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A055081
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Number of positive integers whose harmonic mean with n is a positive integer.
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4
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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a(2^n) = n+1, a(p^n) = 2n+1 if p>=3 is prime. - Benoit Cloitre, Nov 26 2023
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EXAMPLE
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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.
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MAPLE
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seq(nops(select(`<`, numtheory:-divisors(2*n^2), 2*n)), n=1..100); # Robert Israel, Feb 07 2016
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MATHEMATICA
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Count[Divisors[2 #^2], x_ /; x < 2 #] & /@ Range[90] (* Ivan Neretin, May 04 2015 *)
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PROG
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(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
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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