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

Author

Walter Nissen, Jul 09 2004

Keywords

• nonn
• more

Comments

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.

Examples

			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
		

References

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

Links

• 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

Crossrefs

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

Extensions

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