A117494 -id:A117494 - cp's OEIS frontend

A122414 Triangle T(n,k) for 1 <= k <= n read by rows, where T(n,k) = 1 if gcd(n,k) is prime, 0 otherwise.

0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0

Offset: 1

Klaus Brockhaus, Sep 03 2006

			The triangle starts
  0
  0 1
  0 0 1
  0 1 0 0
  0 0 0 0 1
  0 1 1 1 0 0
  0 0 0 0 0 0 1
  0 1 0 0 0 1 0 0
  0 0 1 0 0 1 0 0 0
  0 1 0 1 1 1 0 1 0 0
  0 0 0 0 0 0 0 0 0 0 1
  0 1 1 0 0 0 0 0 1 1 0 0
  0 0 0 0 0 0 0 0 0 0 0 0 1
  0 1 0 1 0 1 1 1 0 1 0 1 0 0
  0 0 1 0 1 1 0 0 1 1 0 1 0 0 0

Cf. A010051 (diagonal), A122415 (sub-triangle).

Row sums are in A117494. [From Klaus Brockhaus, May 29 2009]

A122414 := proc(n,k)
    if isprime(igcd(n,k)) then
        1;
    else
        0;
    end if;
end proc: # R. J. Mathar, Apr 21 2021

row[n_] := Boole[PrimeQ[GCD[n, Range[n]]]]; Array[row, 14] // Flatten (* Amiram Eldar, May 23 2025 *)

{m=14; v=vector(m,x,vector(x)); for(n=1,m,for(k=1,n,if(isprime(gcd(n,k)),v[n][k]=1))); for(n=1,m,for(k=1,n,print1(v[n][k],",")))}

T(n,n) = A010051(n).

T(n,1) = 0.

A375487 a(n) is the number of integers k between 0 and n such that n AND k is a prime number (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 0, 1, 2, 0, 1, 2, 4, 0, 0, 4, 5, 0, 3, 2, 6, 0, 1, 8, 10, 0, 6, 4, 11, 0, 4, 4, 10, 0, 7, 2, 11, 0, 0, 16, 16, 0, 9, 8, 17, 0, 1, 8, 14, 0, 12, 4, 16, 0, 8, 8, 16, 0, 13, 4, 17, 0, 8, 4, 15, 0, 15, 2, 18, 0, 0, 32, 33, 0, 16, 16, 34, 0, 1, 16, 27, 0, 18, 8

Offset: 0

Rémy Sigrist, Aug 17 2024

			The first terms, alongside the corresponding k's, are:
  n   a(n)  k's
  --  ----  ------------------
   0     0  None
   1     0  None
   2     1  2
   3     2  2, 3
   4     0  None
   5     1  5
   6     2  2, 3
   7     4  2, 3, 5, 7
   8     0  None
   9     0  None
  10     4  2, 3, 6, 7
  11     5  2, 3, 6, 7, 11
  12     0  None
  13     3  5, 7, 13
  14     2  2, 3
  15     6  2, 3, 5, 7, 11, 13

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Scatterplot of (n, k) such that 0 <= k <= n <= 1024 and n AND k is prime

Cf. A000720, A102210, A117494, A375485 (XOR variant), A375486 (OR variant).

a(n) = sum(k = 0, n, isprime(bitand(n, k)))

a(n) = 0 iff n = 0 or n belongs to A102210.

a(2^k-1) = A000720(2^k-1) for any k > 0.

cp's OEIS Frontend

A122414 Triangle T(n,k) for 1 <= k <= n read by rows, where T(n,k) = 1 if gcd(n,k) is prime, 0 otherwise.

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