The Art of Thinking

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 funct...

The lazy integer application  Let's apply formula log(w)  =  2 * Sum( ((w-1)/(w+1))^(2*n-1) / (2*n-1)  : n = 1 2 3 ...) for every real w > 0; see https://whmth.blogspot.com/2024/05/the-logarithmic-function.html EXAMPLES For  w := 2  we obtain         log(2)  =  2 * ( 1/3  +  1/(3*3^3)  +  1/(5*3^5)  +  1/(7*3^7)  + ... ) For  w:=3  we obtain         log(3)  =  2 * ( 1/2  +  1/(3*2^3)  +  1/(5*2^5)  +  1/(7*2^7)  +  ... ) For  w:=4  it's more efficient to apply log(4) = 2*log(2) The same goes for all composite integers: log(6) = log(2)+log(3)        log(8) = 3*log(2)        log(9) = 2*log(3) etc. Thus, let's concentrate on primes. We obtain log(5)  =  2 * ( 2/3  +  (2/3)^3 / 3  +  (2/3)^5 / 5  +  (2/3)^7 / 7  +  .....