A095739 -id:A095739 - cp's OEIS frontend

A096366 Known primitive friendly integers.

6, 12, 24, 28, 30, 40, 42, 56, 60, 80, 84, 96, 108, 135, 140, 168, 200, 210, 224, 234, 240, 264, 270, 273, 280, 360, 380, 408, 480, 496

Offset: 1

Walter Nissen, Jul 01 2004

Friends m and n are primitive friendly iff they have no common prime factor of the same multiplicity.
There may be other primitive friendly integers within the range of those given, but they have yet to be calculated.
All perfect numbers are 2-primitive-friendly (since they are all products of distinct powers of 2 and distinct Mersenne primes). - Daniel Forgues, Jun 24 2009
A friendly integer can be both primitive and nonprimitive. For example, consider 30. First, 30 is friendly to 140, but this relation is nonprimitive, because it is 5 times the friendly pair {6, 28}. But then, 30 is also friendly to 6200, and this is a primitive pair (not a scaling of a smaller friendly pair). - Jeppe Stig Nielsen, Dec 07 2022

			While 6 and 28 are not coprime because they share the common factor 2, the factor 2 appears twice in 28 but only once in 6, so they are in the sequence.
From _Suyash Pandit_, Oct 15 2023: (Start)
280 is primitive friendly with 1553357978368 = 2^8*7^2*19^2*37*73*127;
360 is primitive friendly with 155086041146982400 = 2^20*5^2*7^3*13*31*127*337;
380 is primitive friendly with 31701183232 = 2^8*19^2*37*73*127;
408 is primitive friendly with 874453888 = 2^7*7*11*17^2*307. (End)

Claude W. Anderson and Dean Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84 (1977) pp. 65-66.
Dean Hickerson, Friendly number, post to newsgroup sci.math, Jan 31, 2000.
Walter Nissen, Primitive Friendly Integers and Exclusive Multiples, 2004 post to NMBRTHRY mailing list.

Cf. A014567, A074902, A095738, A095739.

Offset 1 from Michel Marcus, Dec 13 2022

Terms 280, 360, 380, and 408 from Suyash Pandit, Sep 16 2023

A095738 Numbers that are coprime to sigma but are not prime powers.

Original entry on oeis.org

21, 35, 36, 39, 50, 55, 57, 63, 65, 75, 77, 85, 93, 98, 100, 111, 115, 119, 129, 133, 143, 144, 155, 161, 171, 175, 183, 185, 187, 189, 201, 203, 205, 209, 215, 217, 219, 221, 225, 235, 237, 242, 245, 247, 253, 259, 265, 275, 279, 291, 299, 301, 305, 309, 319

Offset: 1

Walter Nissen, Jul 08 2004

nonn

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 if and only if they have the same abundancy. E.g., abund(12) = abund(234) = 7 / 3, so 12 and 234 are friends.
Integers which have no friends are called solitary.
The numbers in this sequence are solitary.
Compare abundancy to abundance as defined in A033880.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Claude W. Anderson and Dean Hickerson, Advanced Problem 6020: Friendly Integers, Amer. Math. Monthly, 1977, V84#1p65-6.
Walter Nissen, Primitive Friendly Integers and Exclusive Multiples, 2004 post to NMBRTHRY mailing list

Cf. A014567, A074902, A095739, A000203, A033880.

Select[Range[320], PrimeNu[#] > 1 && GCD[#, DivisorSigma[1, #]] == 1 &] (* Amiram Eldar, Jun 25 2019 *)

isok(n) = (gcd(sigma(n), n) == 1) && (! isprime(n)) && (! (ispower(n, , &p) && isprime(p))); \\ Michel Marcus, Jan 24 2014

Edited by Franklin T. Adams-Watters, Mar 06 2014

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

Original entry on oeis.org

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

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A096366 Known primitive friendly integers.

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A095738 Numbers that are coprime to sigma but are not prime powers.

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

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