A372699 a(1) = 1, j = a(n-1). For n > 1, a(n) is the least novel m such that rad(m*j) is the smallest primorial number divisible by A007947(j).

1, 2, 3, 4, 6, 5, 12, 10, 9, 8, 18, 15, 16, 24, 20, 27, 32, 36, 25, 30, 7, 60, 14, 45, 40, 48, 50, 54, 75, 64, 72, 80, 81, 96, 90, 21, 70, 42, 35, 84, 100, 108, 120, 28, 105, 56, 135, 128, 144, 125, 150, 49, 180, 63, 140, 126, 160, 162, 200, 192, 225, 216, 240

Offset

1,2

Comments

a(n) depends upon if rad(j) = A002110(k) for some k (equivalently A083720(j) = 1), or not. If so a(n) is least novel m such that rad(m*j) = A002110(k+1). Otherwise a(n) = least novel m such that rad(m*j) = A002110(A000720(q)), where q = gpf(j).

Put otherwise, if p = nextprime(q), and A = A083720, then for n > 1 if A(j) = 1, a(n) is the least novel p-smooth number divisible by p, and if A(j) = w > 1, a(n) is the least novel q-smooth number divisible by w.

If j is a term in A002110, a(n) = smallest prime which has not yet appeared in the sequence (e.g., 1-->2, 2-->3, 6-->5, 30-->7, 210-->11, and so on).

Primes are in order and if p is prime and p|a(n) there is an i <= n such that a(i) = p (no multiple of p appears prior to p). Sequence is conjectured to have "Property S" of A368900. Also, for integers x, y with x < y and rad(x) = rad(y), x appears in the sequence before y. Conjecture: Sequence is a permutation of the positive integers which preserves the above mentioned properties of A000027.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log log scatterplot of a(n), n = 1..10800.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..10800, showing primes in red, perfect powers of primes in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, with purple indicating powerful numbers that are not prime powers. Primorials P(k) = A002110(k) and primes are labeled.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^18, with a color function indicating greatest prime factor p = gpf(a(n)), with red indicating p = 2, orange p = 3, ..., magenta p = 29, and pink p = 31.

Formula

a(n) = A002110(k) --> a(n+1) = prime(k+1); a(n+2) = 2*a(n), a(n+3) = 2*prime(k+1) (n >= 1).

Example

a(1) = 1 = A002110(0) so a(2) = 2 since rad(2*1) = 2 = A002110(1).
a(2) = 2 = A002110(1) so a(3) = 3 since then rad(3*2) = 6 = A002110(2).
a(6) = 5 is not primorial so a(7) = 12, since rad(12*5) = 30 = A002110(3) and no smaller novel number has this property.

Mathematica

nn = 120;
c[_] := False; m[_] := 1;
Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2]; j = a[2];
f[x_] := f[x] = FactorInteger[x][[All, 1]];
Do[(If[Divisible[j, Set[q, Times @@ Prime@ Range[#2]]],
       q = Times @@ Prime@ Range[#2 + 1]];
       k = q/#3; mm = m[k];
       While[Or[c[k*mm], ! Divisible[q, Times @@ f[mm]]], mm++];
       k *= mm; While[c[k*m[k]], m[k]++]) & @@
     {#, PrimePi@ Last[#], Times @@ #} &[f[j]];
  Set[{a[n], c[k], j}, {k, True, k}], {n, 3, nn}];
Array[a, nn]

Crossrefs

Cf. A000027, A002110, A007947, A083720, A368900.

Sequence in context: A206710 A175502 A364246 * A368133 A362855 A370740

Adjacent sequences: A372696 A372697 A372698 * A372700 A372701 A372702

Keyword

nonn

Author

David James Sycamore and Michael De Vlieger, May 14 2024

Extensions

More terms from Michael De Vlieger, May 18 2024

Status

approved