A175907 Known friendly squarefree numbers.
6, 30, 42, 66, 78, 102, 114, 138, 174, 186, 210, 222, 246, 258, 273, 282, 318, 330, 354, 366, 390, 402, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 642, 654, 678, 690, 714, 762, 786, 798, 806, 822, 834, 858, 870, 894, 906, 930, 942, 966, 978, 1002, 1038
Offset: 1
Keywords
Examples
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.
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
Programs
-
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
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
Comments