A084440 Number of ways to write the n-th prime as 1+p+p^k, p prime and k > 0.

0, 0, 1, 2, 2, 1, 0, 1, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset

1,4

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(A084441(n)) > 0, a(A084442(n)) = 1, a(A084443(n)) = 0.

Example

a(11) = 2: prime(11) = 31 = 1+5+5^2 = 1+3+3^3.

Maple

N:= 200:
V:= Vector(N):
P:= [seq(ithprime(i), i=1..N)]:
for i from 1 to N do
  p:= P[i];
  for k from 1 do
    q:= 1+p+p^k;
    if q > P[N] then break fi;
    r:= ListTools:-BinarySearch(P, q);
    if r > 0 then V[r]:= V[r]+1 fi;
od od:
convert(V, list); # Robert Israel, Dec 13 2023

Mathematica

f[p_] := Module[{qs = FactorInteger[p - 1][[;; , 1]]}, Sum[Boole[p - q - 1 == q^IntegerExponent[p - q - 1, q]], {q, qs}]]; f[2] = 0; f /@ Prime[Range[100]] (* Amiram Eldar, Mar 25 2025 *)

Crossrefs

Cf. A084441, A084442, A084443.

Sequence in context: A111407 A374065 A382074 * A179287 A016372 A016342

Adjacent sequences: A084437 A084438 A084439 * A084441 A084442 A084443

Keyword

nonn

Author

Reinhard Zumkeller, May 26 2003

Status

approved