A328444 -id:A328444 - cp's OEIS frontend

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.

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.

A328443 Lexicographically earliest sequence of distinct positive numbers such that a(1) = 1, a(2) = 2, and for n > 2, a(n) divides Sum_{i = 1..k} a(i) with k > 0 as small as possible.

Original entry on oeis.org

1, 2, 3, 6, 4, 12, 8, 16, 7, 14, 28, 9, 18, 36, 13, 26, 52, 59, 73, 101, 5, 10, 11, 22, 55, 110, 32, 64, 128, 41, 82, 164, 177, 29, 203, 15, 17, 51, 85, 255, 157, 314, 43, 129, 387, 61, 122, 244, 488, 493, 503, 257, 514, 67, 134, 268, 536, 197, 591, 701, 733

Offset: 1

Rémy Sigrist, Oct 15 2019

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 can also be seen as an irregular table, with first row (1, 2), and for n > 1, the n-th row corresponds to the divisors of the sum of the first n terms not yet in the sequence in ascending order (and the sum of the first n terms is the last term of the n-th row).

			The table begins:
     1, 2;
     3;
     6;
     4, 12;
     8, 16;
     7, 14, 28;
     9, 18, 36;
     ...

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A328443

See A328444 for a similar sequence.

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

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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cp's OEIS Frontend

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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A328443 Lexicographically earliest sequence of distinct positive numbers such that a(1) = 1, a(2) = 2, and for n > 2, a(n) divides Sum_{i = 1..k} a(i) with k > 0 as small as possible.

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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