A340799 a(n) is the smallest prime p such that p + 1 has 2n divisors.

2, 5, 11, 23, 47, 59, 191, 167, 179, 239, 5119, 359, 20479, 2111, 719, 839, 1114111, 1259, 786431, 3023, 2879, 15359, 348127231, 3359, 22031, 266239, 6299, 6719, 22280142847, 5039, 559419490303, 7559, 156671, 7798783, 25919, 10079, 1168231104511, 5505023

Offset

1,1

Comments

The only prime p such that p+1 has an odd number of divisors is 3, because every number with an odd number of divisors is a square. - Robert Israel, Jan 12 2025

Links

Robert Israel, Table of n, a(n) for n = 1..255

Formula

A000005(a(n) + 1) = 2n.

Example

a(4) = 23 because 23 is the smallest prime p such that p + 1 has 2*4 divisors; tau(24) = 8.

Maple

g:= proc(n, k) # lists of integers > k whose product is n
      option remember;
      local F, m;
      if n = 1 then return [[]]
      elif k >= n then return []
      fi;
      F:= select(`>`, numtheory:-divisors(n), k);
      [seq(op(map(t -> [m, op(t)], procname(n/m, m-1))), m=F)]
end proc:
children:= proc(t) local m, nt, S, i, qs, t3s, t1s;
    nt:= nops(t[3]);
    S:= select(i -> t[3][i] <> 2, [$1..nt]);
    if S = [] then m:= nt else m:= min(S) fi;
    qs:= [seq(nextprime(t[3][i]), i=1..m)];
    t3s:= [seq(subsop(i = qs[i], t[3]), i = 1..m)];
    t1s:= [seq(t[1]*(qs[i]/t[3][i])^t[2][i], i=1..m)];
    [seq([t1s[i], t[2], t3s[i]], i=1..m)]
end proc:
f:= proc(d) local pq, s, t, i; uses priqueue;
  initialize(pq);
  for t in g(2*d, 1) do insert([-mul(2^(t[i]-1), i=1..nops(t)), t -~ 1, [2$nops(t)]], pq) od;
  do
     t:= extract(pq);
     if nops(convert(t[3], set)) = nops(t[3]) and isprime(-t[1]-1) then return -t[1]-1 fi;
  for s in children(t) do insert(s, pq) od
  od:
end proc:
map(f, [$1..40]); # Robert Israel, Jan 12 2025

Prog

(Magma) Ax:=func<n|exists(r){m:m in[1..10^7] | IsPrime(m) and #Divisors(m + 1) eq n*#Divisors(m)}select r else 0>; [Ax(n): n in[1..20]];

Crossrefs

Cf. A000005 (tau), A003680, A080371.

Sequence in context: A347309 A174162 A379652 * A186253 A226462 A376193

Adjacent sequences: A340796 A340797 A340798 * A340800 A340801 A340802

Keyword

nonn,look

Author

Jaroslav Krizek, Jan 21 2021

Status

approved