A374404 - OEIS

A374404

a(1) = 1, a(2) = 2. Let i = a(n-2), j = a(n-1). For n > 2 if rad(i*j) is primorial number A002110(t), a(n) is least novel k such that rad(i*j*k) = A002110(t+1). Otherwise, if rad(i*j) is not primorial, a(n) is least novel k such that rad(i*j*k) = A002110(s), where prime(s) = A006530(i*j); rad = 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, 98, 48, 75, 112, 121, 90, 126, 132, 80, 147, 143, 100, 168

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OFFSET

1,2

COMMENTS

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.

LINKS

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.

MATHEMATICA

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}];