A085947 -id:A085947 - cp's OEIS frontend

A328444 Lexicographically earliest sequence of distinct positive numbers such that a(1) = 1, a(2) = 2, and for n > 2, a(n) divides Sum_{i = n-k..n-1} a(i) with k > 0 as small as possible.

1, 2, 3, 5, 4, 9, 13, 11, 6, 17, 23, 8, 31, 39, 7, 46, 53, 33, 43, 19, 62, 27, 89, 29, 59, 22, 81, 103, 92, 15, 107, 61, 12, 73, 85, 79, 41, 10, 51, 34, 95, 129, 14, 143, 157, 20, 177, 197, 187, 16, 203, 219, 211, 86, 99, 37, 68, 21, 18, 24, 42, 66, 36, 102

Offset: 1

Rémy Sigrist, Oct 15 2019

nonn

When computing a(n) for n > 2, there may be candidates for different values of k; we choose the candidate that minimizes k.

This sequence is an infinite variant of A085947; a(n) = A085947(n) for n = 1..39.

			For n = 3:
- the divisors of a(2) = 2 are all already in the sequence,
- 3 is the least divisor of a(1) + a(2) = 1 + 2 = 3 not yet in the sequence,
- so a(3) = 3.
For n = 4:
- the divisors of a(3) = 3 are all already in the sequence,
- 5 is the least divisor of a(2) + a(3) = 2 + 3 = 5 not yet in the sequence,
- so a(3) = 5.
For n = 5:
- the divisors of a(4) = 5 are all already in the sequence,
- 4 is the least divisor of a(3) + a(4) = 3 + 5 = 8 not yet in the sequence,
- so a(5) = 4.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A328444

See A328443 for a similar sequence.

Cf. A085947.

PARI
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\\ See Links section.
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a(n) <= Sum_{k = 1..n-1} a(k) for any n > 2.

A332301 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest divisor of the sum of the previous two terms that has not yet appeared. If all divisors have appeared then a(n) equals the sum.

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 13, 11, 6, 17, 23, 8, 31, 39, 7, 46, 53, 33, 43, 19, 62, 27, 89, 29, 59, 22, 81, 103, 92, 15, 107, 61, 12, 73, 85, 79, 41, 10, 51, 61, 14, 25, 39, 16, 55, 71, 18, 89, 107, 28, 45, 73, 118, 191, 309, 20, 47, 67, 38, 21, 59, 40, 99, 139, 34, 173, 69, 121, 95, 24, 119, 143, 131, 137, 134, 271, 135, 58, 193

Offset: 1

Scott R. Shannon, Feb 09 2020

nonn

This sequence uses the same rules as A085947 except that the restriction of all entries being unique is removed. The first term to repeat is 61 which appears at a(32) and then again at a(40). But in this case as a(31) and a(39) are different the sequence does not form a repeating loop of values. In fact it is easy to show the sequence can never form a repeating loop. If it did it would imply there is a first value A such that the sequence would be of the form ...,A,B,C,D,...,X,Y,A,B,C,D,...,X,Y,A,B,C,... . As the terms in the repeating loop are unchanging it implies every term's divisors have been previously seen, and thus A+B=C and B+C=D and so on, and thus each term in the loop is larger than the previous term. But that leads to a contradiction as from the first loop X and Y are larger than A, but the beginning of the second loop implies X+Y=A. Thus no unchanging series of repeated terms can exist.

In the first 1 million terms the largest value is a(895234) = 25216687, the lowest unseen value is 69006, and the value seen the most frequently is 618083, which occurs six times.

			a(5) = 4 as a(3) + a(4) = 3 + 5 = 8, and the divisors of 8 are 1,2,4,8. 1 and 2 have already appeared so 4 is the least divisor not yet in the sequence.
a(40) = 61 as a(38) + a(39) = 10 + 51 = 61. The divisors of 61 are 1 and 61, both of which have already appeared, at a(1) and a(32), thus a(40) = 61. Note that as a(31) and a(39) differ a(33) and a(41) differ and the sequence does not repeat.

Cf. A085947, A328444.

A140460 a(0) = 2; thereafter a(n) = smallest integer not a multiple of an earlier terms nor a sum of two earlier terms.

Original entry on oeis.org

2, 3, 7, 11, 17, 23, 29, 37, 41, 47, 53, 59, 65, 71, 79, 83, 89, 95, 101, 107, 113, 125, 131, 137, 149, 155, 163, 167, 173, 179, 191, 197, 211, 215, 223, 227, 233, 239, 247, 251, 257, 263, 269, 277, 281, 293, 305, 311, 317, 331, 335

Offset: 0

Russell Y. Webb (r.webb(AT)elec.canterbury.ac.nz), Jul 22 2008

Note that the first composite value is a(12) = 65 = 5 * 13, since 5 and 13 are the first two primes sieved out of the sequence. Similarly, the second composite value is a(17) = 95 = 5 * 19, since 5 and 19 are the first and third primes sieved out of the sequence. - Jonathan Vos Post, Jul 27 2008

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A000040, A002858, A008320, A054016, A075573, A085947.

#include 
#include 
void sieve(){ const int first = 2; const int max = 10000; bool member[max]; for(int i = first; i < max; ++i){ member[i] = true; } for(int i = first; i < max; ++i){ if(member[i]){ for(int x = i + i; x < max; x += i){ member[x] = false; } for(int j = first; j < i; ++j){ if(member[j] && i + j < max){ member[i + j] = false; } } } } int num = 0; for(int i = first; i < max; ++i){ if(member[i]){ printf("%i, ", i); num += 1; } } printf(" num = %i ", num); }

s={2};Do[m=2;Until[Total[Boole[Divisible[m,s]]]==0&&!MemberQ[Total/@Subsets[s,{2}],m],m++];AppendTo[s,m],{n,50}];s (* James C. McMahon, Jul 09 2025 *)

A323890 a(1) = 1, a(2) = 2; thereafter a(n+1) = smallest unused divisor of a(n) if there are any, otherwise a(n) + a(n-1).

Original entry on oeis.org

1, 2, 3, 5, 8, 4, 12, 6, 18, 9, 27, 36, 63, 7, 70, 10, 80, 16, 96, 24, 120, 15, 135, 45, 180, 20, 200, 25, 225, 75, 300, 30, 330, 11, 341, 31, 372, 62, 434, 14, 448, 28, 476, 17, 493, 29, 522, 58, 580, 116, 696, 87, 783, 261, 1044, 174, 1218, 21, 1239, 59, 1298, 22, 1320, 33, 1353

Offset: 1

Ivan Neretin, Sep 02 2019

			a(6) = 4, and all divisors of 4 are already used, hence a(7) = a(6) + a(5) = 8 + 4 = 12. Now the smallest unused divisor of 12 is 6, hence a(8) = 6.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A050196, A085947.

Nest[Append[#, If[(d = Complement[Divisors@#[[-1]], #]) == {}, #[[-1]] + #[[-2]], Min[d]]] &, {1, 2}, 63]

A326791 For n > 1, let f_n be the lexicographically earliest sequence of distinct positive terms such that f_n(1) = 1, f_n(2) = n, and for k > 2, f_n(k) divides f_n(k-2) + f_n(k-1); if f_n is finite, then a(n) is the number of terms of f_n, otherwise a(n) = -1; a(1) = 1.

Original entry on oeis.org

1, 39, 23, 5, 10, 9, 31, 8, 8, 8, 7, 8, 7, 9, 10, 29, 9, 38, 36, 13, 14, 13, 8, 12, 40, 19, 11, 25, 10, 15, 13, 18, 11, 5, 39, 36, 40, 37, 12, 25, 11, 12, 29, 30, 33, 25, 32, 5, 40, 25, 12, 25, 11, 21, 40, 27, 18, 19, 17, 9, 41, 18, 11, 5, 41, 37, 40, 12, 29

Offset: 1

Rémy Sigrist, Oct 19 2019

nonn

Apparently, f_n is finite for any n > 0.
The first records are:
  n        a(n)
  -------  ----
        1     1
        2    39
       25    40
       61    41
       78    45
      266    47
      279    56
      629   102
    95417   103
   331468   104
  1318090   108
  5383290   109

			For n = 2:
- f_2 corresponds to A085947 which is finite with 39 terms,
- hence a(2) = 39.

Rémy Sigrist, PARI program for A326791

Cf. A085947.

PARI
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See Links section.
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A329811 Lexicographically earliest sequence of distinct positive integers such that for any consecutive triple of digits, say (x, y, z), x+y is a multiple of z.

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 11, 12, 13, 14, 15, 6, 17, 8, 19, 21, 31, 23, 51, 32, 57, 18, 91, 52, 7, 35, 16, 71, 41, 53, 25, 72, 111, 112, 131, 45, 34, 151, 61, 74, 152, 73, 54, 37, 29, 121, 123, 58, 132, 134, 153, 47, 141, 56, 171, 81, 92, 1111, 211, 213, 145, 38, 191

Offset: 1

Eric Angelini and Rémy Sigrist, Nov 21 2019

This sequence is infinite as we can always extend it with a repunit not yet used.
All terms are zeroless.
113 is the first zeroless number that cannot appear in this sequence.

			The first terms, alongside the corresponding (x, y, z), are:
  n   a(n)  x  y  z
  --  ----  -  -  -
   1     1  1  2  3
   2     2  2  3  5
   3     3  3  5  4
   4     5  5  4  9
   5     4  4  9  1
   6     9  9  1  1
   7    11  1  1  1
            1  1  2
   8    12  1  2  1
            2  1  3
   9    13  1  3  1
            3  1  4
  10    14  1  4  1
            4  1  5
  11    15  1  5  6
            5  6  1

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored scatterplot of the first 100000 terms (where the color is function of the previous term mod 100)
Rémy Sigrist, PARI program for A329811
Rémy Sigrist, C# program for A329811

Cf. A002275, A052382, A085947.

See Links section.
(C#) See Links section.

A360281 Lexicographically earliest sequence of distinct positive integers such that for any n > 2, a(n) is a divisor or a multiple of a(n-1) + a(n-2).

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 13, 11, 6, 17, 23, 8, 31, 39, 7, 46, 53, 33, 43, 19, 62, 27, 89, 29, 59, 22, 81, 103, 92, 15, 107, 61, 12, 73, 85, 79, 41, 10, 51, 122, 173, 295, 18, 313, 331, 14, 69, 83, 38, 121, 159, 20, 179, 199, 21, 44, 65, 109, 58, 167, 25, 16, 82, 49

Offset: 1

Rémy Sigrist, Feb 01 2023

nonn

This sequence has similarities with A085947, A328444 and A332301:
- these sequences agree for n = 1..39,
- however, a(40) = 122,
     A085947(40) does not exist,
     A328444(40) = 34,
     A332301(40) = 61.

			The first terms, alongside the relationship with the two prior terms, are:
  n   a(n)  Relationship
  --  ----  ------------
   1     1  N/A
   2     2  N/A
   3     3  2+1
   4     5  3+2
   5     4  (5+3)/2
   6     9  4+5
   7    13  9+4
   8    11  (13+9)/2
   9     6  (11+13)/4
  10    17  6+11
  11    23  17+6
  12     8  (23+17)/5

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program.

Cf. A085947, A328444, A332301.

PARI
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\\ See Links section.
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cp's OEIS Frontend

A328444 Lexicographically earliest sequence of distinct positive numbers such that a(1) = 1, a(2) = 2, and for n > 2, a(n) divides Sum_{i = n-k..n-1} a(i) with k > 0 as small as possible.

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Formula

A332301 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest divisor of the sum of the previous two terms that has not yet appeared. If all divisors have appeared then a(n) equals the sum.

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A140460 a(0) = 2; thereafter a(n) = smallest integer not a multiple of an earlier terms nor a sum of two earlier terms.

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C

Mathematica

A323890 a(1) = 1, a(2) = 2; thereafter a(n+1) = smallest unused divisor of a(n) if there are any, otherwise a(n) + a(n-1).

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Mathematica

A326791 For n > 1, let f_n be the lexicographically earliest sequence of distinct positive terms such that f_n(1) = 1, f_n(2) = n, and for k > 2, f_n(k) divides f_n(k-2) + f_n(k-1); if f_n is finite, then a(n) is the number of terms of f_n, otherwise a(n) = -1; a(1) = 1.

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A329811 Lexicographically earliest sequence of distinct positive integers such that for any consecutive triple of digits, say (x, y, z), x+y is a multiple of z.

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PARI

A360281 Lexicographically earliest sequence of distinct positive integers such that for any n > 2, a(n) is a divisor or a multiple of a(n-1) + a(n-2).

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PARI