A088739 -id:A088739 - cp's OEIS frontend

A210615 Least semiprime dividing n, or 0 if no semiprime divides n.

0, 0, 0, 4, 0, 6, 0, 4, 9, 10, 0, 4, 0, 14, 15, 4, 0, 6, 0, 4, 21, 22, 0, 4, 25, 26, 9, 4, 0, 6, 0, 4, 33, 34, 35, 4, 0, 38, 39, 4, 0, 6, 0, 4, 9, 46, 0, 4, 49, 10, 51, 4, 0, 6, 55, 4, 57, 58, 0, 4, 0, 62, 9, 4, 65, 6, 0, 4, 69, 10, 0, 4, 0, 74, 15, 4, 77, 6

Offset: 1

Jonathan Vos Post, Mar 23 2012

Roughly analogous to Least Prime Factor A020639 but with semiprimes rather than primes.

			a(24) = 4 because 24 is divisible by the semiprimes {4,6} of which 4 is the smallest.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001358, A008578, A020639, A088739 (this sequence without the zeros).

Table[If[PrimeQ[n] || n < 2, 0, f = FactorInteger[n]; If[f[[1,2]] > 1, f[[1,1]]^2, f[[1,1]]*f[[2,1]]]], {n, 100}] (* T. D. Noe, Mar 24 2012 *)
Flatten[Table[Select[Divisors[n],PrimeOmega[#]==2&,1],{n,80}]/.{}->{0}] (* Harvey P. Dale, Dec 07 2012 *)

a(n) = min {k such that k|n and k in A001358} else 0 if there exists no such k.

a(p) = 0 iff p in A008578. - Alois P. Heinz, Mar 28 2012

A137428 Positive integers n which have a composite divisor smaller than their largest prime factor.

Original entry on oeis.org

20, 28, 40, 42, 44, 52, 56, 60, 66, 68, 76, 78, 80, 84, 88, 92, 99, 100, 102, 104, 110, 112, 114, 116, 117, 120, 124, 126, 130, 132, 136, 138, 140, 148, 152, 153, 156, 160, 164, 168, 170, 171, 172, 174, 176, 180, 184, 186, 188, 190, 196, 198, 200, 204, 207, 208

Offset: 1

Leroy Quet, Apr 17 2008

nonn

The primitive elements of this sequence are those of the form s*p, where s is a semiprime and p a prime larger than s, cf. A252478. Any multiple of these primitive terms is also in the sequence. - M. F. Hasler, Jan 02 2015

			The positive divisors of 60 are 1,2,3,4,5,6,10,12,15,20,30,60. The divisor 4, a composite, is less than the prime divisor 5. So 60 is in this sequence.

Cf. A006530, A088739.

isA137428 := proc(n) local dvs,p,i ; dvs := sort(convert(numtheory[divisors](n) minus{1},list)) ; for i from 1 to nops(dvs) do if isprime(op(-i,dvs)) then p := op(-i,dvs) ; break ; fi ; od: for i from 1 to nops(dvs) do if op(i,dvs) < p and not isprime(op(i,dvs)) then RETURN(true) ; fi ; od: RETURN(false) ; end: for n from 1 to 400 do if isA137428(n) then printf("%d,",n) ; fi ; od: # R. J. Mathar, Apr 21 2008

a = {}; For[n = 2, n < 300, n++, If[FactorInteger[n][[ -1, 1]] > Min[Select[ Divisors[n], ! PrimeQ[ # ]&& # > 1 &]], AppendTo[a, n]]]; a (* Stefan Steinerberger, Apr 21 2008 *)

is(n)=#(n=factor(n)~)>1&&n[1,#n]>=n[1,1]*if(n[2,1]>1,n[1,1],n[1,2]) \\ M. F. Hasler, Jan 02 2015

More terms from R. J. Mathar and Stefan Steinerberger, Apr 21 2008

A179312 Largest semiprime dividing n, or 0 if no semiprime divides n.

Original entry on oeis.org

0, 0, 0, 4, 0, 6, 0, 4, 9, 10, 0, 6, 0, 14, 15, 4, 0, 9, 0, 10, 21, 22, 0, 6, 25, 26, 9, 14, 0, 15, 0, 4, 33, 34, 35, 9, 0, 38, 39, 10, 0, 21, 0, 22, 15, 46, 0, 6, 49, 25, 51, 26, 0, 9, 55, 14, 57, 58, 0, 15, 0, 62, 21, 4, 65, 33, 0, 34, 69, 35, 0, 9, 0, 74

Offset: 1

Jonathan Vos Post, Jan 11 2011

a(p in primes A000040) = 0; a(k in semiprimes A001358) = k. This is to semiprimes A001358 as A006530 is to primes A000040.

			The smallest semiprime is 4, so a(n<4) = 0.
a(4) = 4, since 4 = 2^2 is semiprime, and 4 | 4 (i.e., 4/4 = 1).
a(5) = 0 because 5 is prime, only 1 and 5 evenly divide 5, no prime (with 1 prime factor) is a semiprimes (with two prime factors, not necessarily distinct).
a(6) = 6, since 6 = 2*3 is semiprime, and 6 | ^ (i.e., 6/6 = 1).
a(8) = 4, since 4 = 2^2 is semiprime, and 4 | 8 (i.e., 8/4 = 2).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A001358, A006530, A034699, A052126, A052369, A061395.

Cf. A088739 (smallest semiprime divisor of n-th composite number).

a:= proc(n) local l;
      if n<4 or isprime(n) then 0
    else l:= sort(ifactors(n)[2], (x, y)-> x[1]>y[1]);
         l[1][1] *l[`if`(l[1][2]>=2, 1, 2)][1]
      fi
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jun 23 2012

semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@ n == 2; f[n_] := Max@ Select[ Divisors@ n, semiPrimeQ] /. {-\[Infinity] -> 0}; Array[f, 55]

a(n) = MAX(0, k in A001358 such that k | n).

A088740 Greatest semiprime divisor of n-th composite number.

Original entry on oeis.org

4, 6, 4, 9, 10, 6, 14, 15, 4, 9, 10, 21, 22, 6, 25, 26, 9, 14, 15, 4, 33, 34, 35, 9, 38, 39, 10, 21, 22, 15, 46, 6, 49, 25, 51, 26, 9, 55, 14, 57, 58, 15, 62, 21, 4, 65, 33, 34, 69, 35, 9, 74, 25, 38, 77, 39, 10, 9, 82, 21, 85, 86, 87, 22, 15, 91, 46, 93, 94, 95, 6, 49, 33, 25

Offset: 1

Reinhard Zumkeller, Oct 12 2003

nonn

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A088739, A002808, A001358.

Reap[Do[If[! PrimeQ[n], fl = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]]; Sow[fl[[-1]]*fl[[-2]]]], {n, 4, 1200}]][[2, 1]]; (* Zak Seidov, Mar 18 2013 *)
gsd[n_]:=Max[Select[Divisors[n],PrimeOmega[#]==2&]]; gsd/@Select[Range[ 200],CompositeQ] (* Harvey P. Dale, May 08 2022 *)

A179938 Third largest prime factor of numbers that are divisible by at least three different primes (A000977).

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 3, 3, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 5, 2, 3, 3, 2, 2, 2, 2, 2, 3, 2, 5, 3, 3, 3, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 3, 5, 2, 2, 3, 3, 3, 2, 2, 2, 2, 3, 5, 2, 2, 3, 2, 3

Offset: 1

Jonathan Vos Post, Jan 12 2011

Third largest prime factor of numbers k such that omega(k) = A001221(k) > 2. The 3rd largest prime factor may equal the second largest. This is not identical to third largest distinct prime factor of numbers that are divisible by at least three different primes. Indices n where a(n) equals 2, 3, 5, 7, 11, 13, 17, 19, 23, ... for the first time are 1, 8, 72, 299, 905, 1718, 3302, 6020, 10330, ... the corresponding numbers from A000977 are 30, 90, 350, 1001, 2431, 4199, 7429, 12673, 20677, ...

			a(1) = 2 because 30 = 2 * 3 * 5 has third largest prime factor 2.
a(2) = 2 because 42 = 2 * 3 * 7 has third largest prime factor 2.
a(3) = 2 because 60 = 2 * 2 * 3 * 5 has both third and fourth largest prime factor 2.
a(8) = 3 because 90 = 2 * 3 * 3 * 5 has both second and third largest prime factor 3.

Cf. A000040, A000977, A001221, A002808, A033992, A007774, A000961, A033993, A051270, A087040, A088739, A179312.

b:= proc(n) option remember; local k;
      if n=1 then 30
      else for k from b(n-1)+1 while
              nops(ifactors(k)[2])<3 do od;
           k
      fi
    end:
a:= n-> sort(map(x-> x[1]$x[2], ifactors(b(n))[2]))[-3]:
seq(a(n), n=1..120);

b[n_] := b[n] = Module[{k}, If[n==1, 30, For[k = b[n-1]+1, PrimeNu[k] < 3, k++]; k]];
a[n_] := (Table[#[[1]], {#[[2]]}]& /@ FactorInteger[b[n]] // Flatten // Sort)[[-3]];
Array[a, 120] (* Jean-François Alcover, Nov 28 2020, after Alois P. Heinz *)

Edited by Alois P. Heinz, Jan 14 2011

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A210615 Least semiprime dividing n, or 0 if no semiprime divides n.

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A137428 Positive integers n which have a composite divisor smaller than their largest prime factor.

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A179312 Largest semiprime dividing n, or 0 if no semiprime divides n.

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A088740 Greatest semiprime divisor of n-th composite number.

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A179938 Third largest prime factor of numbers that are divisible by at least three different primes (A000977).

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