1,2

Equivalently, n*tau(n)/sigma(n) is an integer, where tau(n) (A000005) is the number of divisors of n and sigma(n) is the sum of the divisors of n (A000203).

Equivalently, the average of the divisors of n divides n.

Ore showed that every perfect number (A000396) is harmonic. The converse does not hold: 140 is Harmonic but not perfect. Ore conjectured that 1 is the only odd harmonic number.

G. L. Cohen and Deng Moujie, On a generalization of Ore's harmonic numbers, Nieuw Arch. Wisk. (4), 16 (1998) 161-172.

G. L. Cohen and R. M. Sorli, Harmonic seeds, Fibonacci Quart., 36 (1998) 386-390; errata, 39 (2001) 4.

M. Garcia, On numbers with integral harmonic mean. Amer. Math. Monthly 61, (1954). 89-96.

T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput. 73 (2004), 475-491.

R. K. Guy, Unsolved Problems in Number Theory, B2.

H.-J. Kanold, Uber das harmonische Mittel der Teiler einer natuerlichen Zahl, Math. Ann., 133 (1957) 371-374.

W. H. Mills, On a conjecture of Ore, Proc. Number Theory Conf., Boulder CO, 1972, 142-146.

O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly, 55 (1948), 615-619.

Carl Pomerance, On a problem of Ore: Harmonic numbers (unpublished typescript); see Abstract *709-A5, Notices Amer. Math. Soc., 20 (1973) Abstract A-648.

Andreas and Eleni Zachariou, Perfect, semi-perfect and Ore numbers, Bull. Soc. Math. Grece (N.S.), 13 (1972) 12-22.

T. D. Noe and Klaus Brockhaus, Table of n, a(n) for n=1..170

Eric Weisstein's World of Mathematics, Harmonic Divisor Number

Takeshi Goto, All harmonic numbers less than 10^14

Takeshi Goto, Table of a(n) for n=1..937

Note that harmonic mean of divisors of n = n*tau(n)/sigma(n).

n=140 has Sigma[ 0,140 ]=12 divisors with Sigma[ 1,140 ]=336. Average divisor is 336/12=28, an integer, and divides n: n=5*28. n=496, Sigma[ 0,496 ]=10, Sigma[ 1,496 ]=992: average divisor 99.2 is not an integer, but n/(Sigma_1/Sigma_0)=496/99.2=5 is an integer.

Do[ If[ IntegerQ[ n*DivisorSigma[0, n]/ DivisorSigma[1, n]], Print[n]], {n, 1, 1550000}]

(PARI) a(n)=if(n<0, 0, n=a(n-1); until(0==(sigma(n, 0)*n)%sigma(n, 1), n++); n) - Michael Somos Feb 06 2004

See A003601 for analogues referring to arithmetic mean and A000290 for geometric mean of divisors.

See A001600 and A090240 for the integer values obtained.

sigma_0(n) is the number of divisors of n (A000005).

sigma_1(n) is the sum of the divisors of n [same as sigma(n)] (A000203).

Cf. A090944, A090945, A035527, A007691, A074247, A053783. Not a subset of A003601.

Sequence in context: A117423 A084778 A108051 this_sequence A074247 A053783 A110047

Adjacent sequences: A001596 A001597 A001598 this_sequence A001600 A001601 A001602

nonn,nice

njas

More terms from Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Sep 18 2001