[go: up one dir, main page]
Search a number
harmonic divisor numbers
A number  $n$  is called harmonic divisor number if the harmonic mean of its divisors is an integer. This is equivalent to say that the average of the divisors of  $n$  divides  $n$, i.e.,  $n\tau(n)/\sigma(n)$  is an integer.

Harmonic divisor numbers are also called harmonic numbers, for brevity, or Ore numbers, after O.Ore who studied them.

O.Ore proved that all the perfect numbers are also harmonic and conjectured that 1 is the only odd harmonic number. This conjecture has been verified by G.L.Cohen et al. for  $n<10^{24}$  and if true, it will imply that no odd perfect numbers exist.

Jaycob Coleman has observed that all the Ore numbers up to  $10^{14}$  are also practical numbers and conjectured this holds in general.

T. Goto and K. Okeya have computed a list of the 937 harmonic numbers up to  $10^{14}$.

The first harmonic numbers are 1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664, 117800, 167400, 173600, 237510 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of harmonic numbers

A graph displaying how many harmonic numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of harmonic numbers
Useful links Mathworld, Harmonic Divisor Number
Wikipedia, Harmonic divisor number
OEIS, Sequence A001599
Harmonic numbers can also be... (you may click on names or numbers and on + to get more values)

aban 28 140 270 496 672 5058000640 62487000576 abundant 140 270 672 1638 2970 6200 8190 18600 18620 27846 + 37035180 44660070 45532800 46683000 admirable 140 270 672 1638 6200 30240 32760 105664 2178540 23569920 45532800 alternating 270 496 672 1638 amenable 28 140 496 672 6200 8128 18600 18620 30240 32760 + 825120800 886402440 900463200 995248800 apocalyptic 1638 6200 8128 8190 18600 18620 27846 arithmetic 140 270 672 1638 2970 6200 8190 18600 18620 27846 + 5772200 6051500 8506400 8872200 betrothed 140 binomial 28 496 8128 695520 33550336 8589869056 137438691328 c.nonagonal 28 496 8128 33550336 8589869056 137438691328 congruent 28 270 496 1638 2970 6200 8128 8190 18620 27846 + 4713984 4754880 5772200 8506400 Cunningham 28 32760 Curzon 270 1638 8190 1089270 11981970 81695250 dig.balanced 27846 55860 167400 242060 2178540 2845800 11981970 15495480 164989440 191711520 droll 672 evil 270 1638 8190 27846 30240 32760 55860 173600 237510 360360 + 481572000 500860800 526480500 714954240 fibodiv 28 frugal 33550336 gapful 140 1638 18600 18620 27846 30240 32760 117800 167400 173600 + 87825283840 93419333280 95088913920 95300150400 happy 28 496 8128 2290260 2457000 4713984 Harshad 140 270 1638 2970 6200 8190 18600 30240 32760 167400 + 8628633000 8659696500 8696764800 9866368512 hexagonal 28 496 8128 33550336 8589869056 137438691328 highly composite 332640 hoax 50401728 hyperperfect 28 496 8128 33550336 8589869056 137438691328 iban 140 270 idoneal 28 inconsummate 6200 332640 695520 950976 interprime 270 2970 32760 55860 167400 753480 950976 1539720 2229500 2457000 + 18154500 52141320 56511000 71253000 Jordan-Polya 30240 junction 14303520 23963940 52141320 Lynch-Bell 672 metadrome 28 nialpdrome 6200 8872200 nude 672 8128 32997888 O'Halloran 140 oban 28 odious 28 140 496 672 2970 6200 8128 18600 18620 105664 + 825120800 886402440 900463200 995248800 pandigital 2178540 perfect 28 496 8128 33550336 8589869056 137438691328 pernicious 28 140 496 672 2970 6200 8128 18600 18620 105664 + 950976 2845800 4358600 8506400 plaindrome 28 practical 28 140 270 496 672 1638 2970 6200 8128 8190 + 5772200 6051500 8506400 8872200 pronic 2970 8190 pseudoperfect 28 140 270 496 672 1638 2970 6200 8128 8190 + 726180 753480 950976 33550336 repfigit 28 self 1638 18600 55860 726180 950976 4358600 428972544 758951424 766284288 825120800 Smith 2970 27846 super Niven 140 30240 360360 super-d 8190 105664 237510 332640 753480 superabundant 332640 tau 672 30240 23569920 45532800 164989440 447828480 623397600 714954240 taxicab 46683000 tetrahedral 695520 triangular 28 496 8128 33550336 8589869056 137438691328 uban 28 Ulam 28 8128 18620 360360 unprimeable 18600 18620 32760 55860 173600 237510 360360 695520 753480 950976 + 2290260 2845800 4754880 8872200 untouchable 8190 18600 30240 32760 332640 360360 695520 753480 950976 upside-down 28 wasteful 28 140 270 496 672 1638 2970 6200 8128 8190 + 5772200 6051500 8506400 8872200 Zuckerman 672 2716826112 Zumkeller 28 140 270 496 672 1638 2970 6200 8128 8190 + 27846 30240 32760 55860