A377566 Lexicographically earliest infinite sequence of distinct positive integers such that if j = a(n-1) is primorial, a(n) is the smallest prime not already a term, whereas if j is not primorial a(n) is the smallest novel number > j divisible by rad(j).

1, 2, 3, 6, 5, 10, 20, 30, 7, 14, 28, 42, 84, 126, 168, 210, 11, 22, 44, 66, 132, 198, 264, 330, 660, 990, 1320, 1650, 1980, 2310, 13, 26, 52, 78, 156, 234, 312, 390, 780, 1170, 1560, 1950, 2340, 2730, 5460, 8190, 10920, 13650, 16380, 19110, 21840, 24570, 27300, 30030, 17

Offset: 1

Author

David James Sycamore, Nov 03 2024

Keywords

• nonn

Comments

In other words: j in A002110 implies a(n) = p, next missing prime; j not in A002110 implies a(n) = m*rad(j), with minimal novel m.

Immediately following odd prime term p = prime(n), 2*p occurs, and as the sequence extends, multiples of intervening primes q; 2A101301(n) gives the number of steps (terms) from prime(n) to A002110(n).

Sequence can be generated by the following recursion: If a(t) = prime(n), n > 1 then a(t+k-1) = A060735(k)*prime(n); k = 1,2...A101301(n)+1; see Example.

Examples

			If j = a(n-1) is squarefree then a(n) = 2*j.
a(9) = prime(4) = 7, A101301(4) = 7, so there are 7+1 = 8 terms from 7 to A002110(4) = 210, namely: A060735(7+k-1)*7, k = 1,2,...8; so: 1*7,2*7,4*7,6*7,12*7,18*7,24*7,30*7 = 7,14,28,42,84,126,168,210.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of log_10 a(n), n = 1..2^17.

Crossrefs

Cf. A000040, A002110, A005117, A101301.

Programs

Mathematica

{{1, 2, 3, 6}}~Join~Table[Prime[m + 2]*If[n == 0, 1, Product[Prime[i], {i, n}]]*k, {m, 10}, {n, 0, m}, {k, 1 + Boole[n > 1], If[n == 0, 1, Prime[n + 1]]}] // Flatten
(* faster for large datasets, or *)
nn = 1000; c[] := False; m[] := 1; f[x_] := FactorInteger[x][[All, 1]]; Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2]; j = 2; u = v = 3;
Do[If[Or[IntegerQ@ Log2[j],
     And[EvenQ[j], Union@ Differences@ PrimePi[#] == {1}] ],
     k = v, k = Times @@ #;
     While[c[k m[k]], m[k]++]; k *= m[k]] &[f[j]];
   Set[{a[n], c[k], j}, {k, True, k}];
   If[k == u, While[c[u], u++]];
   If[k == v, While[c[v], v = NextPrime[v] ] ], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Nov 04 2024 *)