A250204 Sierpiński problem in base 6: Least k > 0 such that n*6^k+1 is prime, or 0 if no such k exists.

1, 1, 1, 0, 1, 1, 1, 4, 0, 1, 1, 1, 1, 0, 2, 1, 1, 1, 0, 5, 1, 4, 1, 0, 1, 1, 1, 2, 0, 1, 2, 1, 1, 0, 1, 2, 1, 1, 0, 1, 5, 5, 2, 0, 1, 1, 1, 3, 0, 2, 1, 1, 7, 0, 1, 1, 2, 1, 0, 2, 1, 1, 1, 0, 2, 1, 8, 1, 0, 1, 2, 1, 1, 0, 7, 1, 1, 4, 0, 4, 1, 2, 1, 0, 2, 5, 1, 2, 0, 1, 1, 2, 3, 0, 1, 1, 9, 2, 0, 1, 1, 1, 1, 0, 1, 6, 1, 2, 0, 1, 3, 1, 4, 0, 1, 2, 23, 1, 0, 4

Offset

1,8

Comments

a(5k+4) = 0, since (5k+4)*6^n+1 is always divisible by 5, but there are infinitely many numbers not in the form 5k+4 such that a(n) = 0. For example, a(174308) = 0 since 174308*6^n+1 is always divisible by 7, 13, 31, 37, or 97 (See A123159). Conjecture: if n is not in the form 5k+4 and n < 174308, then a(n) > 0.

However, according to the Barnes link no primes n*6^k+1 are known for n = 1296, 7776 and 46656, so these may be counterexamples. - Robert Israel, Mar 17 2015

Links

Eric Chen, Table of n, a(n) for n = 1..1000

Gary Barnes, Sierpinski conjectures and proofs

Maple

N:= 1000: # to get a(1) to a(N), using k up to 10000
a[1]:= 1:
for n from 2 to N do
  if n mod 5 = 4 then a[n]:= 0
  else
    for k from 1 to 10000 do
    if isprime(n*6^k+1) then
       a[n]:= k;
       break
    fi
    od
  fi
od:
L:= [seq(a[n], n=1..N)]; # Robert Israel, Mar 17 2015

Mathematica

(* m <= 10000 is sufficient up to n = 1000 *)
a[n_] := For[k = 1, k <= 10000, k++, If[PrimeQ[n*6^k + 1], Return[k]]] /. Null -> 0; Table[a[n], {n, 1, 120}]

Prog

(PARI) a(n) = if(n%5==4, 0, for(k = 1, 10000, if(ispseudoprime(n*6^k+1), return(k))))

Crossrefs

Cf. A040076, A046067, A078680, A033809, A123159.

Cf. A250205 (Least k > 0 such that n*6^k-1 is prime).

Keyword

nonn

Author

Eric Chen, Mar 11 2015

Status

approved

A344393 T(n, k) = Eulerian1(n - k, k), for n >= 0 and 0 <= k <= floor(n/2). Triangle read by rows.

1, 1, 1, 0, 1, 1, 1, 4, 0, 1, 11, 1, 1, 26, 11, 0, 1, 57, 66, 1, 1, 120, 302, 26, 0, 1, 247, 1191, 302, 1, 1, 502, 4293, 2416, 57, 0, 1, 1013, 14608, 15619, 1191, 1, 1, 2036, 47840, 88234, 15619, 120, 0, 1, 4083, 152637, 455192, 156190, 4293, 1

Offset

0,8

Comments

The antidiagonal representation of the first order Eulerian numbers (A173018).

Links

Table of n, a(n) for n=0..55.

Example

Triangle starts:
[ 0] [1]
[ 1] [1]
[ 2] [1,    0]
[ 3] [1,    1]
[ 4] [1,    4,     0]
[ 5] [1,   11,     1]
[ 6] [1,   26,    11,     0]
[ 7] [1,   57,    66,     1]
[ 8] [1,  120,   302,    26,    0]
[ 9] [1,  247,  1191,   302,    1]
[10] [1,  502,  4293,  2416,   57, 0]
[11] [1, 1013, 14608, 15619, 1191, 1]

Maple

T := (n, k) -> combinat:-eulerian1(n - k, k):
seq(print(seq(T(n, k), k=0..n/2)), n = 0..11);

Crossrefs

Cf. A000800 (row sums).

Cf. A173018.

Keyword

nonn,tabf

Author

Peter Luschny, May 17 2021

Status

approved