A084833 -id:A084833 - cp's OEIS frontend

A332941 Lexicographically earliest sequence of positive numbers in which no set of consecutive terms sums to a prime.

1, 8, 1, 15, 9, 1, 14, 6, 30, 6, 9, 15, 6, 4, 8, 12, 10, 14, 6, 12, 8, 10, 12, 18, 12, 6, 6, 6, 24, 6, 6, 8, 1, 9, 6, 10, 8, 12, 6, 14, 10, 6, 4, 8, 12, 10, 20, 6, 18, 6, 6, 4, 8, 12, 6, 4, 12, 8, 10, 8, 6, 6, 18, 6, 6, 20, 10, 12, 8, 4, 6, 12, 12, 6, 12, 6, 12

Offset: 1

S. Brunner, Mar 03 2020

nonn

Terms >= 30 seem to be very rare. Up to a(450000), 30 appears only 7 times: at n = 9, 288, 2507, 15902, 54405, 242728, 425707.

For n <= 450000, the largest term is 32; it appears at n = 335308 and 370687.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A254337, A084833.

s:= proc(i, j) option remember; `if`(i>j, 0, a(j)+s(i, j-1)) end:
a:= proc(n) option remember; local k; for k while
      ormap(isprime, [k+s(i, n-1)$i=1..n]) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 23 2020

s[i_, j_] := s[i, j] = If[i > j, 0, a[j] + s[i, j-1]];
a[n_] := a[n] = Module[{k}, For[k = 1, AnyTrue[k+Table[s[i, n-1], {i, 1, n}], PrimeQ], k++]; k];
Array[a, 100] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

def A(ee):
    a=[1]
    print(1)
    n=1
    while n<=ee:
        i=1
        while i>0:
            ii=i
            iz=c=0
            while iz<=len(a):
                c=0
                if ii>2:
                    for j in range(2, int((ii)**0.5+1.5)):
                        if ii%j==0:
                            c=1
                            break
                if c==0 and ii>1:
                    break
                else:
                    iz += 1
                    ii=ii+a[n-iz]
            if c==1:
                n += 1
                a.append(i)
                print(i)
                break
            if i<4:
                i=4
            else:
                i += 1
    return a

A084834 a(n) is the smallest number not previously used such that a(n)+a(n-1), a(n)+a(n-1)+a(n-2), ..., a(n)+...+a(1) are not prime.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 4, 6, 30, 38, 10, 36, 14, 34, 42, 39, 21, 69, 27, 33, 45, 20, 16, 24, 50, 25, 51, 66, 18, 72, 54, 60, 74, 22, 8, 19, 41, 28, 48, 44, 40, 78, 57, 35, 58, 102, 12, 63, 65, 64, 56, 46, 96, 68, 76, 114, 80, 52, 84, 90, 99, 55, 2, 93, 75, 100, 62, 120, 98

Offset: 1

Jon Perry, Jun 06 2003

nonn

No sum of a continuous subsequence is ever prime. Is every integer used? Can an odd number ever be surrounded by two even numbers (and similarly for even numbers)?

			a(4)=7 because 7 is the smallest unused number such that 1+3+5+x, 3+5+x and 5+x are all composite

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A084833, A254337.

checkprime(a,b)=local(fl); fl=0; for (i=1,b-1,if (isprime(a+s[i]),fl=1; break)); if (fl==0, for (j=1,b-1,if (a==p[j],fl=1; break))); fl
p=vector(300); p[1]=1; pc=2; while (pc<300, x=1; s=vector(300); 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

A360019 Lexicographically earliest increasing sequence of positive numbers in which no nonempty subsequence of consecutive terms sums to a triangular number.

Original entry on oeis.org

2, 5, 7, 11, 12, 14, 16, 17, 18, 19, 20, 22, 25, 26, 30, 31, 34, 35, 37, 42, 46, 49, 52, 54, 59, 63, 64, 68, 72, 73, 77, 80, 81, 84, 85, 87, 92, 93, 94, 98, 100, 101, 108, 113, 115, 117, 118, 121, 122, 123, 125, 129, 130, 132, 133, 134, 141, 142, 143, 146, 149

Offset: 0

Ctibor O. Zizka, Jan 21 2023

nonn

The sequence cannot contain any triangular numbers.

			a(0) = 2 by the definition of the sequence. The next number > a(0) is 3, but it is a triangular number, so we try 4, but 2 + 4 = 6 is a triangular number. Then we try 5; {5, 2 + 5} are not triangular numbers, thus a(1) = 5. a(2) cannot be 6, so we try 7; {7, 5 + 7, 2 + 5 + 7} are not triangular numbers, thus a(2) = 7.

Cf. A000217, A084833, A332941.

q:= proc(n) option remember; issqr(8*n+1) end:
s:= proc(i, j) option remember; `if`(i>j, 0, a(j)+s(i, j-1)) end:
a:= proc(n) option remember; local k; for k from 1+a(n-1) while
      ormap(q, [k+s(i, n-1)$i=0..n]) do od; k
    end: a(-1):=-1:
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 21 2023

triQ[n_] := IntegerQ @ Sqrt[8*n + 1]; a[0] = 2; a[n_] := a[n] = Module[{k = a[n - 1] + 1, t = Accumulate @ Table[a[i], {i, n - 1, 0, -1}]}, While[triQ[k] || AnyTrue[t + k, triQ], k++]; k]; Array[a, 61, 0] (* Amiram Eldar, Jan 21 2023 *)

More terms from Jon E. Schoenfield, Jan 21 2023

cp's OEIS Frontend

A332941 Lexicographically earliest sequence of positive numbers in which no set of consecutive terms sums to a prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

A084834 a(n) is the smallest number not previously used such that a(n)+a(n-1), a(n)+a(n-1)+a(n-2), ..., a(n)+...+a(1) are not prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A360019 Lexicographically earliest increasing sequence of positive numbers in which no nonempty subsequence of consecutive terms sums to a triangular number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions