A329414 Lexicographically earliest sequence of distinct positive numbers such that among the pairwise sums of any six consecutive terms there are exactly two prime sums.

1, 2, 3, 7, 13, 19, 5, 8, 9, 17, 16, 40, 4, 6, 11, 12, 10, 14, 22, 18, 15, 20, 24, 26, 25, 29, 28, 52, 30, 35, 21, 23, 33, 31, 32, 27, 39, 37, 38, 43, 36, 48, 44, 46, 34, 45, 42, 50, 41, 54, 49, 69, 51, 47, 57, 60, 53, 55, 59, 58, 61, 56, 66, 65, 63, 67, 62, 78, 68, 70, 64, 71, 72, 73, 75, 81, 82, 80

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Nov 14 2019

Keywords

• nonn

Comments

Condition a(1) = 1 follows from minimality. Conjectured to be a permutation of the positive integers: a(10^6) = 999994 and all numbers up to there have appeared at that point. - M. F. Hasler, Nov 15 2019

Examples

			a(1) = 1 by minimality.
a(2) = 2 as 2 is the smallest available integer not leading to a contradiction. Note that as 1 + 2 = 3 we already have one prime sum (on the required two) with the sextuplet {1,2,a(3),a(4),a(5),a(6)}.
a(3) = 3 as 3 is the smallest available integer not leading to a contradiction. Note that as 2 + 3 = 5 we now have the two prime sums required with the sextuplet {1,2,3,a(4),a(5),a(6)}.
a(4) = 7 as a(4) = 4, 5 or 6 would lead to a contradiction: indeed, the sextuplets {1,2,3,4,a(5),a(6)}, {1,2,3,5,a(5),a(6)} and {1,2,3,6,a(5),a(6)} will produce more than the two required prime sums. With a(4) = 7 we have no contradiction as the sextuplet {1,2,3,7,a(5),a(6)} has now exactly two prime sums: 1 + 2 = 3 and 2 + 3 = 5.
a(5) = 13 as a(5) = 4, 5, 6, 8, 9, 10, 11 or 12 would again lead to a contradiction (more than 2 prime sums with the sextuplet); in combination with any other term before it, a(5) = 13 will produce only composite sums.
a(6) = 19 as 19 is the smallest available integer not leading to a contradiction: indeed, the sextuplet {1,2,3,7,13,19} shows exactly the two prime sums we are looking for: 1 + 2 = 3 and 2 + 3 = 5.
a(7) = 5 as 5 is the smallest available integer not leading to a contradiction; indeed, the sextuplet {2,3,7,13,19,5} shows exactly two prime sums, which are 2 + 3 = 5 and 2 + 5 = 7.
And so on.
		

Links

• Jean-Marc Falcoz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A329333 (3 consecutive terms, exactly 1 prime sum). See also A329450, A329452 onwards.

Programs

PARI

A329414(n,show=0,o=1,N=2,M=5,p=[],U,u=o)={for(n=o,n-1,show&&print1(o",");U+=1<<(o-u);U>>=-u+u+=valuation(U+1,2); p=concat(if(#p>=M,p[^1],p),o);my(c=N-sum(i=2,#p,sum(j=1,i-1, isprime(p[i]+p[j])))); if(#pM. F. Hasler, Nov 15 2019