1,1

Abundancy is defined as the ratio of the multiplicative sum-of-divisors function to the integer itself: abund(n) = sigma(n)/n. E.g. abund ( 10 ) = sigma ( 10 ) / 10 = (1+2+5+10) / 10 = 1.8 = 9 / 5.

Integers m and n are friendly iff they have the same abundancy. E.g. abund ( 12 ) = abund ( 234 ) = 7 / 3 ===> 12 and 234 are friends.

Integers which have no friends are called solitary.

"It is believed that 10, 14, 15, 20, 22, 26, 33, 34, 38, 44, 46, 51, 54, 58, 62, 68, 69, 70, 72, 74, 76, 82, 86, 87, 88, 90, 91, 92, 94, 95, 99, 104, 105, 106, and many others are also solitary, although a proof appears to be extremely difficult." Quote from Eric Weisstein. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 02 2006

Anderson, Claude W. and Hickerson, Dean; Advanced Problem 6020, "Friendly Integers", Amer. Math. Monthly, 1977, V84#1p65-6.

Eric Weisstein's World of Mathematics, Solitary Number.

Cf. A095738, A074902.

Sequence in context: A045264 A049067 A114814 this_sequence A055577 A002798 A124388

Adjacent sequences: A095736 A095737 A095738 this_sequence A095740 A095741 A095742

nonn

Walter Nissen (wnissen(AT)tfn.net), Jul 08 2004

More terms from Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 02 2006