A383134 Array read by ascending antidiagonals: A(n,k) is the length of the arithmetic progression of only primes having difference n and first term prime(k).

2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 2, 3, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Stefano Spezia, Apr 17 2025

Keywords

• nonn
• tabl

Examples

			The array begins as:
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 3, 2, 1, 2, 1, 2, 1, 1, 2, ...
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 3, 1, 2, 1, 2, 1, 2, 1, 1, ...
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 5, 3, 4, 2, 3, 1, 2, 1, ...
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 3, 2, 1, 2, 1, 1, 1, 2, 2, ...
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 3, 1, 2, 1, 2, 1, 2, 1, 1, ...
  ...
A(2,2) = 3 since 3 primes are in arithmetic progression with a difference of 2 and the first term equal to the 2nd prime: 3, 5, and 7.
A(6,3) = 5 since 5 primes are in arithmetic progression with a difference of 6 and the first term equal to the 3rd prime: 5, 11, 17, 23, and 29.
		

References

• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 139.

Crossrefs

Cf. A000012, A006512, A040976, A054977, A175191 (n=2).

Cf. A088430, A206045, A237453.

Programs

Mathematica

A[n_,k_]:=Module[{count=1,sum=Prime[k]},While[PrimeQ[sum+=n], count++]; count]; Table[A[n-k+1,k],{n,13},{k,n}]//Flatten

Formula

A(A006512(n),k) = 1 for n > 1.

A(A040976(n),k) = A054977(k+1).