

A364246


a(1) = 1. Thereafter, a(n) is the least novel multiple of either prime(k+1) if rad(a(n1)) = A002110(k), or Product_{prime q; q < gpf(a(n1)); and q!a(n1)} q otherwise.


1



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
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OFFSET

1,2


COMMENTS

In other words, if the squarefree kernel of a(n1) is a primorial term then a(n) is the least novel multiple of the smallest prime which does not divide a(n1). Otherwise a(n) is the least novel multiple of the product of all primes < gpd(a(n1)) which do not divide a(n1). Primes >= 5 arrive late (as least unused term), and a(k) is prime(m) iff a(k1) is A002110(m1). 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(k1)/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


LINKS



EXAMPLE

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.


MATHEMATICA

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}];


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



