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

%I A376396 #12 Sep 22 2024 18:27:36
%S A376396 5,13,17,89,97,101,389,397,401,409,2593,2609,2617,2621,2633,11593,
%T A376396 11597,11617,11621,11633,11657,11593,11597,11617,11621,11633,11657,
%U A376396 11677,11593,11597,11617,11621,11633,11657,11677,11681,11593,11597,11617,11621,11633,11657,11677,11681,11689
%N A376396 Triangle read by rows: the n-th row gives the least sequence of n consecutive primes all of the form 4*m + 1.
%C A376396 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 A376396 R. K. Guy, Unsolved Problems in Number Theory, 2nd. ed., Section A4.
%D A376396 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 11593 at. p. 173.
%H A376396 Jens Kruse Andersen, <a href="http://primerecords.dk/congruent-primes.htm">Consecutive Congruent Primes</a>.
%H A376396 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 A376396 The triangle begins as:
%e A376396        5;
%e A376396       13,     17;
%e A376396       89,     97,    101;
%e A376396      389,    397,    401,    409;
%e A376396     2593,   2609,   2617,   2621,   2633;
%e A376396    11593,  11597,  11617,  11621,  11633,  11657;
%e A376396    11593,  11597,  11617,  11621,  11633,  11657,  11677;
%e A376396    11593,  11597,  11617,  11621,  11633,  11657,  11677,  11681;
%e A376396    11593,  11597,  11617,  11621,  11633,  11657,  11677,  11681,  11689;
%e A376396   373649, 373657, 373661, 373669, 373693, 373717, 373721, 373753, 373757, 373777;
%e A376396   ...
%t A376396 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 A376396 Cf. A002144, A016813, A057624 (1st column), A145986 (right hand column).
%K A376396 nonn,tabl
%O A376396 1,1
%A A376396 _Stefano Spezia_, Sep 22 2024