A329456 -id:A329456 - cp's OEIS frontend

A329565 For all n >= 0, exactly five sums are prime among a(n+i) + a(n+j), 0 <= i < j < 6; lexicographically earliest such sequence of distinct nonnegative numbers.

0, 1, 2, 3, 6, 24, 4, 5, 8, 9, 10, 11, 7, 13, 12, 17, 16, 14, 15, 19, 22, 18, 21, 20, 26, 23, 25, 27, 33, 34, 28, 29, 32, 38, 39, 30, 31, 41, 40, 36, 35, 42, 61, 44, 43, 66, 37, 52, 45, 47, 46, 51, 50, 57, 48, 49, 53, 55, 56, 59, 54, 58, 72, 95, 62, 65, 67, 63, 84, 64, 60, 68, 89, 71, 69, 73, 80, 78, 70, 79, 87, 76, 75, 74, 88, 77, 81, 82, 189, 85

Offset: 0

M. F. Hasler, Feb 09 2020

nonn

That is, there are 5 primes, counted with multiplicity, among the 15 pairwise sums of any 6 consecutive terms.
Conjectured to be a permutation of the nonnegative integers.
If so, then the restriction to [1..oo) is a permutation of the positive integers.

			For n = 0, we consider pairwise sums of the first 6 terms a(0..5) = (0, 1, 2, 3, 6, 24): We have (a(i) + a(j), 0 <= i < j < 6) = (1; 2, 3; 3, 4, 5; 6, 7, 8, 9; 24, 25, 26, 27, 30) among which there are 5 primes, counted with repetition. If one tries to take a(4) equal to 4 or 5, this yields already 6 primes among the pairwise sums of the first 5 terms, so the smallest possible choice is a(4) = 6, and thereafter any a(5) less than 24 would again yield too many prime sums. So (0, 1, 2, 3, 6, 24) is indeed the start of the lexicographically earliest nonnegative sequence with the required properties.
Then one finds that a(6) = 4 is possible, giving also 6 prime sums for n = 1, so this is the correct continuation (modulo later confirmation that the sequence can be continued without contradiction given this choice).
Next one finds that a(7) = 5 is also possible, and so on.

M. F. Hasler, Prime sums from neighboring terms, OEIS wiki, Nov. 23, 2019

Cf. A329425 (6 primes using 5 consecutive terms), A329566 (6 primes using 6 consecutive terms).
Cf. A329449 (4 primes using 4 consecutive terms), A329456 (4 primes using 5 consecutive terms).
Cf. A329454 (3 primes using 4 consecutive terms), A329455 (3 primes using 5 consecutive terms).
Cf. A329411 (2 primes using 3 consecutive terms), A329452 (2 primes using 4 consecutive terms), A329453 (2 primes using 5 consecutive terms).
Cf. A329333 (1 (odd) prime using 3 terms), A128280 & A055265 (1 prime using 2 terms); A055266 & A253074 (0 primes using 2 terms), A329405 & A329450 (0 primes using 3 terms), A329406 ff: other variants.

{A329565(n,show=0,o=0,N=5/*#primes*/,M=5,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

A329567 For all n >= 0, exactly seven sums are prime among a(n+i) + a(n+j), 0 <= i < j < 6: lexicographically earliest such sequence of distinct nonnegative numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 5, 11, 26, 15, 9, 32, 14, 17, 20, 21, 27, 10, 16, 19, 7, 12, 13, 24, 6, 23, 35, 25, 37, 18, 36, 22, 31, 61, 28, 30, 39, 40, 43, 33, 64, 38, 45, 34, 29, 63, 50, 44, 53, 42, 59, 47, 54, 48, 41, 90, 49, 55, 52, 108, 58, 46, 51, 121, 73, 78, 76, 100, 79, 81, 151, 60, 67, 112, 70, 69, 82, 62, 87, 57, 80, 111, 56, 71, 66, 68, 86, 83, 65

Offset: 0

M. F. Hasler, Feb 09 2020

nonn

That is, there are 7 primes, counted with multiplicity, among the 15 pairwise sums of any 6 consecutive terms.
Conjectured to be a permutation of the nonnegative integers. The restriction to [1,oo) is then a permutation of the positive integers with the same property, but not the smallest one which is A329576 = (1, 2, 3, 4, 5, 8, 11, ...).
For n > 5, a(n) is the smallest number not used earlier such that the set a(n) + {a(n-5), ..., a(n-1)} has the same number of primes as a(n-6) + {a(n-5), ..., a(n-1)}. Such a number always exists, by definition of the sequence. (If it would not exist for a given n, this means the term a(n-1) (or earlier) "was wrong and must be corrected", so to say. Of course this only refers to an incorrect computation.)

			Using the smallest possible 5 initial terms a(0..4) = (0, 1, 2, 3, 4), we have a total of 6 primes among the pairwise sums, namely 0+2, 0+3, 1+2, 1+4, 2+3 and 3+4. To satisfy the definition for n = 0, the next term a(5) must give exactly one more prime when added to these 5 initial terms. The smallest number with this property is 6, but this choice of a(5) would make it impossible to find a suitable a(7): Indeed, for n = 1 we must consider the pairwise sums of (1, 2, 3, 4, a(5), a(6)). If we had (1, 2, 3, 4, 6, a(6)), the first 5 terms would give 5 prime sums 1+2, 1+4, 1+6, 2+3 and 3+4. Then a(6) should give two more prime sums, which is easily possible, either with even a(6) such that 1+a(6) and 3+a(6) are prime, or odd a(6) such that two among {2, 4, 6} + a(6) are prime. Thereafter, for n = 2, we drop the 1 and include a(7) instead, which must produce the same number of prime sums when added to {2, 3, 4, 6, a(6)} as was the case for 1. For even a(6) this was 4 (1+2, 1+4, 1+6 and 1+a(6)), which is impossible to achieve with a(7) > 1, since 2+x, 4+x and 6+x can't be all prime for x > 1. For odd a(6) it is 3 (1+2, 1+4 and 1+6), which is also impossible as well for odd a(7) (same reason as before) as for even a(7) (since only 3 and a(6) are odd and can give a prime sum).
This shows that we can't take a(5) equal to 6, and must consider the next larger possibility, which is a(5) = 8 (with prime sum 3+8 = 11, while 7 would give more than one, 0+7 and 4+7). Now we find that the smallest possible a(6) = 5 yields a solution and all subsequent terms can also be chosen greedily.

Éric Angelini, Prime sums from neighbouring terms, SeqFan list, Nov 11 2019.
Éric Angelini, Prime sums from neighbouring terms, SeqFan list, Nov 11 2019. [Cached copy, with permission]
Eric Angelini, Prime sums from neighbouring terms, personal blog "Cinquante signes" [Cached copy of html file, with permission]
Eric Angelini, Prime sums from neighbouring terms, personal blog "Cinquante signes" [Cached copy of pdf file, with permission]
M. F. Hasler, Prime sums from neighboring terms, OEIS wiki, Nov. 23, 2019.

Cf. related sequences with N prime sums using M consecutive terms, labeled (N,M): A329425 (6,5), A329566 (6,6), A329449 (4,4), A329456 (4,5), A329454 & A329416 (3,4), A329455 (3,5), A329411 (2,3), A329452 (2,4), A329453 (2,5), A329333 (1,3), A128280 & A055265 (1,2); A055266 & A253074 (0,2), A329405 & A329450 (0,3), A329406 - A329416: (1,4) ... (2,10).

{A329567(n,show=1,o=0,N=7,M=5,X=[[4,6]],p=[],u=o,U)=for(n=o,n-1, 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

cp's OEIS Frontend

A329565 For all n >= 0, exactly five sums are prime among a(n+i) + a(n+j), 0 <= i < j < 6; lexicographically earliest such sequence of distinct nonnegative numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI