 7, 10, 3, 20, 3, 6, 9, 6, 10, 16, 3, 6, 9, 13, 3, 20, 3, 9, 17, 6, 3, 16, 5, 10, 9, 9, 3, 15, 9, 13, 9, 6, 3, 30, 3, 6, 16, 7, 9, 20, 3, 9, 9, 19, 3, 22, 3, 6, 16, 9, 10, 19, 3, 16, 9, 6, 3, 30, 9, 6, 9, 13, 3, 33 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also the number of factors of 2n^2 which are less than 2n, since the harmonic mean of n and 2n^2/k-n is 2n-k and these are all positive integers iff k<2n is a factor of 2n^2. So a(n)=3 iff n=4 or n is an odd prime.

For any n>2, there are three distinct trivial Diophantine solutions of H(n,x)=y, H being the harmonic mean: [x=n,y=n],[x=n(n-1),y=2(n-1)],[x=n(2n-1),y=2n-1]. Existence of any other solution proves that n is not a prime. - Stanislav Sykora, Feb 03 2016

a(n)=4 only for n=8. a(n)=5 iff n is 16 or the square of an odd prime. - Robert Israel, Feb 07 2016

LINKS

Ivan Neretin, Table of n, a(n) for n = 1..10000

FORMULA

a(n) >= min(n,3). - Stanislav Sykora, Feb 03 2016

EXAMPLE

a(6)=7 since the pairwise harmonic means of 6 with 2, 3, 6, 12, 18, 30 and 66 are 3, 4, 6, 8, 9, 10 and 11 respectively.

MAPLE

seq(nops(select(`<`, numtheory:-divisors(2*n^2), 2*n)), n=1..100); # Robert Israel, Feb 07 2016

MATHEMATICA

Count[Divisors[2 #^2], x_ /; x < 2 #] & /@ Range[90] (* Ivan Neretin, May 04 2015 *)

PROG

(PARI) a(n) = {my(c=0); for(y=1, 2*n-1, if((y*n)%(2*n-y)==0, c++)); return(c); } \\ Stanislav Sykora, Feb 03 2016

CROSSREFS

The smallest and largest positive integers whose harmonic means with n are positive integers are A053626 and A000384 with harmonic means of A053627 and A004273.

Sequence in context: A168113 A170895 A141479 * A275379 A109833 A132005

Adjacent sequences:  A055078 A055079 A055080 * A055082 A055083 A055084

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jun 13 2000

STATUS

approved

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Last modified June 22 13:46 EDT 2021. Contains 345380 sequences. (Running on oeis4.)