|
%I #25 Jun 24 2022 23:38:31
%S 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,
%T 32,33,34,35,36,37,38,39,40,41,42,43,44,45,47,48,49,50,51,52,53,54,55,
%U 56,57,58,59,60,61,62,64,65,66,67
%N The smallest number in a group of four consecutive mutually coprime integers such that the sum of three of them plus the square of the fourth is prime.
%C 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.
%H Robert Israel, <a href="/A129350/b129350.txt">Table of n, a(n) for n = 1..10000</a>
%F 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.
%e Consider the quadruple {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.
%p filter:= proc(n)
%p local S,k,Ss,cands;
%p S:= {n};
%p for k from n+1 while nops(S) < 4 do
%p if max(map2(igcd,k,S))=1 then S:= S union {k}
%p fi
%p od;
%p Ss:= convert(S,`+`);
%p cands:= {seq(S[i]^2 - S[i] + Ss, i=1..4)};
%p ormap(isprime,cands)
%p end proc:
%p select(filter, [$1..100]); # _Robert Israel_, Dec 15 2014
%K nonn
%O 1,2
%A _J. M. Bergot_, May 28 2007
%E Edited by _R. J. Mathar_, Dec 17 2014
|
