A211677 - OEIS

%I #30 Feb 16 2025 08:33:17

%S 1,28,496,1638,24384,2886100,13035330,29410290,4426793280

%N First number k whose value of sigma(k)/k appears n times.

%C The values of sigma(k)/k are 1, 2, 2, 8/3, 8/3, 96/35, 32/9, 32/9, 32/7. Note that these values are nondecreasing. Is that always the case? In the table below, all numbers in the same row are friendly to each other.

%C a(10) <= 27477725184. a(11) <= 88071903612. a(12) <= A027687(12). - _Donovan Johnson_, Aug 06 2012

%C For n>1, these are the smallest numbers to appear consecutively (n-1) times in A050973. - _Michel Marcus_, Jan 28 2014

%H Claude W. Anderson and Dean Hickerson, <a href="https://www.jstor.org/stable/2318325">Problem 6020: Friendly Integers</a>, Amer. Math. Monthly 84 (1977) pp. 65-66.

%H Achim Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/mpn.html">Multiply Perfect Numbers</a> (sigma(k)/k is an integer)

%H Tom De Medts, <a href="http://mathoverflow.net/questions/56376">MathOverflow: Recovering n from sigma(n)/n</a>

%H Carl Pomerance, <a href="http://www.math.dartmouth.edu/~carlp/PDF/paper13.pdf">Multiply perfect numbers, Mersenne primes and effective computability</a>, Math. Ann. 226 (1977), 195-206.

%H Eric W. Weisstein, <a href="https://mathworld.wolfram.com/FriendlyNumber.html">Friendly number</a>

%H Eric W. Weisstein, <a href="https://mathworld.wolfram.com/FriendlyPair.html">Friendly pair</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Friendly_number">Friendly number</a>

%e These are the values of k such that sigma(k)/k appears n times:

%e n   k values

%e 1:  1

%e 2:  6, 28

%e 3:  6, 28, 496

%e 4:  84, 270, 1488, 1638

%e 5:  84, 270, 1488, 1638, 24384

%e 6:  210, 17360, 43400, 284480, 2229500, 2886100

%e 7:  3780, 66960, 167400, 406224, 1097280, 6656832, 13035330

%e 8:  3780, 66960, 167400, 406224, 1097280, 6656832, 13035330, 29410290

%e 9:  164989440, 270138960, 318729600, 326781000, 481572000, 623397600, 675347400, 995248800, 4426793280 - _Donovan Johnson_, Aug 06 2012

%e These numbers appear in A211679.

%Y Cf. A000203 (sigma), A050973, A211679.

%K nonn,hard,more

%O 1,2

%A _T. D. Noe_, May 09 2012

%E a(7)-a(8) from _Donovan Johnson_, May 10 2012

%E a(9) from _Michel Marcus_ and _Donovan Johnson_, Aug 06 2012