A369690 a(n) = max(A119288(n), A053669(n)).

2, 3, 2, 3, 2, 5, 2, 3, 2, 5, 2, 5, 2, 7, 5, 3, 2, 5, 2, 5, 7, 11, 2, 5, 2, 13, 2, 7, 2, 7, 2, 3, 11, 17, 7, 5, 2, 19, 13, 5, 2, 5, 2, 11, 5, 23, 2, 5, 2, 5, 17, 13, 2, 5, 11, 7, 19, 29, 2, 7, 2, 31, 7, 3, 13, 5, 2, 17, 23, 5, 2, 5, 2, 37, 5, 19, 11, 5, 2, 5, 2

Offset

1,1

Comments

Equivalently, a(n) is the largest p such that p is the 2nd smallest prime dividing n or the smallest prime not dividing n.

If squarefree n is such that a(n) = p, then a(k) = p for k in the infinite sequence { k = m*n : rad(m) | n }. Consequence of the fact that both A119288(n) and A053669(n) do not depend on multiplicity of prime divisors p | n.

Links

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

Formula

a(n) <= A003557(n) for n > 4 in A246547 and for n in A361098.

Numbers n that set records include 1, 2, and squarefree semiprimes, i.e., (A100484 \ {4}) U {1, 2}.

Example

Let p be the second least prime factor of n or 1 if n is a prime power, and let q be the smallest prime that does not divide n.
a(1) = 2 since max(p, q) = max(1, 2) = 2.
a(2) = 3 since max(p, q) = max(1, 3) = 3.
a(4) = 3 since max(p, q) = max(1, 3) = 3.
a(6) = 5 since max(p, q) = max(3, 5) = 5.
a(9) = 2 since max(p, q) = max(1, 2) = 2.
a(15) = 5 since max(p, q) = max(5, 2) = 5.
a(36) = 5 since max(p, q) = max(3, 5) = 5.
Generally,
a(n) = 2 for n in A061345 = union of {1} and sequences { m*p : prime p > 2, rad(m) | p }.
a(n) = 3 for n in A000079 = { 2*m : rad(m) | 2 }.
a(n) = 5 for k in { k = m*d : rad(m) | d, d in {6, 10, 15} }.
a(n) = 7 for k in { k = m*d : rad(m) | d, d in {14, 21, 30, 35} }.
a(n) = 11 for k in { k = m*d : rad(m) | d, d in {22, 33, 55, 77, 210} }, etc.

Mathematica

{2}~Join~Array[If[PrimePowerQ[#],
  q = 2; While[Divisible[#, q], q = NextPrime[q]]; q,
  q = 2; While[Divisible[#, q], q = NextPrime[q]];
    Max[FactorInteger[#][[2, 1]], q]] &, 120, 2]

Crossrefs

Cf. A000079, A002110, A003557, A007947, A024619, A053669, A061345, A096015 (smallest instead of 2nd smallest), A100484, A0119288, A246547, A361098.

Sequence in context: A060395 A016026 A086757 * A166985 A046215 A256113

Adjacent sequences: A369687 A369688 A369689 * A369691 A369692 A369693

Keyword

nonn,easy

Author

Peter Munn and Michael De Vlieger, Feb 18 2024

Status

approved