A329580 For every n >= 0, exactly 10 sums are prime among a(n+i) + a(n+j), 0 <= i < j < 8: lexicographically earliest such sequence of distinct nonnegative numbers.

0, 1, 2, 3, 4, 5, 6, 90, 7, 11, 8, 9, 10, 12, 13, 30, 29, 31, 14, 16, 15, 17, 22, 42, 19, 25, 18, 24, 20, 23, 28, 33, 43, 35, 36, 38, 26, 21, 32, 27, 34, 71, 37, 39, 40, 44, 63, 64, 68, 41, 46, 183, 50, 45, 333, 51, 98, 47, 58, 62, 69, 65, 48, 101, 66, 49, 61, 78, 57, 53, 180, 52, 55, 96, 631, 54, 56, 83, 75, 95, 74, 116, 60

Offset: 0

Author

M. F. Hasler, Nov 17 2019

Keywords

• nonn

Comments

That is, there are 10 primes, counted with multiplicity, among the 28 pairwise sums of any 8 consecutive terms.

Is this a permutation of the nonnegative integers?

If so, then the restriction to [1..oo) is a permutation of the positive integers, but not the lexicographically earliest one with this property, which starts (1, 2, 3, 4, 5, 6, 7, 19, 10, 8, 9, 12, 11, 18, 13, 29, ...).

We remark the surprisingly large numbers 333 and 631 among the first terms.

Examples

			In P(7) := {0, 1, 2, 3, 4, 5, 6} there are already S(7) := 10 primes 0+2, 0+3, 0+5, 1+2, 1+4, 1+6, 2+3, 2+5, 3+4, 5+6 among the pairwise sums, so the next term a(7) must not produce any more primes when added to elements of P(7). We find that a(7) = 90 is the smallest possible term.
Then in P(8) = {1, 2, 3, 4, 5, 6, 90} there are S(8) = 7 primes among the pairwise sums, so a(8) must produce 3 more primes when added to elements of P(8). We find a(8) = 7 is the smallest possibility (with 4+7, 6+7 and 90+7).
And so on.
		

Crossrefs

Cf. A329579 (9 primes using 7 consecutive terms), A329425 (6 primes using 5 consecutive terms).

Cf. A329455 (4 primes using 5 consecutive terms), A329455 (3 primes using 5 consecutive terms), A329453 (2 primes using 5 consecutive terms), A329452 (2 primes using 4 consecutive terms).

Cf. A329577 (7 primes using 7 consecutive terms), A329566 (6 primes using 6 consecutive terms), A329449 (4 primes using 4 consecutive terms).

Cf. A329454 (3 primes using 4 consecutive terms), A329411 (2 primes using 3 consecutive terms), A329333 (1 odd prime using 3 terms), A329450 (0 primes using 3 terms).

Cf. A329405 ff: other variants defined for positive integers.

Programs

PARI

A329580(n,show=0,o=0,N=10,M=7,p=[],U,u=o)={for(n=o,n-1, if(show>0,print1(o", "), show<0,listput(L,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(#p