A074902 -id:A074902 - cp's OEIS frontend

A095751 Conjectured list of integers known to be friendly but not known to be primitive friendly.

66, 78, 102, 114, 120, 132, 138, 150, 174, 186, 204, 222, 228, 246, 252, 258, 276, 282, 294, 300, 308, 312, 318, 330, 348, 354, 364, 366, 372

Offset: 1

Walter Nissen, Jul 09 2004

There may be other integers in the sequence within the range of those given, but they have yet to be calculated and moreover, some of these given may prove to be primitive friendly.
Abundancy is defined as the ratio of the multiplicative sum-of-divisors function to the integer itself: abund(n) = sigma(n)/n. E.g., abund(10) = sigma(10) / 10 = (1+2+5+10)/10 = 1.8 = 9/5.
Integers m and n are friendly iff they have the same abundancy. E.g., abund(12) = abund(234) = 7/3 ===> 12 and 234 are friends.
Friends m and n are primitive friendly iff they have no common prime factor of the same multiplicity.

			66 is a friend of 308, 5456, 89408 and 369053696, but all of these are divisible by 11 and not 121, while 66 is not known to be primitive friendly.
280 is not a term because although 280 = 2^3*5*7 and 1553357978368 = 2^8*7^2*19^2*37*73*127 have the same abundancy they have no common prime factors of the same multiplicity and so are primitive friendly. It should be noted that 18620 = 2^2*5*7^2*19 also has the same abundancy. - _Suyash Pandit_, Sep 24 2023

Hickerson, Dean; "Re: Friendly number", post to sci.math newsgroup, 2000, available through groups.google.com.

Claude W. Anderson and Dean Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84 (1977) pp. 65-66.
Walter Nissen, Primitive Friendly Integers and Exclusive Multiples, 2004 post to NMBRTHRY mailing list

Cf. A014567, A074902, A095738, A095739, A096366.

Terms 280 and 360 removed by Suyash Pandit, Sep 24 2023

Added "Conjectured" to definition following comments from the Editors. - N. J. A. Sloane, Oct 09 2023

A140688 List of numbers which are both amicable and friendly.

Original entry on oeis.org

12285, 67095, 71145, 87633, 142310

Offset: 1

Walter Nissen, Jul 11 2008

There are almost certainly many other amicables both within and without the displayed range which also are friendly, but they have not yet been identified.
The usual OEIS policy is to display sequences only as far as they are known to be complete. How far is this sequence known to be complete? - N. J. A. Sloane, Jul 12 2008. The answer appears to be that nothing is known for certain. It may well be that 12285 is not even the first term! - N. J. A. Sloane, Dec 04 2008
It is not known if the two smallest amicable numbers, 220 and 284, are friendly. In fact it is not even known if 10 is friendly. - Walter Nissen, Dec 02 2008
The following numbers are all known to be members of this sequence: 12285, 67095, 71145, 87633, 142310, 525915, 863835, 947835, 1125765, 1798875, 3606850, 5357625, 5684679, 5730615, 6088905, 9206925, 9478910, 9491625, 10634085, 12361622, 13671735, 14426230, 17041010, 17257695, 17754165, 20308995, 20955645, 22227075, 22508145, 23111055, 23389695, 25132545, 34765731, 35115795, 36939357, 43266285, 53011395, 66595130, 74769345, 80422335, 82824255, 82977345, 84591405. - Dean Hickerson, Dec 02 2008 (communicated by Walter Nissen). However, until we have more definite information about the correctness of the first five terms (there could be additional terms less than 142310), there is no point in adding these terms to the "DATA" line. - N. J. A. Sloane, Nov 23 2011

			(69615, 87633) are an amicable pair with sigma = 157248. { 14445, 87633 } are a friendly pair of abundancy = 192/107. Therefore 87633 is a member of the sequence.
The smallest friend of 3606850 is 7521154875.

Dean Hickerson, "Re: Friendly number", post to sci.math newsgroup, 2000, available through groups.google.com.
Ore, Oystein, Number Theory and Its History, McGraw-Hill, 1948, reprinted 1988, section 5-3, pp 96-100.

Claude W. Anderson and Dean Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84 (1977) pp. 65-66.
Walter Nissen, Abundancy : Some Resources
Walter Nissen, Primitive Friendly Integers and Exclusive Multiples
J. O. M. Pedersen, Known Amicable Pairs [Broken link]
Eric Weisstein's World of Mathematics, Friendly Pair

Cf. A063990, A074902.

Edited by N. J. A. Sloane, Dec 04 2008 and Nov 23 2011

A175907 Known friendly squarefree numbers.

Original entry on oeis.org

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

Thomas Kellar, Oct 14 2010, Oct 15 2010

nonn

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)

			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.

Oystein Ore, Number Theory and Its History, McGraw-Hill, 1948, reprinted 1988, section 5-3, pp. 96-100.

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

Cf. A005117, A014567, A074902 (known friendly numbers), A095751, A096366, A140688.

{ 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

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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A095751 Conjectured list of integers known to be friendly but not known to be primitive friendly.

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A140688 List of numbers which are both amicable and friendly.

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A175907 Known friendly squarefree numbers.

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