A365834 a(n) is the first number k with n prime divisors such that, if m is the next number with n prime divisors, k + m and k - m also have n prime divisors. In each case the divisors are counted with multiplicity.

51, 130, 260, 520, 1040, 3120, 21440, 42880, 85760, 171520, 343040, 686080, 1372160, 2744320, 5488640, 10977280, 21954560, 43909120, 87818240, 175636480, 351272960, 702545920, 1405091840, 2810183680, 5620367360, 11240734720, 22481469440, 44962938880, 89925877760, 179851755520, 359703511040

Offset

2,1

Comments

The offset is 2 because it is impossible for the sum and difference of primes to both be prime.

Links

Robert Israel, Table of n, a(n) for n = 2..1000

Formula

a(n) = 2^(n-2) * 5 * 67 for all n >= 8, the next number with n prime divisors being 2^(n-2) * 3 * 113.

Proof:

k, m, k + m and m - k all have n prime divisors iff 2*k, 2*m, 2*k + 2*m and 2*m - 2*k all have n+1 prime divisor. 2*m is the next number after 2*k with n+1 prime divisors iff m is the next number after k with n prime divisors and between 2*k and 2*m there are no odd numbers with n+1 prime divisors. But the smallest odd number with n+1 prime divisors is 3^(n+1), and 3^(n+1) > 2^(n-1) * 3 * 113 for n >= 10. To complete the proof, we verify computationally the claim for n = 8, 9 and 10.

Example

a(3) = 130 because  130 = 2 * 5 * 13 is a triprime, the next triprime is 138 = 2 * 3 * 23, and 130 + 138 = 268 = 2^2 * 67 and 138 - 130 = 8 are triprimes, and this is the first triprime that works.

Maple

with(priqueue);
f:= proc(n) local M, a, b, v, q, w;
  M:= 10^1000:
  a:= 2^n:
  initialize(Aprimes);
  insert([-2^n, 0, 2], Aprimes);
  do
    v:= extract(Aprimes);
    if v[2] = n then
      b:= -v[1];
      if numtheory:-bigomega(b-a) = n and numtheory:-bigomega(b+a)= n then return a
      fi;
      a:= b;
    else
      insert(v+[0, 1, 0], Aprimes);
      q:= nextprime(v[3]);
      w:= v[1]*(q/v[3])^(n-v[2]);
      if w >= -M then insert([w, v[2], q], Aprimes) fi
    fi
  od:
end proc:
map(f, [$2..60]);

Crossrefs

Cf. A001222, A365833.

Sequence in context: A063314 A129298 A065010 * A097644 A353987 A044302

Adjacent sequences: A365831 A365832 A365833 * A365835 A365836 A365837

Keyword

nonn

Author

Zak Seidov and Robert Israel, Sep 19 2023

Status

approved