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A175907
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Known friendly squarefree numbers.
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0
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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
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listen;
history;
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internal format)
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OFFSET
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1,1
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COMMENTS
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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)
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REFERENCES
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Oystein Ore, Number Theory and Its History, McGraw-Hill, 1948, reprinted 1988, section 5-3, pp. 96-100.
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LINKS
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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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
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STATUS
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approved
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