A376396 Triangle read by rows: the n-th row gives the least sequence of n consecutive primes all of the form 4*m + 1.

5, 13, 17, 89, 97, 101, 389, 397, 401, 409, 2593, 2609, 2617, 2621, 2633, 11593, 11597, 11617, 11621, 11633, 11657, 11593, 11597, 11617, 11621, 11633, 11657, 11677, 11593, 11597, 11617, 11621, 11633, 11657, 11677, 11681, 11593, 11597, 11617, 11621, 11633, 11657, 11677, 11681, 11689

Offset: 1

Author

Stefano Spezia, Sep 22 2024

Keywords

• nonn
• tabl

Comments

Guy writes that the terms of the 9th row have been found by De Haan. Moreover, Guy gives the terms of the 11th row: 766261, 766273, 766277, 766301, 766313, 766321, 766333, 766357, 766361, 766369, 766373.

Examples

			The triangle begins as:
       5;
      13,     17;
      89,     97,    101;
     389,    397,    401,    409;
    2593,   2609,   2617,   2621,   2633;
   11593,  11597,  11617,  11621,  11633,  11657;
   11593,  11597,  11617,  11621,  11633,  11657,  11677;
   11593,  11597,  11617,  11621,  11633,  11657,  11677,  11681;
   11593,  11597,  11617,  11621,  11633,  11657,  11677,  11681,  11689;
  373649, 373657, 373661, 373669, 373693, 373717, 373721, 373753, 373757, 373777;
  ...
		

References

• R. K. Guy, Unsolved Problems in Number Theory, 2nd. ed., Section A4.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 11593 at. p. 173.

Links

• Jens Kruse Andersen, Consecutive Congruent Primes.
• D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359-373.

Crossrefs

Cf. A002144, A016813, A057624 (1st column), A145986 (right hand column).

Programs

Mathematica

kold=1; row[n_]:=Module[{r={}}, k=kold; While[Mod[Prime[k],4]!=1, k++]; While[Product[Boole[Mod[Prime[k+i],4]==1], {i,0,n-1}]!=1, k++]; kold=k; Table[Prime[i+k], {i,0,n-1}]]; Array[row,9]//Flatten