Sums, but other...
Smallest sum of n primes ending with N zeroes
10=2+3 + 5 (n=3, N=1)
100=2+...+ 23 (n=9, N=2)
63731000=2+...+ 35677 (n=3795, N=3)
515530000=2+...+ 106853 (n=10183, N=4)
15570900000=2+...+ 632501 (n=51531, N=5)
29057028000000=2+...+ 31190879 (n=1926965, N=6)
98078160000000=2+...+ 58369153 (n=3471031, N=7)
12606879200000000=2+...+707712517, (n=36631619, N=8)
(n=??, N=9)
Zak At The Prime Service

--- In primenumbers@yahoogroups.com, "Zak Seidov" <seidovzf@y...>
wrote:
> (*
> sorry, Jud,
> i've prepared this message
> right before I saw your remarkable one...
> *)
>
> Mark,
> I've checked your observation for n0<=23 and nf<=800,000,
> and n0 up to 200 for nf<=100,000 with no new result.
> (* sure this sentence's obsolete*)
>
> Then I relaxed the rule,
> asking the sum to be a factor of prime[n0],
> then i guess
> for any n0 we can find such a sum for sure!
>
> So,
> for each n0 up to 300 (that is prime0 = 1987),
> i've found FIRST sum
> (from prime[no] up to prime[nf]) which a factor of prime0),
> i present some:
>
> {n0,p[n0],nf,p[nf],sum/p[n0]}
>
> each n0 up to 10:
>
> {1,2,3,5,5},
> {2,3,4,7,5},
> {3,5,9,23,19},
> {4,7,10,29,17},
> {5,11,11,31,13},
> {6,13,12,37,13},
> {7,17,11,31,7},
> {8,19,16,53,17},
> {9,23,13,41,7},
> {10,29,40,173,103}
>
> Notice that all sum/p0 are prime!!
> But n0=11 ruthlessly stopped this nice rule,
> and from first 300 only 57 sum/p0 are prime...
>
> each 10-th n0 up to 300:
>
> {10,29,40,173,103},
> {20,71,33,137,20},
> {30,113,166,983,643},
> {40,173,136,769,259},
> {50,229,182,1091,375},
> {60,281,378,2593,1571},
> {70,349,217,1327,352},
> {80,409,603,4441,2988},
> {90,463,280,1811,463},
> {100,541,1058,8461,7643},
> {110,601,1333,10979,11310},
> {120,659,837,6449,3756},
> {130,733,743,5651,2592},
> {140,809,360,2423,437},
> {150,863,387,2671,480},
> {160,941,236,1487,99},
> {170,1013,1512,12653,8757},
> {180,1069,2280,20149,20061},
> {190,1151,2082,18169,15321},
> {200,1223,673,5021,1192},
> {210,1291,364,2459,225},
> {220,1373,1045,8329,2850},
> {230,1451,4154,39461,53117},
> {240,1511,3056,28001,26495},
> {250,1583,2617,23539,18138},
> {260,1657,2769,25087,19544},
> {270,1733,1699,14503,6504},
> {280,1811,1716,14653,6351},
> {290,1889,2263,20011,11086},
> {300,1987,2837,25763,17132}
>
> to be clear,
> i repeat notations for the last line:
>
> {n0=300,prime[n0]=1987,nf=2837,prime[nf]=25763,
> (1987+...+25763)/1987=17132}}
>
> Zak
>
>
> --- In primenumbers@yahoogroups.com, Jud McCranie <judmccr@b...>
> wrote:
> > At 10:44 PM 7/19/2003, Mark Underwood wrote:
> > >I found a pretty result which I thought might also have promise:
> > >
> > >The sum of a sequence of consecutive primes at some point equals
> the
> > >product of the first and last primes!
> > >
> > >For instance,
> > >
> > >2+3+5 = 2*5
> > >
> > >3+5+7+11+13 = 3*13
> > >
> > >5+7+11+13+17+19+23+29+31 = 5*31
> > >
> > >7+11+13+17+19+23+29+31+37+41+43+47+53 = 7*53
> > >
> > >This is such a strong result that I was almost floored when I
> > >discovered that it fails for all the rest of the primes I tested
> > >under 200 !
> >
> >
> > See sequences A055514 and A055233, and
> > http://www.primepuzzles.net/puzzles/puzz_098.htm
> >
> >
> >
> > [Non-text portions of this message have been removed]