1,1

The sequence is not known to be complete up to 372, since there are many small numbers, including 10, 14, 15 and 20, which have not been proved to be solitary. If any other numbers up to 372 are friendly, the smallest corresponding values of m are > 10^30.

A positive integer n is 'friendly' if abundancy(n) = abundancy(m) for some positive integer m not equal to n, where abundancy(n) = sigma(n)/n (cf. A000203); otherwise n is 'solitary'. (The name "friendly" is also sometimes mistakenly used with other meanings; cf. A063990 and A007770.)

All perfect numbers are friendly numbers, but they are only friendly with each other (a perfect number being defined as having abundancy index of 2.) - Daniel Forgues, Jun 23 2009

Triangle A211679 has rows that list the first numbers that have n-1 smaller friends. Sequence A211677 lists just the last number in each row. - T. D. Noe, May 10 2012

Claude W. Anderson and Dean Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84 (1977) pp. 65-66.

Tapas Chatterjee, Sagar Mandal, and Sourav Mandal, On characterizing potential friends of 20, Ann. West Univ. Timisoara Math. Comp. Sci. (2025) Vol. 61, Iss. 1, 205-229. See p. 206.

Sagar Mandal, Prime Divisors of 10's Friends: A Generalization of Prior Bounds, arXiv:2412.02701 [math.GM], 2024. See p. 10.

Sagar Mandal, Exploring the relationships between the divisors of friends of 10, Bull. Calcutta Math. Soc. (2025) Vol. 48, No. 1-3, 21-32.

Sagar Mandal and Sourav Mandal, Upper bounds for the prime divisors of friends of 10-II, arXiv:2412.02701 [math.GM], 2024. See p. 8.

Eric Weisstein's World of Mathematics, Friendly Pair

Eric Weisstein's World of Mathematics, Friendly Number

24 is in the sequence since abundancy(24) = abundancy(91963648) = 5/2.

Union of A050972 and A050973. Cf. A014567.

Sequence in context: A081512 A096387 A094185 * A096366 A247145 A188158

Adjacent sequences: A074899 A074900 A074901 * A074903 A074904 A074905

N. J. A. Sloane, Sep 15 2002

Edited by Dean Hickerson, Sep 19 2002

approved