A332301 -id:A332301 - cp's OEIS frontend

A352774 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that does not share a factor with a(n-2) + a(n-1) or a(n-2)*a(n-1).

1, 2, 5, 3, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 9, 35, 41, 27, 43, 47, 49, 53, 55, 59, 61, 67, 15, 71, 73, 65, 77, 51, 79, 83, 85, 89, 91, 97, 33, 101, 95, 39, 103, 107, 109, 113, 115, 119, 121, 127, 21, 125, 131, 57, 137, 139, 133, 45, 143, 149, 63, 145, 151, 69, 157, 155, 161, 81, 163, 167

Offset: 1

Scott R. Shannon, Apr 02 2022

nonn

As a(2) is even, which forces a(3) and a(4) to be odd, all following terms will be odd as the sum of two odd terms is even. Beyond a(5) = 7 all subsequent primes appear in their natural order.

			a(2) = 5 as a(1) + a(2) = 3, a(1)*a(2) = 2, and 5 is the smallest unused number that does not share a factor with 3 or 2.

Scott R. Shannon, Image of the first 5000 terms. The green line is y = n.

Cf. A352768, A352790, A332301, A351001, A349493, A251622.

A352790 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that is a divisor of either a(n-2)+a(n-1) or a(n-2)*a(n-1).

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 6, 15, 7, 11, 18, 22, 8, 10, 16, 13, 26, 39, 65, 52, 20, 12, 24, 32, 14, 23, 37, 30, 67, 97, 41, 46, 29, 25, 27, 45, 36, 54, 72, 21, 28, 42, 35, 49, 84, 19, 38, 57, 95, 76, 171, 114, 285, 90, 50, 60, 40, 48, 44, 33, 66, 99, 55, 77, 121, 198, 242, 88, 110, 80, 100, 64, 82, 73

Offset: 1

Scott R. Shannon, Apr 03 2022

nonn

The sequence shows large variations in its values due to some terms being required to equal the product of two large primes as all other divisors of both the sum and product have been used, e.g., a(25893) = 68485691. In the first 50000 terms there are 28233 occasions where the chosen term is a divisor of the sum and 22777 occasions where it is a divisor of the product. In the same range the fixed points are 1, 2, 3, 42, 3674, 11520, 18515, 39501 - it is likely more exist. The smallest number not to appear is 761, while the primes do not occur in their natural order. It is possible the sequence is finite as two terms could appear whose sum and product divisors have all been used. If not then the sequence is likely a permutation of the positive integers.
From Michael De Vlieger, Apr 03 2022: (Start)
Let S = a(n-1)+a(n+2) and let P = a(n-1)*a(n-2). Let u be the smallest missing number in a(1..n-1).
It is possible that S is prime, but P is prime iff either a(n-1)=1 or a(n-2)=1; since a(1) = 1 is given and followed by 2, for n > 2, P is always composite.
The axiom a(n) = (k | S) or (k | P) implies (k <= S) or (k <= P). Consequently, u <= k <= max(S,P). Let D contain divisors {d : d | S and d >= u} and let E contain factors {d : d | P and d >= u}. A solution k must appear in T = D U E.
For sufficiently large n, S is large, but P is large and composite.
In this sequence we may have equality of (one) input and output, since input S or P does not necessarily already exist in a.
(End)

			a(4) = 5 as a(2)+a(3) = 5 and a(2)*a(3) = 6. The combined divisors of these two numbers are 1,2,3,5,6, and of those 5 is the smallest that has not appeared.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..2^16, showing records in red, local minima in blue, fixed points in gold, and accentuating primes in green.
Scott R. Shannon, Line graph of the first 50000 terms.

Cf. A352768, A332301, A351001, A349493, A251622.

nn = 74; c[] = 0; u = a[1] = c[1] = 1; a[2] = c[2] = 2; While[c[u] > 0, u++]; Do[k = 1; t = TakeWhile[Reverse@ Union[Divisors[#1 + #2], Divisors[#1*#2]], # >= u &] & @@ {a[i - 2], a[i - 1]}; While[c[t[[-k]]] > 0, k++]; Set[{a[i], c[t[[-k]]]}, {t[[-k]], i}]; If[a[i] == u, While[c[u] > 1, u++]], {i, Length[s] + 1, nn}]; Array[a, nn] (* _Michael De Vlieger, Apr 03 2022 *)

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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A352774 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that does not share a factor with a(n-2) + a(n-1) or a(n-2)*a(n-1).

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A352790 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that is a divisor of either a(n-2)+a(n-1) or a(n-2)*a(n-1).

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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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Author

Keywords

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Crossrefs

Programs

PARI