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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.
0
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
1,1
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.
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.
EXAMPLE
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;
...
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
CROSSREFS
Cf. A002144, A016813, A057624 (1st column), A145986 (right hand column).
Sequence in context: A350379 A145040 A361563 * A195549 A294132 A125146
KEYWORD
nonn,tabl
AUTHOR
Stefano Spezia, Sep 22 2024
STATUS
approved