A211679 -id:A211679 - cp's OEIS frontend

A074902 Known friendly numbers.

6, 12, 24, 28, 30, 40, 42, 56, 60, 66, 78, 80, 84, 96, 102, 108, 114, 120, 132, 135, 138, 140, 150, 168, 174, 186, 200, 204, 210, 222, 224, 228, 234, 240, 246, 252, 258, 264, 270, 273, 276, 280, 282, 294, 300, 308, 312, 318, 330, 348, 354, 360, 364, 366, 372

Offset: 1

N. J. A. Sloane, Sep 15 2002

nonn

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

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

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

Union of A050972 and A050973. Cf. A014567.

Edited by Dean Hickerson, Sep 19 2002

A211677 First number k whose value of sigma(k)/k appears n times.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, May 09 2012

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.
a(10) <= 27477725184. a(11) <= 88071903612. a(12) <= A027687(12). - Donovan Johnson, Aug 06 2012
For n>1, these are the smallest numbers to appear consecutively (n-1) times in A050973. - Michel Marcus, Jan 28 2014

			These are the values of k such that sigma(k)/k appears n times:
n   k values
1:  1
2:  6, 28
3:  6, 28, 496
4:  84, 270, 1488, 1638
5:  84, 270, 1488, 1638, 24384
6:  210, 17360, 43400, 284480, 2229500, 2886100
7:  3780, 66960, 167400, 406224, 1097280, 6656832, 13035330
8:  3780, 66960, 167400, 406224, 1097280, 6656832, 13035330, 29410290
9:  164989440, 270138960, 318729600, 326781000, 481572000, 623397600, 675347400, 995248800, 4426793280 - _Donovan Johnson_, Aug 06 2012
These numbers appear in A211679.

Claude W. Anderson and Dean Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84 (1977) pp. 65-66.
Achim Flammenkamp, Multiply Perfect Numbers (sigma(k)/k is an integer)
Tom De Medts, MathOverflow: Recovering n from sigma(n)/n
Carl Pomerance, Multiply perfect numbers, Mersenne primes and effective computability, Math. Ann. 226 (1977), 195-206.
Eric W. Weisstein, Friendly number
Eric W. Weisstein, Friendly pair
Wikipedia, Friendly number

Cf. A000203 (sigma), A050973, A211679.

a(7)-a(8) from Donovan Johnson, May 10 2012

a(9) from Michel Marcus and Donovan Johnson, Aug 06 2012

cp's OEIS Frontend

A074902 Known friendly numbers.

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