A372699 - OEIS

%I #51 May 25 2024 15:42:42

%S 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,

%T 50,54,75,64,72,80,81,96,90,21,70,42,35,84,100,108,120,28,105,56,135,

%U 128,144,125,150,49,180,63,140,126,160,162,200,192,225,216,240

%N 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).

%C 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).

%C 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.

%C 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).

%C 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.

%H Michael De Vlieger, <a href="/A372699/b372699.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A372699/a372699.png">Log log scatterplot of a(n)</a>, n = 1..10800.

%H Michael De Vlieger, <a href="/A372699/a372699_1.png">Log log scatterplot of a(n)</a>, 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.

%H Michael De Vlieger, <a href="/A372699/a372699_2.png">Log log scatterplot of a(n)</a>, 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.

%F 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).

%e a(1) = 1 = A002110(0) so a(2) = 2 since rad(2*1) = 2 = A002110(1).

%e a(2) = 2 = A002110(1) so a(3) = 3 since then rad(3*2) = 6 = A002110(2).

%e 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.

%t nn = 120;

%t c[_] := False; m[_] := 1;

%t Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2]; j = a[2];

%t f[x_] := f[x] = FactorInteger[x][[All, 1]];

%t Do[(If[Divisible[j, Set[q, Times @@ Prime@ Range[#2]]],

%t        q = Times @@ Prime@ Range[#2 + 1]];

%t        k = q/#3; mm = m[k];

%t        While[Or[c[k*mm], ! Divisible[q, Times @@ f[mm]]], mm++];

%t        k *= mm; While[c[k*m[k]], m[k]++]) & @@

%t      {#, PrimePi@ Last[#], Times @@ #} &[f[j]];

%t   Set[{a[n], c[k], j}, {k, True, k}], {n, 3, nn}];

%t Array[a, nn]

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

%K nonn

%O 1,2

%A _David James Sycamore_ and _Michael De Vlieger_, May 14 2024

%E More terms from _Michael De Vlieger_, May 18 2024