A084834 -id:A084834 - cp's OEIS frontend

A254337 Lexicographically earliest sequence of distinct numbers such that no sum of consecutive terms is prime.

0, 1, 8, 6, 10, 14, 12, 4, 20, 16, 24, 18, 22, 28, 26, 34, 30, 32, 36, 40, 42, 46, 38, 44, 52, 48, 54, 50, 58, 56, 62, 64, 60, 66, 68, 72, 70, 74, 80, 76, 78, 86, 82, 84, 90, 92, 94, 88, 98, 96, 104, 100, 102, 108, 110, 112, 114, 106, 116, 122, 118, 120, 124, 126, 130, 132, 134, 128, 138, 136, 142, 140, 144, 146, 148, 150, 154, 152, 156, 158

Offset: 0

M. F. Hasler, Jan 28 2015

In other words, no sum a(i)+a(i+1)+a(i+2)+...+a(n) may be prime. In particular, the sequence may not contain any primes.
I conjecture that the sequence contains all even numbers > 2 and no odd number beyond 1. If so, we must simply ensure that the sum a(1)+...+a(n) is not prime, which is always possible for one of the three consecutive even numbers {2n, 2n+2, 2n+4}. As a consequence, it would follow that a(n) ~ 2n.
Is there even a proof that the smallest odd composite number, 9, does not appear?
The variant A254341 has the additional restriction of alternating parity, which avoids excluding the odd numbers.
The least odd composite number a'(n+1) that could occur as the next term after a(n) and such that sum(a(i),i=k...n)+a'(n+1) is composite for all k <= n is (for n = 0, 1, 2,...): 9, 9, 25, 21, 39, 25, 69, 65, 45, 119, 95, 77, 55, 27, 595, 561, 531, 865, 1519, 1479, 1437, 1391, 1353, 1309, 1257, 1209, 1155, 1105, 1047, 2317, 2255, 2191, 3565, 5719, 13067, 12995, 12925, 12851, 12771, 12695, 12617, 12531, 12449, 12365, 12275, ... The growth of this sequence shows how it is increasingly unlikely that an odd number could occur, since the next possible even term is only about 2n.

			To explain the beginning of the sequence, observe that starting with the smallest possible terms 0, 1 does not appear to lead to a contradiction (and in fact never does), so we start there.
The next composite would be 4 but 1+4=5 is prime, as is 1+6, but 1+8=9 is not, so we take a(2) = 8 to be the next term.
4 is impossible for a(3) since 1+8+4=13 is prime, but neither 1+8+6=15 nor 8+6 is prime, so a(3)=6.

Robert G. Wilson v and M. F. Hasler, Table of n, a(n) for n = 0..5000 (terms 0..999 from M. F. Hasler)

Cf. A254341, A153136, A254211, A002808, A253073, A253074, A054408, A084834.

Cf. A025044 (no pairwise sum is prime), A025043 (no pairwise difference is prime).

f[lst_List] := Block[{k = 1}, While[ PrimeQ@ k || MemberQ[lst, k] || Union@ PrimeQ@ Accumulate@ Reverse@ Join[lst, {k}] != {False}, k++]; Append[lst, k]]; Nest[f, {0}, 70] (* Robert G. Wilson v, Jan 31 2015 *)

a=[];u=0; for(i=1,99, a=concat(a,0); until( ! isprime(s) || ! a[i]++, while( isprime(a[i]) || bittest(u,a[i]), a[i]++); s=a[k=i]; while( k>1 && ! isprime( s+=a[k--]),)); u+=2^a[i]; print1(a[i]","))

It appears that a(n) ~ 2n.

A025043 a(n) is not of the form prime + a(k), k < n.

Original entry on oeis.org

0, 1, 9, 10, 25, 34, 35, 49, 55, 85, 91, 100, 115, 121, 125, 133, 145, 155, 169, 175, 187, 195, 205, 217, 235, 247, 253, 259, 265, 289, 295, 301, 309, 310, 319, 325, 335, 343, 355, 361, 375, 385, 391, 395, 403, 415, 425, 445, 451, 469, 475, 481, 485, 493, 505, 511, 515

Offset: 1

David W. Wilson

nonn

Since this sequence includes 0 no terms are prime. - Charles R Greathouse IV, Jul 25 2013
Lexicographically earliest sequence of distinct natural numbers such that no two terms differ by a prime. - Peter Munn, Jun 19 2017
Congruence analysis from Peter Munn, Jun 30 2017: (Start)
If a(k) is in congruence class q mod p for some prime p, a(k) + p is the only higher number in this class that can be written as prime + a(k). Thus the ways a number m can be written as prime + a(k) for some k are much constrained if m shares membership of one or more such congruence classes with all except a few of the smaller terms in the sequence.
Of the first 100 terms, congruence class 1 mod 2 (odd numbers) contains 95, 1 mod 3 contains 76, and 0 mod 5 contains 53. No other congruence class modulo a prime contains more than 23.
The only even terms up to a(10000) are 0, 10, 34, 100, 310; of which 10, 100 and 310 are congruent to 10 mod 30, therefore to both 1 mod 3 and 0 mod 5. Note an initial sparseness of terms not congruent to either 1 mod 3 or 0 mod 5: this subsequence starts 9, 309, 527, 899, 989, 999. It becomes less sparse: as a proportion of the main sequence it is 0.04, 0.086 and 0.1555 of the first 100, 1000 and 10000 terms respectively.
Conjecture: there are only finitely many even terms.
(End)

David W. Wilson, Table of n, a(n) for n = 1..10000.

Cf. A025044, A072545, A084834, A254337.

A072545 a(0) = 1, a(n) for n > 0 is the smallest number > a(n-1) such that a(n)-a(k) is nonprime for 0 <= k < n.

Original entry on oeis.org

1, 2, 10, 11, 26, 35, 36, 50, 56, 86, 92, 101, 116, 122, 126, 134, 146, 156, 170, 176, 188, 196, 206, 218, 236, 248, 254, 260, 266, 290, 296, 302, 310, 311, 320, 326, 336, 344, 356, 362, 376, 386, 392, 396, 404, 416, 426, 446, 452, 470, 476, 482, 486, 494

Offset: 0

Amarnath Murthy, Aug 04 2002

nonn

a(0) = 1, a(3) = 11, a(5) = 35, a(11) = 101 and a(33) = 311 are the only odd elements <= 10^6 and probably the only ones. If so, then for n >= 34, a(n) is the smallest even k >= a(n-1)+4 for which none of k-1, k-11, k-35, k-101 or k-311 is prime. - David W. Wilson, Dec 14 2006

			26 is the smallest number > 11 which differs from 1, 2, 10, 11 by a nonprime (25, 24, 16, 15), so 26 is the next term after 11.

D. W. Wilson, Table of n, a(n) for n = 0..10000

Cf. A025043, A025044, A068638, A084834, A254337.

print1(a=1,","); v=[1]; n=1; while(n<55,a++; k=1; while(k<=n&&!isprime(a-v[k]), k++); if(k>n,n++; v=concat(v,a); print1(a,",")))

Edited and extended by Klaus Brockhaus, Aug 09 2002

A084833 a(n) is the smallest number such that a(n) + a(n-1), a(n) + a(n-1) + a(n-2), ..., a(n) + ... + a(1) are nonprime.

Original entry on oeis.org

1, 3, 5, 1, 15, 9, 15, 6, 3, 6, 21, 6, 3, 6, 15, 6, 3, 6, 3, 9, 3, 9, 15, 6, 3, 6, 3, 9, 9, 9, 3, 9, 9, 9, 3, 6, 3, 3, 3, 3, 3, 6, 15, 6, 3, 3, 3, 6, 9, 6, 3, 6, 9, 9, 3, 6, 3, 6, 15, 6, 3, 6, 3, 9, 3, 9, 3, 9, 9, 6, 3, 6, 9, 6, 3, 3, 3, 6, 3, 9, 3, 6, 3, 3, 3, 6, 3, 6, 3, 6, 9, 6, 3, 3, 3, 6, 9, 6, 3

Offset: 1

Jon Perry, Jun 06 2003

nonn

No sum of a continuous subsequence is ever prime. Does the sequence consist only of multiples of 3 after a(4)?

			a(5) = 15 as 1+3+5+1+15 = 25 is composite, 3+5+1+15 = 24 is composite, 5+1+15 = 21 is composite, and 1+15 = 16 is composite, and no smaller number has this property.

Cf. A084834.

{ checkprime(a,b)=local(fl); fl=0; for (i=1,b-1,if (isprime(a+s[i]),fl=1; break)); fl }
{ p=vector(100); p[1]=1; pc=2; while (pc<100, x=1; s=vector(100); for (i=1,pc-1,s[i]=sum(k=i,pc-1,p[k])); i=1; while (checkprime(x,pc),x++); p[pc]=x; pc++); p }

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A254337 Lexicographically earliest sequence of distinct numbers such that no sum of consecutive terms is prime.

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A025043 a(n) is not of the form prime + a(k), k < n.

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A072545 a(0) = 1, a(n) for n > 0 is the smallest number > a(n-1) such that a(n)-a(k) is nonprime for 0 <= k < n.

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A084833 a(n) is the smallest number such that a(n) + a(n-1), a(n) + a(n-1) + a(n-2), ..., a(n) + ... + a(1) are nonprime.

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