A206854 Smallest integer m such that m is a product of 2n-1 consecutive primes and a sum of 2n-1 consecutive primes.

2, 33263, 2775683761181, 52139749485151463, 31359251876786281892441299570699, 2385018819218440287149, 23509572623777698757692123744388316389653416929069870587, 436178570920976645136650311902311012102337977560516289614008518576769313, 166345108784858794943225366868487068031523855419640057875257310044811

Offset: 1

Author

Zak Seidov, Feb 13 2012

Keywords

• nonn

Comments

n=1: m = 2 (trivial case: product and sum of single prime, 2);

n=2: m = 33263 = product{29, 31, 37} = sum{11083, 11087, 11093};

n=3: m = 2775683761181 = product({293, 307, 311, 313, 317}) = sum({555136752211, 555136752221, 555136752227, 555136752251, 555136752271});

n=4: m = 52139749485151463=product({229, 233, 239, 241, 251, 257, 263})= sum({7448535640735789, 7448535640735843, 7448535640735867, 7448535640735877, 7448535640735991, 7448535640736009, 7448535640736087});

n=5: m = 31359251876786281892441299570699 = product({3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191}) = sum({3484361319642920210271255507593, 3484361319642920210271255507619, 3484361319642920210271255507719, 3484361319642920210271255507767, 3484361319642920210271255507923, 3484361319642920210271255507937, 3484361319642920210271255507941, 3484361319642920210271255508067, 3484361319642920210271255508133});

n=6: m = 2385018819218440287149 = product({67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109}) = sum({216819892656221844131, 216819892656221844133, 216819892656221844139, 216819892656221844169, 216819892656221844307, 216819892656221844331, 216819892656221844347, 216819892656221844373, 216819892656221844397, 216819892656221844401, 216819892656221844421}).

Crossrefs

Cf. A203619.

Programs

Maple

scp:= proc(x,n) local P,i,s;
  P:= Vector(n);
  P[1]:= nextprime(ceil(x/n));
  for i from 2 to n do P[i]:= nextprime(P[i-1]) od;
  s:= convert(P,`+`);
  while s > x do
    s:= s - P[n];
    P[2..n]:= P[1..n-1];
    if P[2] = 2 then return false fi;
    P[1]:= prevprime(P[2]);
    s:= s + P[1];
  od;
  evalb(s=x)
end proc:
f:= proc(n) local i,P,r;
     P:= ;
     r:= convert(P,`*`);
     while not scp(r,2*n-1) do
       r:= r/P[1];
       P[1..2*n-2]:= P[2..2*n-1];
       P[2*n-1]:= nextprime(P[2*n-2]);
       r:= r*P[2*n-1];
     od;
end proc:
f(1):= 2:
map(f, [$1..8]); # Robert Israel, Mar 13 2023

Extensions

a(7)-a(9) from Robert Israel, Mar 13 2023