A129350 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.

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

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

Robert Israel, Table of n, a(n) for n = 1..10000

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

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.

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 * A368365 A004746 A188301

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