Baroque numbers (a new start)

Introduction 

Perfect numbers were recognized and fairly popular for millennia, especially the first perfect number  6 where

2*6  =  1+2+3+6

-- here to the right of sign = we see all divisors of  6.

Less popular were the multiply perfect numbers such as 120, where

3 * 120   =   1+2+4+8  +  3+6+12+24  +  5+10+20+40  +  15+30+60+120

-- here to the right of sign = we see all divisors of  120; indeed:

1+2+4+8  +  3+6+12+24  +  5+10+20+40  +  15+30+60+120  =

(1+2+4+8) * (1+3+5+15)  =

15 * 24  =  360  =

3 * 120

The other common names in the literature of  multiply perfect numbers  are  multiperfect number,  and  pluperfect number. However, years ago in place of these names I have introduced:

baroque numbers

I'll provide the related definitions and elementary properties in what follows.


The sum of divisors function s(n)


Given a natural number  n  (as 1 2 3 ...), let  s(n)  stand for the sum of all divisors of  n.  For instance:

  • s(1)  =  1
  • s(2)  =  1+2  =  3
  • s(3)  =  1+3  =  4
  • s(4)  =  1 + 2 + 4  =  7
  • s(5)  = 1 + 5  =  6
  • s(6)  =  1 + 2 + 3 + 6  =  12
  •                 etc.
If  n > 1  then  s(n) > 1 + n = n+1.

If  p  is prime (hence  p>1) then  s(p) = 1+p.

If  p  is prime,  and  f  is a non-negative integer (i.e.  f = 1  or  f  is natural)  then

s(p^f)   =   1 + p + ... + p^f  =  (p^(f+1)-1) / (p-1)
For instance:
  • s(8)  =  s(2^3)  =  1+2+4+8  =   15
  • s(16)  =  s(2^4)  =  1+2+4+8+16  =  31
  • s(32)  =  s(2^4)  =  1+2+4+8+16+32  =  63
  •  
  • s(9)  =  s(3^2)  =  1+3+9  =  13
  • s(27)  =  s(3^3)  =  1+3+9+27  =  40
  • s(81)  =  s(3^4)  =  1+3+9+27+81  =  121
  • s(243)  =  s(3^5)  =  1+3+9+27+81+243  =  364
  •  
  • s(25)  =  s(5^2)  =  1+5+25  =  31
  • s(125)  =  s(5^3)  = 1+5+25+125  =  156
  •  
  • s(49)  =  s(7^2)  =  1+7+49  =  57
  • s(343)  =  s(7^3)  =  1+7+49+343  =   400
  •  
  •                       etc.

Remark  If natural  n > 1  is not prime,  and  f  is natural, then

s(n^f)  >  1 + n + ... + n^f

Indeed, there is natural  d  such that. d | n  (i.e.  d  divides  n)  such that. 1 < d < n.  Thus,  s(n^f)  is greater or equal:

s(n^f)  >/  d + (1 + n + ... + n^f)  >  1 + n + ... + n^ f

Moreover:

s(n^f)  >/  (1+d)*(1 + n + ... + n^(f-1)) + n^f

Divisors

It's time to introduce divisors explicitly.

Let  d =/= 0  and  n  be arbitrary integers. Then an ancient Greek theorem tells us that there is a unique pair  of integers q  r  such that:

n  =  q*d + r
and
0 \< r < |d|

Thus, we say that  d  divides  n, or that  d. is a divisor of  n, symbolically   d | n  <=:=>  r=0.

I other words,  integer. d =/= 0  divides integer  <==>  there exists an integer  q  such that  n = q*d (such  q  if it exists is unique).

We see that:
  • Integer  1  is a divisor of every integer  n, i.e.  1 | n;
  • integer  -1  is a divisor of every integer  n, i.e.  -1 | n;
  • if  0 | n  then  n=0  (for arbitrary integer  n);
  • n | n  for arbitrary integer  n;
  • if  k | m  and  m | n  then  k | n  (for arbitrary integers  k  m  n);
  • if  d | n  then  -d | n  (for arbitrary integers  d n);
  • if  d | n > 0  then  |d| \< n   (for arbitrary integers  d n);

gcd(m n) and coprime numbers

Let  m n  be arbitrary natural numbers. Natural number. d  is said to be a common divisor of  m  and  n  <=:=>  d|m  and  d | n.

A common divisor  d  of  m  and  n  is said to be the greatest common divisor of  m  and  n,  or  d := gcm(m n)  <=:=>  for every common divisor  c  of  m  and  n, integer  c  divides  d,  i.e.  c | d.

Multiplicativity of s(n)



(under construction)


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