A379044 - OEIS

A379044

a(1) = 1, a(2) = 2. For n > 2 a(n) is the smallest positive integer k, not already a term in the sequence, such that A053669(k) = A006530(a(n-1)).

1, 2, 3, 4, 5, 6, 8, 7, 30, 12, 10, 18, 14, 60, 24, 16, 9, 20, 36, 22, 210, 90, 42, 120, 48, 26, 2310, 420, 150, 54, 28, 180, 66, 630, 240, 72, 32, 11, 840, 270, 78, 4620, 1050, 300, 84, 330, 1260, 360, 96, 34, 30030, 6930, 1470, 390, 9240, 1680, 450, 102, 60060, 11550, 1890, 480, 108, 38, 510510

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OFFSET

1,2

COMMENTS

For n > 2, a(n) is the smallest novel number whose least non divisor prime is the greatest prime divisor of a(n-1). Let p be the smallest prime which does not divide a(n-1). If rad(a(n-1)) = A002110(k), p = prime(k+1), otherwise p is the smallest prime < Gpf(a(n-1) which does not divide a(n-1).

The terms of the sequence can be extracted from the table in A125703, where the k-th row gives numbers whose least non divisor prime p is the k-th prime. From a(5) onwards the first appearance of q = Gpf(a(n-1)) = prime(k) gives a(n) = A002110(k-1), the first term in the k-th row of A125703. Subsequent appearances of same q selects a(n) from the same row. The primes q = Gpf(a(n-1)) decrement from term to term if rad(a(n-1) is in A002110, and occasionally we see a full cycle of q from some higher prime terminating at q = 2 (see Example). However this is rare, because of interruptions when rad(a(n-1)) is not primorial, causing q to increment before decrementing again. This retards the arrival of odd numbers in the sequence. Whereas we may suppose that all numbers in A125703 will eventually appear, the blocking of full decremetation cycles of q produces a "firewall" effect opposing this possibility. Conjecture: Full cycles of decrementing q conclude with a(n) a power of 2 (producing the next odd term).

LINKS

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

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.

Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..2^16

Michael De Vlieger, Log log scatterplot of a(n), n = 1..1024 showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple also represents powerful numbers that are not prime powers.

EXAMPLE

a(3) = 3 since the greatest prime factor of a(2)is 2, and 3 is the smallest novel number whose smallest non divisor prime is 2.

a(5) = 5 implies a(6) = 6 because 6 is the smallest novel term whose smallest non divisor prime is 5.

a(16) = 16 (Gpf = 2) so a(17) = 9, the smallest novel odd (composite) number.

a(37) = 32 (Gpf = 2) so a(38) = 11 the smallest novel odd (prime) number.

a(13,14,15,16) = 14,60,24,16 with decrementing Gpf's 7,5,3,2 respectively, leading to a(17) = 9. Likewise a(33,34,35,36,37) = 66,630,240,72,32 with decrementing Gpf's 11,7,5,3,2 respectively, leading to a(38) = 11 (next odd number).

MATHEMATICA

nn = 120; c[_] := False; m[_] := 1;

a[1] = 1; a[2] = j = p = 2; facs = {2}; c[1] = c[2] = True;

f[x_] := Block[{q = 2}, While[Divisible[x, q], q = NextPrime[q]]; q];

Do[p = Max[facs];

If[IntegerQ@ Log2[j], k = m[1]; While[Or[c[k], f[k] != p], k++];

If[k > m[1], While[c[m[1]], m[1]++]],

k = Product[Prime[i], {i, -1 + PrimePi[p]}];

While[Or[c[#], f[#] != p] &[k*m[k]], m[k]++]; k *= m[k]];

Set[{a[n], c[k], j, facs},

{k, True, k, FactorInteger[k][[All, 1]] } ], {n, 3, nn}];

Array[a, nn] (* Michael De Vlieger, Dec 17 2024 *)