### Primitive Friendly Integers and Exclusive Multiples

Date:     Fri, 2004-07-02 19:36:26 +0000
From:     Walter Nissen
To:       nmbrthry@listserv.nodak.edu
Subject:  Primitive Friendly Integers and Exclusive Multiples

Greetings, all.

Primitive Friendly Integers and Exclusive Multiples

- - -                                 "I hope to make many new friends".
- - -                                                - Marty, a mandrill

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.

All of the perfects are friends of one another.  In particular, they all
have abundancy 2 and thus, by definition, are all said to be 2-perfect.

Integers which have no friends are called solitary.

Multiperfect integers are those of integer abundancy.  n-multiperfects
are proper multiperfects iff n > 2.  E.g., abund ( 120 ) = 3 and thus
120 is 3-perfect, a proper multiperfect, more particularly, the
smallest proper multiperfect.

A fundamental concept is the primitive friendly pair.  Friends m and n
are primitive friendly iff they have no common prime factor of the same
multiplicity.

E.g., 6 and 28 are primitive friendly; while they are not coprime
because they share the common factor 2, the factor 2 appears twice in 28
but only once in 6.  More particularly, they are 2-primitive-friendly.
30 and 140 are 2.4-friendly, but not primitive friendly, because both
are divisible by 5 and not by 25, i.e., the factor 5 appears in both
with identical multiplicity = 1.  Generalizing, all the even perfects
are 2-primitive-friendly.

With this background, the 3 tables of Anderson and Hickerson (see
reference below) become:

The six tables:

Solitary because prime powers:
(1), 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32,
37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97

Solitary because coprime to sigma, but not prime powers:
21, 35, 36, 39, 50, 55, 57, 63, 65, 75, 77, 85, 93, 98, 100

Solitary, but not coprime to sigma:
18, 45, 48, 52

Primitive Friendly:
6, 12, 24, 28, 30, 40, 42, 56, 60, 80, 84, 96

Friendly, not known to be Primitive Friendly:
66, 78

Unknown, not known to be friendly nor solitary:
10, 14, 15, 20, 22, 26, 33, 34, 38, 44, 46, 51, 54, 58, 62, 68, 69, 70,
72, 74, 76, 82, 86, 87, 88, 90, 91, 92, 94, 95, 99

If j and k are friendly, and m is coprime to both,
then mj and mk are friendly.
(regardless of whether j is coprime to k)
(regardless of how many different prime factors m has)
So every primitive friendly pair generates a huge family of friendly
pairs.
And every friendly pair is either primitive or a member of a family
generated by a primitive pair.

The concept which underlies this is "exclusive multiple".  E.g.,
mj is an exclusive multiple of j, iff m is coprime to j.
The family generated by a primitive friendly pair consists of the
exclusive multiples of the pair.  To generate such a family completely
from a friendly pair which is not a primitive friendly pair requires
stripping out the prime powers of identical exponent from each of the
integers, thus producing a primitive friendly pair, and then generating
all of the exclusive multiples of that derived primitive friendly pair.

Based on a less fundamental concept, it is possible to generate a family
of friendly pairs from any friendly pair by producing exclusive
multiples of the pair.  But then you can diddle the identical
exponent(s) and pretty soon you are back to the family generated from a
primitive friendly pair consisting of its exclusive multiples.
Hence, the term "primitive friendly pair".

The primitive friendly pair of least known abundancy is the
1.06371191135734072-primitive-friendly pair {45847; 17927087081}, in
epprep, the (7100000-2)-primitive-friendly pair {00000002;127;
00000004;151;911}.  (For epprep, see reference below).  It plays an
important role in substitutions in multiperfects.

Fred Helenius reports that the following 2 numbers are 8-perfect, making
them 8-primitive-friendly.
2^47 3^26 5^9 7^10 11^4 13^5 17 19 23 29 31^3 37^2 43^2 47 61 67^2 71 73
79^2 97 109 137 157 179 241 257 281 337 379 433 521 631 673 757 821 1123
3221 8209 293459 ,
2^85 3^25 5^11 7^7 11^3 13^7 17^2 19^3 23^2 29^2 31 37 43 53 59 61^2 67
79 83^2 97^2 107 127 181 193 307 317 331 367 431 601^2 761 769 811 1069
1201 1621 3169 4639 9277 9719 14281 19993 252283 398581^2 797161 2099863
32668561 8831418697 2932031007403 .

Letting fp be a friendly pair and pfp a primitive friendly pair, here
are a few questions which naturally arise:
What values of abund() are possible?
What values of abund() can possibly give rise to friends?
What values of abund() can possibly give rise to primitive friends?
Are there infinitely many pfps?
What 3-perfects are exclusive multiples of 15 (or 21)?
Which multiperfect has the strangest "shape"?  I.e., shape of its
exponents?  (Needs definition of "shape").  Compare to "shape" of
factorials, superabundants, etc.
abund ( fp ) can be arbitrarily large, but can abund ( pfp ) be
arbitrarily large?
Every pair of multiperfects of the same abund(), divested of their
common factors having the same multiplicity, generates a pfp.  But are
there pfps not so generated?

References:

Anderson, Claude W. and Hickerson, Dean;  Advanced Problem 6020,
"Friendly Integers",  Amer. Math. Monthly, 1977, V84#1p65-6.

Hickerson, Dean; "Re: friendly/solitary numbers [was: typos]", post to
seqfan mailing list, 2002, courtesy Olivier Gerard (G'erard).

Hickerson, Dean; "Re: Friendly number", post to sci.math newsgroup,

Nissen, "Exponential Prime Power Representation", post to sci.math,
1995, available through groups.google.com, describes epprep, which
facilitates calculation and communication.



The article above originally was posted to the NMBRTHRY mailing list .

The six tables of the article are represented in Neil J. A. Sloane's On-Line Encyclopedia of Integer Sequences :
Solitary because prime powers
Solitary because coprime to sigma, but not prime powers
Solitary, but not coprime to sigma
Primitive Friendly
Friendly, not known to be Primitive Friendly
but not : Unknown, not known to be friendly nor solitary

1093 small primitive friendly pairs appear as a supplement to this article .

Some pfps can be generated from a formula .

Better Web links for some references above :

Anderson, C. W. and Hickerson, Dean
"Friendly Integers"
Amer. Math. Monthly, 1977, V84#1p65-6.
Available from JSTOR .

Dean Hickerson
Re: friendly number
2000 Jan 31

Walter Nissen
Exponential Prime Power Representation
post to sci.math , 1995

More references :

L. Alaoglu , Erd"os P'al = Paul Erdos
On Highly Composite and Similar Numbers
Transactions of the American Math. Society , 1944 Nov , V56#3pp448-469

Erd"os P'al = Paul Erdos
Remarks on number theory, II ; Some problems on the sigma function
Acta Arithmetica , V5 , 1959 , pp171-177

Richard F. Ryan
Results concerning uniqueness for
{\sigma (x)/x = \sigma (p^n q^m )/(p^n q^m)} and related topics
International Math. J. , 2002 , V2#5pp497-514

Wm. G. Stanton and Judy A. Holdener
Abundancy "Outlaws" of the Form (sigma(N) + t)/N
Journal of Integer Sequences , 2007 , V10 , Article 07.9.6

2000 Mathematics Subject Classification : Primary 11A25 ; Secondary 11A05 , 11A41 , 11A51 , 11N25 , 11Y55 , 11Y70

Walter Nissen
posted 2008-01-29