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Some prime sequences (Posted on 2018-07-04) Difficulty: 3 of 5
What numbers N are the sum of a sequence of consecutive primes such that the product of the first and last numbers in the sequence is N?

No Solution Yet Submitted by Ady TZIDON    
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Here are some | Comment 1 of 11

Choosing amongst only the first 9592 primes: 

 10 =  2 +  3 +  5

 39 =  3 +  5 +  7 + 11 + 13

 155 =  5 +  7 + 11 + 13 + 17 + 19 + 23 + 29 + 31

 371 =  7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53

I imagine an upper limit can be analytically determined where the product surpasses any possible sum by virtue of the magnitude of the 1st prime exceeding the sqrt of the sum of any possible span of primes up to the 2nd prime of the product (he said, waving his hands frantically...).   



        program sss

        implicit none

        integer i,j,k,primes(1000000),pcnt,lim,len,

        1 ll,last,l,maxlen,ibeg

        integer*16 prod,sum


c

c generate primes from 1 to 10^i

c

        primes(1)=2

        pcnt=1

2       print*,'k?'

        read*,k

        if(k.lt.2.or.k.gt.7)go to 2

            do 1 j=3,10**k,2

            lim=sqrt(1.*j)+1

                do i=3,lim,2

                if(1.*i*int((1.*j)/i).eq.j)go to 1         

                enddo

            pcnt=pcnt+1

            primes(pcnt)=j

c           print*,j

c           if(j.gt.30)stop

1           enddo

        

            do ibeg=1,pcnt-1

c           print*,'ibeg= ',ibeg

            maxlen=pcnt-ibeg+1

                do len=2,maxlen

                last=ibeg+len-1

                sum=0

                    do l=ibeg,last

                    sum=sum+primes(l)

                    enddo

                prod=primes(ibeg)*primes(last)

                    if(sum.eq.prod)then

c                   print*,' sum ',sum

                    print 3,sum,(primes(ll),ll=ibeg,last)

3                   format(i4,' = ',30(i2,1x,'+ '))

                    endif

                enddo

            enddo

c           do i=1,20

c           print*,primes(i)

c           enddo

        print*,pcnt

        end

rabbit-3:~ lord$ sss

 k?

5

  10 =  2 +  3 +  5 + 

  39 =  3 +  5 +  7 + 11 + 13 + 

 155 =  5 +  7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 

 371 =  7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 


Edited on July 5, 2018, 1:50 pm
  Posted by Steven Lord on 2018-07-04 11:27:34

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