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Triangle read by rows: the n-th row gives the least sequence of n consecutive primes all of the form 4*m + 1.
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%I #12 Sep 22 2024 18:27:36

%S 5,13,17,89,97,101,389,397,401,409,2593,2609,2617,2621,2633,11593,

%T 11597,11617,11621,11633,11657,11593,11597,11617,11621,11633,11657,

%U 11677,11593,11597,11617,11621,11633,11657,11677,11681,11593,11597,11617,11621,11633,11657,11677,11681,11689

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

%C 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.

%D R. K. Guy, Unsolved Problems in Number Theory, 2nd. ed., Section A4.

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 11593 at. p. 173.

%H Jens Kruse Andersen, <a href="http://primerecords.dk/congruent-primes.htm">Consecutive Congruent Primes</a>.

%H D. K. L. Shiu, <a href="http://dx.doi.org/10.1112/S0024610799007863">Strings of Congruent Primes</a>, J. Lond. Math. Soc. 61 (2) (2000) 359-373.

%e The triangle begins as:

%e 5;

%e 13, 17;

%e 89, 97, 101;

%e 389, 397, 401, 409;

%e 2593, 2609, 2617, 2621, 2633;

%e 11593, 11597, 11617, 11621, 11633, 11657;

%e 11593, 11597, 11617, 11621, 11633, 11657, 11677;

%e 11593, 11597, 11617, 11621, 11633, 11657, 11677, 11681;

%e 11593, 11597, 11617, 11621, 11633, 11657, 11677, 11681, 11689;

%e 373649, 373657, 373661, 373669, 373693, 373717, 373721, 373753, 373757, 373777;

%e ...

%t 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

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

%K nonn,tabl

%O 1,1

%A _Stefano Spezia_, Sep 22 2024