A362889 -id:A362889 - cp's OEIS frontend

A374379 a(1) = 1, a(2) = 2. Let i = a(n-2), j = a(n-1), then for n > 2 a(n) is the least novel k such that rad(k) !| rad(ij) whereas rad(ij*k) is a primorial number (A002110), where rad is A007947.

1, 2, 3, 5, 4, 6, 10, 7, 9, 20, 14, 12, 15, 21, 8, 25, 18, 28, 30, 11, 35, 24, 22, 70, 27, 33, 140, 13, 66, 105, 26, 44, 210, 39, 55, 42, 52, 110, 63, 65, 88, 84, 40, 77, 36, 45, 49, 16, 60, 56, 99, 50, 91, 132, 75, 98, 121, 90, 112, 143, 120, 119, 286, 135, 126, 154, 80, 48, 147

Offset: 1

David James Sycamore, Jul 06 2024

nonn

For n > 2 a(n) is prime iff rad(i*j) is a primorial number, for then k is the smallest prime which does not divide rad(i*j).
Initially same terms as A362889, but divergence occurs at a(69) = 147.
Conjectured to be a permutation of the natural numbers with primes in order.

			a(3) = 3 since rad(3) !| rad(1*2) whereas rad (1*2*3) = 6 = A002110(2).
a(5,6) = {4,6}—> a(7) = 10 since rad(10) !| rad(24), rad(4*6*10) = 30 = A002110(3) and there is no smaller novel term with this property.
From _Michael De Vlieger_, Jul 06 2024: (Start)
Table of a(3..18) showing prime decomposition of rad(i*j) and rad(i*j*k):
   n  a(n)   rad(i*j)   rad(i*j*k)
  -------------------------------
   3    3    2          2 3
   4    5    2 3        2 3 5
   5    4    . 3 5      2 3 5
   6    6    2 . 5      2 3 5
   7   10    2 3        2 3 5
   8    7    2 3 5      2 3 5 7
   9    9    2 . 5 7    2 3 5 7
  10   20    . 3 . 7    2 3 5 7
  11   14    2 3 5      2 3 5 7
  12   12    2 . 5 7    2 3 5 7
  13   15    2 3 . 7    2 3 5 7
  14   21    2 3 5      2 3 5 7
  15    8    . 3 5 7    2 3 5 7
  16   25    2 3 . 7    2 3 5 7
  17   18    2 . 5      2 3 5
  18   28    2 3 5      2 3 5 7   (End)

Michael De Vlieger, Table of n, a(n) for n = 1..2050
Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..2048.

Cf. A002110, A007947, A055932, A362889.

nn = 1200; c[] := False; rad[n] := rad[n] = Times @@ FactorInteger[n][[All, 1]];
Array[Set[{a[#], c[#]}, {#, True}] &, 2];
f[n_] := Or[IntegerQ@ Log2[n],
  And[EvenQ[n], Union@ Differences@ Map[PrimePi, FactorInteger[n][[All, 1]]] == {1}]]; i = a[1]; j = a[2]; u = 3;
Monitor[Do[r = rad[i*j]; k = u;
  While[Or[c[k], Divisible[#, rad[k]], ! f[#  k]] &[i*j], k++];
  Set[{a[n], c[k], i, j}, {k, True, j, k}];
  If[k == u, While[c[u], u++]], {n, 3, nn}], n];
Array[a, nn] (* Michael De Vlieger, Jul 06 2024 *)

A364246 a(1) = 1. Thereafter, a(n) is the least novel multiple of either prime(k+1) if rad(a(n-1)) = A002110(k), or Product_{prime q; q < gpf(a(n-1)); and q!|a(n-1)} q otherwise.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 12, 10, 9, 8, 15, 14, 30, 7, 60, 21, 20, 18, 25, 24, 35, 36, 40, 27, 16, 33, 70, 39, 770, 42, 45, 22, 105, 26, 1155, 28, 75, 32, 48, 50, 51, 10010, 54, 55, 84, 65, 462, 80, 57, 170170, 63, 90, 49, 120, 56, 135, 34, 15015, 38, 255255, 44, 210

Offset: 1

David James Sycamore, Jul 15 2023

nonn

In other words, if the squarefree kernel of a(n-1) is a primorial term then a(n) is the least novel multiple of the smallest prime which does not divide a(n-1). Otherwise a(n) is the least novel multiple of the product of all primes < gpd(a(n-1)) which do not divide a(n-1). Primes >= 5 arrive late (as least unused term), and a(k) is prime(m) iff a(k-1) is A002110(m-1). The pattern around a prime is P(k), prime(k+1), 2*P(k), m*prime(k+1) for some multiplier m, where P(k) = A002110(k). The sequence is conjectured to be a permutation of the positive integers, with primes in natural order.

A common mode in this sequence is alternation of squarefree semiprime q(j)*q(k), j < k, followed by P(k-1)/q(j). The alternation often occurs in runs such that each iteration increments k. Example: a(241..246): q(2)*q(17) -> P(16)/q(2) -> q(2)*q(18) -> P(17)/q(2) -> q(2)*q(19) -> P(18)/q(2). a(16539..16572) represents a run of 17 alternations. - Michael De Vlieger, Jul 17 2023

			a(5) = 6 a primorial number so the next term is the smallest prime not dividing 6, thus a(7) = 5.
a(26) = 33 = 3*11 and the product of primes < 11 which do not divide 11 is 2*5*7 = 70, which has not occurred previously, therefore a(27) = 70.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..2^16, highlighting prime a(n) in red.

Cf. A002110, A006530, A007947, A362889, A364154.

nn = 120; c[] := False; m[] := 1; a[1] = j = 1; c[1] = True;
f[x_] := If[# == Prime@ Range[PrimePi@ #[[-1]]], Prime[PrimePi@ #[[-1]] + 1],
  Times @@ Complement[Prime@ Range[PrimePi@ #[[-1]] - 1], #]] &[
  FactorInteger[x][[All, 1]]];
Do[While[Set[k, f[j]]; c[k m[k]], m[k]++]; k *= m[k];
  Set[{a[n], c[k], j}, {k, True, k}], {n, 2, nn}];
Array[a, nn] (* Michael De Vlieger, Jul 17 2023 *)

More terms from Michael De Vlieger, Jul 17 2023

A374404 a(1) = 1, a(2) = 2. Let i = a(n-2), j = a(n-1). For n > 2 if rad(ij) is primorial number A002110(t), a(n) is least novel k such that rad(ijk) = A002110(t+1). Otherwise, if rad(ij) is not primorial, a(n) is least novel k such that rad(ijk) = A002110(s), where prime(s) = A006530(i*j); rad = A007947.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 10, 7, 9, 20, 14, 12, 15, 21, 8, 25, 18, 28, 30, 11, 35, 24, 22, 70, 27, 33, 140, 13, 66, 105, 26, 44, 210, 39, 55, 42, 52, 110, 63, 65, 88, 84, 40, 77, 36, 45, 49, 16, 60, 56, 99, 50, 98, 48, 75, 112, 121, 90, 126, 132, 80, 147, 143, 100, 168

Offset: 1

David James Sycamore and Michael De Vlieger, Jul 07 2024

nonn

Definition is subtly different from those of A362889 and A374379, which share the same initial terms of this sequence (divergence at a(53) = 98). If rad(i*j) is primorial = A002110(t), a(n) is least novel prime(t+1)-smooth number divisible by prime(t+1). And if rad(i*j) is not primorial, a(n) is the least novel prime(s)-smooth multiple of A002110(s)/rad(i*j). a(n) is prime iff rad(i*j) is a primorial number not seen earlier as kernel of the product of any prior pair of consecutive terms. It follows from the definition that for any consecutive three terms i,j,k, rad(i*j*k) is always a primorial number.

Conjectured to be a permutation of the positive integers A000027, with primes in order.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, 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 representing powerful numbers that are not prime powers.

Cf. A000027, A002110, A006530, A007947, A055932, A362889, A374379.

nn = 120; c[] := False; m[] := 1;
Do[Set[{a[n], c[n], m[n]}, {n, True, 2}], {n, 3}]; i = a[2]; j = a[3];
f[x_] := f[x] = FactorInteger[x][[All, 1]];
q[x_] := Or[IntegerQ@ Log2[x], And[EvenQ[x], Union@ Differences@ PrimePi@ f[x] == {1}]];
Do[If[q[i*j],
  s = NextPrime@Last@f[i*j]; k = 1;
    While[Or[c[k*s], ! q[i*j*k*s]], k++]; k *= s,
  t = Product[Prime[r], {r, PrimePi@ Last@ f[i*j]}];
    s = t/Apply[Times, f[i*j]]; k = 1;
    While[Or[c[k*s], Times @@ f[i*j*k*s] != t], k++]; k *= s];
  Set[{a[n], c[k], i, j}, {k, True, j, k}], {n, 4, nn}];
Array[a, nn] (* Michael De Vlieger, Jul 12 2024 *)

cp's OEIS Frontend

A374379 a(1) = 1, a(2) = 2. Let i = a(n-2), j = a(n-1), then for n > 2 a(n) is the least novel k such that rad(k) !| rad(ij) whereas rad(ij*k) is a primorial number (A002110), where rad is A007947.

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A364246 a(1) = 1. Thereafter, a(n) is the least novel multiple of either prime(k+1) if rad(a(n-1)) = A002110(k), or Product_{prime q; q < gpf(a(n-1)); and q!|a(n-1)} q otherwise.

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cp's OEIS Frontend

A374379 a(1) = 1, a(2) = 2. Let i = a(n-2), j = a(n-1), then for n > 2 a(n) is the least novel k such that rad(k) !| rad(i*j) whereas rad(i*j*k) is a primorial number (A002110), where rad is A007947.

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Mathematica

A364246 a(1) = 1. Thereafter, a(n) is the least novel multiple of either prime(k+1) if rad(a(n-1)) = A002110(k), or Product_{prime q; q < gpf(a(n-1)); and q!|a(n-1)} q otherwise.

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Mathematica

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Mathematica

A374379 a(1) = 1, a(2) = 2. Let i = a(n-2), j = a(n-1), then for n > 2 a(n) is the least novel k such that rad(k) !| rad(ij) whereas rad(ij*k) is a primorial number (A002110), where rad is A007947.