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 Reversal creates average (Posted on 2018-10-10)
Let Sp(n) be a sum of all primes from p(1) to p(n) inclusive.
Let m(n) be the average value of all those primes, i.e. m(n)= Sp(n)/n.

Find the n-th prime N such that m(n) equals the reversal of N.

Example for n=6: Sp(6)=2+3+5+7+11+13= 41; m(6)=41/6; not the reversal of 13, that is 31 .
So 13 is not our prime.
Another 2-digit number is.

Find it.

Are there any additional i.e. "numbers over 100" solutions?

 No Solution Yet Submitted by Ady TZIDON No Rating

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 computer soln | Comment 1 of 7
For n=23, p(n)=83, Sp(n)/n=38

The code lists all integer Sp(n)/n for p(n) < 100,000 and this is the only one that complies. (I do not see how to prove there are no more.)

n         P(n)     Sp(n)/n
----------------------------------------------------
23          83   38.0000000

53         241   110.000000

853        6599   3066.00000

5034       48989   23066.0000

5233       51137   24091.0000

6162       61223   28921.0000

6658       66809   31535.0000

6752       67783   32033.0000

6974       70379   33213.0000

7098       71707   33874.0000

7177       72559   34295.0000

7228       73091   34567.0000

7231       73133   34583.0000

7396       75017   35465.0000

7698       78439   37084.0000

7706       78539   37127.0000

7738       78889   37299.0000

7840       80051   37848.0000

7975       81547   38575.0000

8006       81883   38742.0000

8127       83233   39394.0000

8793       90901   43007.0000

8970       92849   43972.0000

9050       93787   44408.0000

9085       94169   44599.0000

program pp

implicit none

integer i,n,cnt,sum,iave

real ave

sum=2

cnt=1

do i=3,100000,2

call isprime(i,n)

if(n.eq.1)then

cnt=cnt+1

sum=sum+i

ave=(1.*sum)/(1.*cnt)

iave=ave

if(abs(iave-ave).lt.0.001)print*,cnt,i,ave

endif

enddo

end

subroutine isprime(i,n)

implicit none

integer i,j,k,l,m,n

n=0

k=sqrt(1.*i)

do j=3,k

m=(1.*i)/(1.*j)

l=m*j

if(l.eq.i)go to 1

enddo

n=1

1       return

end

Edited on October 10, 2018, 8:32 pm
 Posted by Steven Lord on 2018-10-10 08:57:25

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