

A129350


The smallest number in a group of four consecutive mutually coprime integers such that the sum of three of the them plus the square of the fourth is prime.


0



1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 23, 24, 25, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67
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OFFSET

1,2


COMMENTS

Groups of 4 coprime integers (a,b,c,d) are defined by fixing a, then searching for the smallest b>a such that b is coprime to a, searching for the smallest c>b coprime to a and b, then searching for the smallest d>c coprime to a, b and c. If at least one of the 4 sums a^2+b+c+d, a+b^2+c+d, a+b+c^2+d or a+b+c+d^2 is prime, a is in the sequence.


LINKS

Table of n, a(n) for n=1..60.


FORMULA

The groups of four coprimes increase from 1,2,3,5 as the first group, 2,3,5,7 as the second, 3,4,5,7 as the third and so forth. Each group has four chances of producing a prime by squaring one and adding the rest.


EXAMPLE

Take the quadruplet 17, 18, 19, 23: 17^2 +18 + 19 + 23 = 349; 18^2 + 17+19+23 = 383; 19^2 + 17+18+23 = 419. Three sum to primes but only one is required put 17 in the sequence.


MAPLE

filter:= proc(n)
local S, k, Ss, cands;
S:= {n};
for k from n+1 while nops(S) < 4 do
if max(map2(igcd, k, S))=1 then S:= S union {k}
fi
od;
Ss:= convert(S, `+`);
cands:= {seq(S[i]^2  S[i] + Ss, i=1..4)};
ormap(isprime, cands)
end proc:
select(filter, [$1..100]); # Robert Israel, Dec 15 2014


CROSSREFS

Sequence in context: A020664 A055570 A083114 * A004746 A188301 A178160
Adjacent sequences: A129347 A129348 A129349 * A129351 A129352 A129353


KEYWORD

nonn


AUTHOR

J. M. Bergot, May 28 2007


EXTENSIONS

Edited by R. J. Mathar, Dec 17 2014


STATUS

approved



