A096015 -id:A096015 - cp's OEIS frontend

A096014 a(n) = (smallest prime factor of n) * (least prime that is not a factor of n), with a(1)=2.

2, 6, 6, 6, 10, 10, 14, 6, 6, 6, 22, 10, 26, 6, 6, 6, 34, 10, 38, 6, 6, 6, 46, 10, 10, 6, 6, 6, 58, 14, 62, 6, 6, 6, 10, 10, 74, 6, 6, 6, 82, 10, 86, 6, 6, 6, 94, 10, 14, 6, 6, 6, 106, 10, 10, 6, 6, 6, 118, 14, 122, 6, 6, 6, 10, 10, 134, 6, 6, 6, 142, 10, 146, 6, 6, 6, 14, 10, 158, 6, 6, 6

Offset: 1

Reinhard Zumkeller, Jun 15 2004

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001747, A020639, A053669, A073582, A096015.

f:= proc(n) local p;
p:= 3;
if n::even then
  while type(n/p,integer) do p:= nextprime(p) od;
else
  while not type(n/p,integer) do p:= nextprime(p) od:
fi;
2*p;
end proc:
f(1):= 2:
map(f, [$1..100]); # Robert Israel, Jun 22 2018

PrimeFactors[n_] := Flatten[ Table[ #[[1]], {1} ] & /@ FactorInteger[n]]; f[1] = 2; f[n_] := Block[ {k = 1}, While[ Mod[ n, Prime[k]] == 0, k++ ]; Prime[k]PrimeFactors[n][[1]]]; Table[ f[n], {n, 83}] (* Robert G. Wilson v, Jun 15 2004 *)
spfn[n_]:=Module[{fi=FactorInteger[n][[;;,1]],k=2},While[MemberQ[fi,k],k=NextPrime[k]];fi[[1]]*k]; Array[spfn,90] (* Harvey P. Dale, Sep 22 2024 *)

dnd(n) = forprime(p=2, , if (n % p, return(p)));
lpf(n) = if (n==1, 1, forprime(p=2, , if (!(n % p), return(p))));
a(n) = dnd(n)*lpf(n); \\ Michel Marcus, Jun 22 2018

a(n) = A020639(n)*A053669(n);
A096015(n) = a(n)/2.
If n (mod 6) = 2, 3 or 4, then a(n) = 6. If n (mod 6) = 0, 1 or 5, then a(n) belongs to A001747 less the first three terms or belongs to A073582 less the first two terms. - Robert G. Wilson v, Jun 15 2004
From Bill McEachen, Jul 26 2024: (Start)
a(n) <= 2*n, except when n = 2.
a(n) = 2*n for n an odd prime. (End)

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

Original entry on oeis.org

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

Peter Munn and Michael De Vlieger, Feb 18 2024

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.

			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.

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

Cf. A000079, A002110, A003557, A007947, A024619, A053669, A061345, A096015 (smallest instead of 2nd smallest), A100484, A119288, A246547, A361098.

{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]

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}.

cp's OEIS Frontend

A096014 a(n) = (smallest prime factor of n) * (least prime that is not a factor of n), with a(1)=2.

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