A095739 Numbers known to be solitary but not coprime to sigma.

18, 45, 48, 52, 136, 148, 160, 162, 176, 192, 196, 208, 232, 244, 261, 272, 292, 296, 297, 304, 320, 352, 369

Offset: 1

Author

Walter Nissen, Jul 08 2004

Keywords

• nonn

Comments

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 W. Weisstein. - Franklin T. Adams-Watters, Feb 02 2006

Links

• Claude W. Anderson and Dean Hickerson, Advanced Problem 6020: Friendly Integers, Amer. Math. Monthly, 1977, V84#1p65-6.
• Eric Weisstein's World of Mathematics, Solitary Number.

Crossrefs

Cf. A095738, A074902.

Extensions

More terms from Franklin T. Adams-Watters, Feb 02 2006