OFFSET
1,1
COMMENTS
From Walter Nissen, May 28 2011: (Start)
As with most aspects of friendly and solitary numbers, this sequence is not known to be complete. A friend could possibly be found for 10, for example; same doubtful status as an odd perfect number.
Note that not all friendly numbers will be found among the primitive friendly numbers listed in link "Primitive Friendly Pairs", and this would be true even if those were not limited to small examples.
Other terms are 1330, 1995, and 49166.
(End)
REFERENCES
Oystein Ore, Number Theory and Its History, McGraw-Hill, 1948, reprinted 1988, section 5-3, pp. 96-100.
LINKS
Claude W. Anderson and Dean Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84 (1977) pp. 65-66.
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.
Walter Nissen, Primitive Friendly Pairs
Walter Nissen, Abundancy: some resources
Eric Weisstein's World of Mathematics, Solitary Number
EXAMPLE
6, being 2 * 3, is squarefree. Having abundancy = 2, 6 is friendly with all the other perfect numbers. Ergo, it is in the sequence. ( 1 ), 2, 3, and 5, being prime powers, are solitary. 4 is a square. Ergo, a(1) is 6.
PROG
(PARI) { for (j=1, 2000, if (issquarefree(j), t=sigma(j)/j; for (i=1, 1000000, p=sigma(i)/i; if(p == t && j != i, print(j, " ", i); ); ); ); ); quit; } \\ provides useful suggestions, but not definitive, Walter Nissen, May 28 2011
CROSSREFS
KEYWORD
nonn
AUTHOR
Thomas Kellar, Oct 14 2010, Oct 15 2010
EXTENSIONS
Added 273 as it is friendly with 2876211; 273 is a counterexample to the conjecture that 6 divides a(n). - Walter Nissen, May 28 2011
Added 806 as it is friendly with 2449562488893. - Suyash Pandit, Jan 24 2024
STATUS
approved