A300813 -id:A300813 - cp's OEIS frontend

A365059 a(1) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that is a multiple of A008472(a(n-1)), the sum of the distinct primes dividing a(n-1).

2, 4, 6, 5, 10, 7, 14, 9, 3, 12, 15, 8, 16, 18, 20, 21, 30, 40, 28, 27, 24, 25, 35, 36, 45, 32, 22, 13, 26, 60, 50, 42, 48, 55, 64, 34, 19, 38, 63, 70, 56, 54, 65, 72, 75, 80, 49, 77, 90, 100, 84, 96, 85, 44, 39, 112, 81, 33, 98, 99, 126, 108, 95, 120, 110, 144, 105, 135, 88, 52, 150, 130, 140

Offset: 1

Scott R. Shannon, Aug 19 2023

nonn

In the first 500000 terms the only fixed points are 38 and 209, although it is likely more exist. In the same range the smallest missing numbers are 311, 313, 337. The sequence is conjectured to be a permutation of the integers >= 2.

			a(3) = 6 as a(2) = 4 and A008472(4) = 2, and 6 is the smallest unused number that is a multiple of 2.
a(11) = 15 as a(10) = 12 and A008472(12) = 5, and 15 is the smallest unused number that is a multiple of 5.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 500000 terms. The green line is a(n) = n.

Cf. A008472, A365060, A300813.

A365060 a(1) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that has a common factor with A008472(a(n-1)), the sum of the distinct primes dividing a(n-1).

Original entry on oeis.org

2, 4, 6, 5, 10, 7, 14, 3, 9, 12, 15, 8, 16, 18, 20, 21, 22, 13, 26, 24, 25, 30, 28, 27, 33, 32, 34, 19, 38, 35, 36, 40, 42, 39, 44, 52, 45, 46, 50, 49, 56, 48, 55, 54, 60, 58, 31, 62, 11, 66, 64, 68, 57, 70, 63, 65, 51, 72, 75, 74, 69, 76, 77, 78, 80, 84, 81, 87, 82, 43, 86, 85, 88, 91, 90, 92, 95

Offset: 1

Scott R. Shannon, Aug 19 2023

nonn

In the first 100000 terms the only fixed point is 9, and it is likely no more exist. In the same range the smallest missing numbers are 503, 839, 877. The sequence is conjectured to be a permutation of the integers >= 2.

			a(3) = 6 as a(2) = 4 and A008472(4) = 2, and 6 is the smallest unused number that shares a factor with 2.
a(8) = 3 as a(7) = 14 and A008472(14) = 9, and 3 is the smallest unused number that shares a factor with 9.

Scott R. Shannon, Table of n, a(n) for n = 1..10000.
Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.

Cf. A008472, A365059, A300813.

A367711 a(1) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that has a common factor k > 1 with A001414(a(n-1)), the sum of the primes dividing a(n-1), with repetition.

Original entry on oeis.org

2, 4, 6, 5, 10, 7, 14, 3, 9, 8, 12, 21, 15, 16, 18, 20, 24, 27, 30, 22, 13, 26, 25, 28, 11, 33, 32, 34, 19, 38, 35, 36, 40, 44, 39, 42, 45, 55, 46, 50, 48, 66, 52, 17, 51, 54, 77, 56, 65, 57, 58, 31, 62, 60, 63, 78, 64, 68, 49, 70, 72, 69, 74, 75, 91, 76, 23, 92, 81, 80, 104, 95, 82, 43, 86

Offset: 1

Scott R. Shannon, Nov 28 2023

nonn

In the first 100000 terms the only fixed point is 9, and it is likely no more exist. In the same range the smallest missing numbers are 4073, 5039, 5261. The sequence is conjectured to be a permutation of the integers >= 2.
From Michael De Vlieger, Nov 28 2023: (Start)
In scatterplot, composites fall in a cototient trajectory just above the line a(n)/n, while primes fall into several trajectories well below the line a(n)/n. This is an effect of finding the next term a(n) such that gcd(a(n), a(n-1)) = 1.
The trajectories T of primes a(n) arrange according to a(n+1)/a(n) = m. Hence, for example, T(m), m = 2 includes {2, 5, 7, 13, 19, 31, 43, ...}, T(3) includes {3, 11, 17} and may be finite, T(4) includes {23, 41, 47, 71, 83, 101, ...}, but T(m) for m in {5, 16, 17, ...} does not appear in the first 2^20 terms. It is evident that the trajectories T(m) are nonlinear.
The smallest missing number in a(1..1048576) is prime(3912) = 36899, followed by primes with indices 3995, 4151, 4179, etc. The smallest missing composite is 1116967. (End)

			a(10) = 8 as a(9) = 9 and A001414(9) = 6, and 8 is the smallest unused number that shares a factor with 6. This is the first term to differ from A365060.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20, showing primes in red, composites in dark blue.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, composite prime powers in gold, squarefree composites in green, numbers neither squarefree nor prime powers in blue, accentuating numbers of the last category that are also squareful in light blue.
Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.

Cf. A001414, A365059, A365060, A300813.

nn = 120; c[_] := False;
  f[x_] := f[x] = Total[Times @@@ FactorInteger[x]]; f[1] = 1;
  a[1] = j = 2; c[2] = True; u = 3;
  Do[k = u; While[Or[c[k], CoprimeQ[j, k]], k++];
    Set[{a[n], c[k], j}, {k, True, f[k]}];
    If[k == u, While[c[u], u++]], {n, 2, nn}];
Array[a, nn] (* Michael De Vlieger, Nov 28 2023 *)

A376548 a(1)=1,a(2)=2. Let p be the smallest prime factor of j=a(n-1), then for n>2 a(n) is the smallest novel multiple of Sopfr(j) if j is in A046363, or of p if it is not.

Original entry on oeis.org

1, 2, 4, 6, 5, 10, 7, 14, 8, 12, 21, 3, 9, 15, 18, 16, 20, 22, 13, 26, 24, 28, 11, 33, 27, 30, 32, 34, 19, 38, 36, 40, 44, 42, 46, 48, 55, 25, 35, 45, 66, 50, 52, 17, 51, 39, 54, 77, 49, 56, 65, 60, 58, 31, 62, 64, 68, 70, 72, 74, 76, 23, 69, 57, 63, 78, 80, 91

Offset: 1

David James Sycamore, Sep 27 2024

nonn

If j=a(n-1) is in A046363 then a(n) = A001414(j) = prime q, and a(n+1) is the least novel multiple of q. Otherwise a(n) is the least novel multiple of the smallest prime factor of j. After a prime term a(n) the sequence produces a string of terms each divisible by the smallest prime factor of a(n+1) until arriving at a term in A046363, whereupon a new prime appears and the process repeats.

Conjectured to be a permutation of the positive integers A000027, in which the primes do not appear in order (prime order starts:2,5,7,3,13,11,19,17,31,23,43..).

			a(2)=2 is not a term in A046360, and has smallest prime factor 2, so a(3) = 4, the least novel multiple of 2. Likewise a(4)=6 since a(3)=4 is not in A046360 and the smallest prime factor of 4 is 2.
a(4)=6 is a term in A046360, so a(5)=A001414(6)=5.
a(6)=10 since 5 is the smallest prime factor of 5, and 10 is the smallest novel multiple of 5.
If a(n-1) = prime p, a(n) is the least novel multiple of p, for example a(12) = 3 and since a(4) = 6 it follows that a(13) = 9. Likewise a(19) = 13, and since no prior term is divisible by 13, a(20) = 36.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, with purple additionally representing powerful numbers that are not squarefree.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20, with the same color function as immediately above. Note various trajectories of primes.

Cf. A000027, A001414, A300813.

nn = 120; c[] := False; m[] := 1;
a[1] = 1; j = a[2] = 2; c[1] = c[2] = True; m[1] = m[2] = 2;
f[x_] := f[x] = Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[x]];
Do[(If[PrimeQ[#2], k = #2, k = #1]; While[c[k*m[k]], m[k]++]; k *= m[k]) & @@
  {FactorInteger[j][[1, 1]], f[j]};
  Set[{a[i], c[k], j}, {k, True, k}], {i, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Sep 28 2024 *)

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A365059 a(1) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that is a multiple of A008472(a(n-1)), the sum of the distinct primes dividing a(n-1).

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A365060 a(1) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that has a common factor with A008472(a(n-1)), the sum of the distinct primes dividing a(n-1).

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A367711 a(1) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that has a common factor k > 1 with A001414(a(n-1)), the sum of the primes dividing a(n-1), with repetition.

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A376548 a(1)=1,a(2)=2. Let p be the smallest prime factor of j=a(n-1), then for n>2 a(n) is the smallest novel multiple of Sopfr(j) if j is in A046363, or of p if it is not.

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Mathematica