A070647 -id:A070647 - cp's OEIS frontend

A006881 Squarefree semiprimes: Numbers that are the product of two distinct primes.

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205

Offset: 1

N. J. A. Sloane, Robert Munafo, Simon Plouffe

Numbers k such that phi(k) + sigma(k) = 2*(k+1). - Benoit Cloitre, Mar 02 2002
Numbers k such that tau(k) = omega(k)^omega(k). - Benoit Cloitre, Sep 10 2002 [This comment is false. If k = 900 then tau(k) = omega(k)^omega(k) = 27 but 900 = (2*3*5)^2 is not the product of two distinct primes. - Peter Luschny, Jul 12 2023]
Could also be called 2-almost primes. - Rick L. Shepherd, May 11 2003
From the Goldston et al. reference's abstract: "lim inf [as n approaches infinity] [(a(n+1) - a(n))] <= 26. If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6." - Jonathan Vos Post, Jun 20 2005
The maximal number of consecutive integers in this sequence is 3 - there cannot be 4 consecutive integers because one of them would be divisible by 4 and therefore would not be product of distinct primes. There are several examples of 3 consecutive integers in this sequence. The first one is 33 = 3 * 11, 34 = 2 * 17, 35 = 5 * 7; (see A039833). - Matias Saucedo (solomatias(AT)yahoo.com.ar), Mar 15 2008
Number of terms less than or equal to 10^k for k >= 0 is A036351(k). - Robert G. Wilson v, Jun 26 2012
Are these the numbers k whose difference between the sum of proper divisors of k and the arithmetic derivative of k is equal to 1? - Omar E. Pol, Dec 19 2012
Intersection of A001358 and A030513. - Wesley Ivan Hurt, Sep 09 2013
A237114(n) (smallest semiprime k^prime(n)+1) is a term, for n != 2. - Jonathan Sondow, Feb 06 2014
a(n) are the reduced denominators of p_2/p_1 + p_4/p_3, where p_1 != p_2, p_3 != p_4, p_1 != p_3, and the p's are primes. In other words, (p_2*p_3 + p_1*p_4) never shares a common factor with p_1*p_3. - Richard R. Forberg, Mar 04 2015
Conjecture: The sums of two elements of a(n) forms a set that includes all primes greater than or equal to 29 and all integers greater than or equal to 83 (and many below 83). - Richard R. Forberg, Mar 04 2015
The (disjoint) union of this sequence and A001248 is A001358. - Jason Kimberley, Nov 12 2015
A263990 lists the subsequence of a(n) where a(n+1)=1+a(n). - R. J. Mathar, Aug 13 2019

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Zervos, Marie: Sur une classe de nombres composés. Actes du Congrès interbalkanique de mathématiciens 267-268 (1935)

T. D. Noe, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pimtz and Y. Yildirim, "Small Gaps Between Primes or Almost Primes", arXiv:math/0506067 [math.NT], March 2005.
G. T. Leavens and M. Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy)
R. J. Mathar, Series of reciprocal powers of k-almost primes arXiv:0803.0900, table 6 k=2 shows sum 1/a(n)^s.
Eric Weisstein's World of Mathematics, Semiprime
Index to sequences related to prime signature

Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.
Cf. A030229, A051709, A001221 (omega(n)), A001222 (bigomega(n)), A001358 (semiprimes), A005117 (squarefree), A007304 (squarefree 3-almost primes), A213952, A039833, A016105 (subsequences), A237114 (subsequence, n != 2).
Subsequence of A007422.
Cf. A259758 (subsequence), A036351, A363923.

a006881 n = a006881_list !! (n-1)
a006881_list = filter chi [1..] where
   chi n = p /= q && a010051 q == 1 where
      p = a020639 n
      q = n `div` p
-- Reinhard Zumkeller, Aug 07 2011

[n: n in [1..210] | EulerPhi(n) + DivisorSigma(1,n) eq 2*(n+1)]; // Vincenzo Librandi, Sep 17 2015

N:= 1001: # to get all terms < N
Primes:= select(isprime, [2,seq(2*k+1,k=1..floor(N/2))]):
{seq(seq(p*q,q=Primes[1..ListTools:-BinaryPlace(Primes,N/p)]),p=Primes)} minus {seq(p^2,p=Primes)};
# Robert Israel, Jul 23 2014
# Alternative, using A001221:
isA006881 := proc(n)
     if numtheory[bigomega](n) =2 and A001221(n) = 2 then
        true ;
    else
        false ;
    end if;
end proc:
A006881 := proc(n) if n = 1 then 6; else for a from procname(n-1)+1 do if isA006881(a) then return a; end if; end do: end if;
end proc: # R. J. Mathar, May 02 2010
# Alternative:
with(NumberTheory): isA006881 := n -> is(NumberOfPrimeFactors(n, 'distinct') = 2 and NumberOfPrimeFactors(n) = 2):
select(isA006881, [seq(1..205)]); # Peter Luschny, Jul 12 2023

mx = 205; Sort@ Flatten@ Table[ Prime[n]*Prime[m], {n, Log[2, mx/3]}, {m, n + 1, PrimePi[ mx/Prime[n]]}] (* Robert G. Wilson v, Dec 28 2005, modified Jul 23 2014 *)
sqFrSemiPrimeQ[n_] := Last@# & /@ FactorInteger@ n == {1, 1}; Select[Range[210], sqFrSemiPrimeQ] (* Robert G. Wilson v, Feb 07 2012 *)
With[{upto=250},Select[Sort[Times@@@Subsets[Prime[Range[upto/2]],{2}]],#<=upto&]] (* Harvey P. Dale, Apr 30 2018 *)

for(n=1,214,if(bigomega(n)==2&&omega(n)==2,print1(n,",")))

for(n=1,214,if(bigomega(n)==2&&issquarefree(n),print1(n,",")))

list(lim)=my(v=List()); forprime(p=2,sqrt(lim), forprime(q=p+1, lim\p, listput(v,p*q))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from sympy import factorint
def ok(n): f=factorint(n); return len(f) == 2 and sum(f[p] for p in f) == 2
print(list(filter(ok, range(1, 206)))) # Michael S. Branicky, Jun 10 2021

from math import isqrt
from sympy import primepi, primerange
def A006881(n):
    def f(x): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 15 2024

def A006881_list(n) :
    R = []
    for i in (6..n) :
        d = prime_divisors(i)
        if len(d) == 2 :
            if d[0]*d[1] == i :
                R.append(i)
    return R
A006881_list(205)  # Peter Luschny, Feb 07 2012

A000005(a(n)^(k-1)) = A000290(k) for all k>0. - Reinhard Zumkeller, Mar 04 2007
A109810(a(n)) = 4; A178254(a(n)) = 6. - Reinhard Zumkeller, May 24 2010
A056595(a(n)) = 3. - Reinhard Zumkeller, Aug 15 2011
a(n) = A096916(n) * A070647(n). - Reinhard Zumkeller, Sep 23 2011
A211110(a(n)) = 3. - Reinhard Zumkeller, Apr 02 2012
Sum_{n >= 1} 1/a(n)^s = (1/2)*(P(s)^2 - P(2*s)), where P is Prime Zeta. - Enrique Pérez Herrero, Jun 24 2012
A050326(a(n)) = 2. - Reinhard Zumkeller, May 03 2013
sopf(a(n)) = a(n) - phi(a(n)) + 1 = sigma(a(n)) - a(n) - 1. - Wesley Ivan Hurt, May 18 2013
d(a(n)) = 4. Omega(a(n)) = 2. omega(a(n)) = 2. mu(a(n)) = 1. - Wesley Ivan Hurt, Jun 28 2013
a(n) ~ n log n/log log n. - Charles R Greathouse IV, Aug 22 2013
A089233(a(n)) = 1. - Reinhard Zumkeller, Sep 04 2013
From Peter Luschny, Jul 12 2023: (Start)
For k > 1: k is a term <=> k^A001221(k) = k*A007947(k).
For k > 1: k is a term <=> k^A001222(k) = k*A007947(k).
For k > 1: k is a term <=> A363923(k) = k. (End)
a(n) ~ n log n/log log n. - Charles R Greathouse IV, Jan 13 2025

Name expanded (based on a comment of Rick L. Shepherd) by Charles R Greathouse IV, Sep 16 2015

A270650 Min(i, j), where p(i)*p(j) is the n-th term of A006881.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 2, 1, 3, 1, 2, 1, 2, 3, 2, 1, 1, 3, 2, 1, 4, 1, 3, 1, 2, 4, 2, 1, 3, 1, 2, 3, 1, 4, 1, 2, 2, 4, 1, 2, 1, 5, 3, 1, 3, 1, 2, 4, 1, 2, 1, 2, 3, 5, 1, 2, 1, 4, 3, 1, 5, 2, 1, 3, 4, 1, 2, 6, 1, 3, 2, 6, 2, 5, 1, 4, 1, 3, 2, 1, 1, 4, 2, 3, 1

Offset: 1

Clark Kimberling, Apr 25 2016

			A006881 = (6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, ... ), the increasing sequence of all products of distinct primes.  The first 4 factorizations are 2*3, 2*5, 2*7, 3*5, so that (a(1), a(2), a(3), a(4)) = (1,1,1,2).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000040, A006881, A096916, A070647, A270652, A270003.

mx = 350; t = Sort@Flatten@Table[Prime[n]*Prime[m], {n, Log[2, mx/3]}, {m, n + 1, PrimePi[mx/Prime[n]]}]; (* A006881, Robert G. Wilson v, Feb 07 2012 *)
u = Table[FactorInteger[t[[k]]][[1]], {k, 1, Length[t]}];
u1 = Table[u[[k]][[1]], {k, 1, Length[t]}]  (* A096916 *)
PrimePi[u1]  (* A270650 *)
v = Table[FactorInteger[t[[k]]][[2]], {k, 1, Length[t]}];
v1 = Table[v[[k]][[1]], {k, 1, Length[t]}]  (* A070647 *)
PrimePi[v1]  (* A270652 *)
d = v1 - u1  (* A176881 *)
Map[PrimePi[FactorInteger[#][[1, 1]]] &, Select[Range@ 240, And[SquareFreeQ@ #, PrimeOmega@ # == 2] &]] (* Michael De Vlieger, Apr 25 2016 *)

A270652 Max(i,j), where p(i)*p(j) is the n-th term of A006881.

Original entry on oeis.org

2, 3, 4, 3, 4, 5, 6, 5, 7, 4, 8, 6, 9, 7, 5, 8, 10, 11, 6, 9, 12, 5, 13, 7, 14, 10, 6, 11, 15, 8, 16, 12, 9, 17, 7, 18, 13, 14, 8, 19, 15, 20, 6, 10, 21, 11, 22, 16, 9, 23, 17, 24, 18, 12, 7, 25, 19, 26, 10, 13, 27, 8, 20, 28, 14, 11, 29, 21, 7, 30, 15, 22

Offset: 1

Clark Kimberling, Apr 25 2016

			A006881 = (6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, ... ), the increasing sequence of all products of distinct primes.  The first 4 factorizations are 2*3, 2*5, 2*7, 3*5, so that (a(1), a(2), a(3), a(4)) = (2,3,4,3).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000040, A006881, A096916, A270650, A070647, A270003.

mx = 350; t = Sort@Flatten@Table[Prime[n]*Prime[m], {n, Log[2, mx/3]}, {m, n + 1, PrimePi[mx/Prime[n]]}]; (* A006881, Robert G. Wilson v, Feb 07 2012 *)
u = Table[FactorInteger[t[[k]]][[1]], {k, 1, Length[t]}];
u1 = Table[u[[k]][[1]], {k, 1, Length[t]}]  (* A096916 *)
PrimePi[u1]  (* A270650 *)
v = Table[FactorInteger[t[[k]]][[2]], {k, 1, Length[t]}];
v1 = Table[v[[k]][[1]], {k, 1, Length[t]}]  (* A070647 *)
PrimePi[v1]  (* A270652 *)
d = v1 - u1  (* A176881 *)
Map[PrimePi[FactorInteger[#][[-1, 1]]] &, Select[Range@ 240, And[SquareFreeQ@ #, PrimeOmega@ # == 2] &]] (* Michael De Vlieger, Apr 25 2016 *)

from math import isqrt
from sympy import primepi, primerange, primefactors
def A270652(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    return primepi(max(primefactors(bisection(f,n,n)))) # Chai Wah Wu, Oct 23 2024

A096916 Lesser prime factor of n-th product of two distinct primes.

Original entry on oeis.org

2, 2, 2, 3, 3, 2, 2, 3, 2, 5, 2, 3, 2, 3, 5, 3, 2, 2, 5, 3, 2, 7, 2, 5, 2, 3, 7, 3, 2, 5, 2, 3, 5, 2, 7, 2, 3, 3, 7, 2, 3, 2, 11, 5, 2, 5, 2, 3, 7, 2, 3, 2, 3, 5, 11, 2, 3, 2, 7, 5, 2, 11, 3, 2, 5, 7, 2, 3, 13, 2, 5, 3, 13, 3, 11, 2, 7, 2, 5, 3, 2, 2, 7, 3, 5, 2, 13, 7, 2, 3, 5, 3, 2, 11, 3, 17, 2, 3

Offset: 1

Reinhard Zumkeller, Jul 15 2004

a(n)*A070647(n) = A006881(n); a(n) < A070647(n);

a(n) = A020639(A006881(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A084126, A195758.

a096916 = a020639 . a006881 -- Reinhard Zumkeller, Sep 23 2011

f[n_]:=Last/@FactorInteger[n]=={1,1};f1[n_]:=Min[First/@FactorInteger[n]];f2[n_]:=Max[First/@FactorInteger[n]];lst={};Do[If[f[n],AppendTo[lst,f1[n]]],{n,0,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 10 2010 *)

go(x)=my(v=List()); forprime(p=2, sqrtint(x\1), forprime(q=p+1, x\p, listput(v, [p*q,p]))); apply(v->v[2], vecsort(Vec(v),1)) \\ Charles R Greathouse IV, Sep 14 2015

A176170 Smallest prime-factor of n-th product of 4 distinct primes.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 10 2010

nonn

FactorInteger[210]=2*3*5*7,...

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A020639, A046386, A006881, A007304, A070647, A096916, A096917, A096918, A096919

f0[n_]:=Last/@FactorInteger[n]=={1,1,1,1};f1[n_]:=Min[First/@FactorInteger[n]];f2[n_]:=First/@FactorInteger[n][[2,1]];f3[n_]:=First/@FactorInteger[n][[3,1]];f4[n_]:=Max[First/@FactorInteger[n]];lst={};Do[If[f0[n],AppendTo[lst,f1[n]]],{n,0,2*7!}];lst

n=0; i=0; while(i<10000,n++;if((4 == bigomega(n))&&(4 == omega(n)),i++;write("b176170.txt", i, " ", factor(n)[1,1]))); \\ Antti Karttunen, Dec 06 2017

a(n) = A020639(A046386(n)). - Antti Karttunen, Dec 06 2017

A176881 a(n)=p-q for n-th product of 2 distinct primes p and q (q

Original entry on oeis.org

1, 3, 5, 2, 4, 9, 11, 8, 15, 2, 17, 10, 21, 14, 6, 16, 27, 29, 8, 20, 35, 4, 39, 12, 41, 26, 6, 28, 45, 14, 51, 34, 18, 57, 10, 59, 38, 40, 12, 65, 44, 69, 2, 24, 71, 26, 77, 50, 16, 81, 56, 87, 58, 32, 6, 95, 64, 99, 22, 36, 101, 8, 68, 105, 38, 24, 107, 70, 4, 111, 42, 76, 6, 80

Offset: 1

Juri-Stepan Gerasimov, Apr 27 2010

nonn

Where products of two distinct primes are in A006881.

If Polignac's conjecture is true, then every even positive integer occurs infinitely many times in this sequence. - Clark Kimberling, Apr 25 2016

			a(1)=1 because 1=3-2 for A006881(1)=6=3*2; a(2)=3 because 3=5-2 for A006881(2)=10=5*2.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A006881, A096916, A070647.

A006881 := proc(n) if n = 1 then 6; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 2 and nops(numtheory[factorset](a)) =2 then return a; end if; end do: end if; end proc:
A020639 := proc(n) numtheory[factorset](n) ; min(op(%)) ; end proc:
A006530 := proc(n) numtheory[factorset](n) ; max(op(%)) ; end proc:
for n from 1 to 130 do c := A006881(n) ; printf("%d,",A006530(c)-A020639(c)) ; end do:
# R. J. Mathar, May 01 2010

mx = 350; t = Sort@Flatten@Table[Prime[n]*Prime[m], {n, Log[2, mx/3]}, {m, n + 1, PrimePi[mx/Prime[n]]}]; (* A006881, Robert G.Wilson v, Feb 07 2012 *)
u = Table[FactorInteger[t[[k]]][[1]], {k, 1, Length[t]}];
u1 = Table[u[[k]][[1]], {k, 1, Length[t]}]  (* A096916 *)
PrimePi[u1]  (* A270650 *)
v = Table[FactorInteger[t[[k]]][[2]], {k, 1, Length[t]}];
v1 = Table[v[[k]][[1]], {k, 1, Length[t]}]  (* A070647 *)
PrimePi[v1]  (* A270652 *)
d = v1 - u1  (* A176881 *)  (* Clark Kimberling, Apr 25 2016 *)

Entries checked by R. J. Mathar, May 01 2010

A195759 Greater prime factor of n-th Blum number.

Original entry on oeis.org

7, 11, 19, 23, 11, 31, 43, 19, 47, 23, 59, 67, 19, 71, 31, 79, 83, 23, 43, 103, 107, 47, 31, 127, 131, 59, 139, 23, 151, 67, 43, 163, 71, 167, 47, 179, 79, 191, 83, 31, 199, 211, 59, 223, 227, 31, 239, 103, 67, 107, 251, 71, 263, 271, 43, 283, 79, 127, 47

Offset: 1

Reinhard Zumkeller, Sep 23 2011

nonn

A195758(n) * a(n) = A016105(n); a(n) > A195758(n);

a(n) = A006530(A016105(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Wikipedia, Blum integer

Cf. A070647.

Haskell
```
a195759 = a006530 . a016105
```

A176171 Second prime-factor of n-th product of 4 distinct primes.

Original entry on oeis.org

3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 5, 3, 3, 3, 3, 5, 5, 3, 3, 3, 3, 3, 3, 5, 5, 3, 5, 3, 3, 3, 3, 5, 3, 3, 5, 3, 3, 3, 5, 3, 3, 3, 5, 7, 3, 5, 3, 5, 3, 5, 5, 3, 5, 3, 3, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 5, 7, 3, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 7, 5, 3, 5, 3, 3, 7, 3, 5, 3, 5, 3, 3, 5, 5, 3, 5, 3, 5, 3, 3, 3

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 10 2010

nonn

FactorInteger[210]=2*3*5*7,...

Cf. A006881, A007304, A070647, A096916, A096917, A096918, A096919, A176170

f0[n_]:=Last/@FactorInteger[n]=={1,1,1,1};f1[n_]:=Min[First/@FactorInteger[n]];f2[n_]:=First/@FactorInteger[n][[2,1]];f3[n_]:=First/@FactorInteger[n][[3,1]];f4[n_]:=Max[First/@FactorInteger[n]];lst={};Do[If[f0[n],AppendTo[lst,f2[n]]],{n,0,2*7!}];lst

A176172 3rd prime-factor of n-th product of 4 distinct primes.

Original entry on oeis.org

5, 5, 5, 7, 5, 7, 5, 5, 7, 7, 7, 11, 5, 7, 5, 7, 5, 11, 7, 7, 7, 5, 11, 5, 7, 13, 7, 7, 5, 11, 13, 11, 7, 5, 7, 7, 5, 7, 13, 7, 5, 11, 11, 17, 7, 7, 11, 5, 7, 11, 11, 5, 11, 7, 5, 13, 7, 13, 17, 5, 7, 13, 11, 13, 7, 5, 11, 7, 7, 11, 19, 5, 11, 11, 7, 11, 7, 13, 5, 11, 17, 7, 13, 11, 7, 5, 7, 7, 5

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 10 2010

nonn

FactorInteger[210]=2*3*5*7,...

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

Cf. A006881, A007304, A070647, A096916, A096917, A096918, A096919, A176170, A176171

N:= 10000: # to use products <= N
Primes:= select(isprime, [2,seq(i,i=3..N/30)]):
P4:= NULL:
for ia from 1 to nops(Primes) do
  a:= Primes[ia];
  for ib from 1 to ia-1 do
    b:= Primes[ib];
    if 6*a*b > N then break fi;
    for ic from 1 to ib-1 do
      c:= Primes[ic];
      if 2*a*b*c > N then break fi;
      for id from 1 to ic-1 do
        d:= Primes[id];
        if a*b*c*d > N then break fi;
        R[a*b*c*d]:= b;
        P4:= P4, a*b*c*d;
od od od od:
P4:= sort([P4]):
map(t -> R[t], P4); # Robert Israel, May 14 2019

f0[n_]:=Last/@FactorInteger[n]=={1,1,1,1};f1[n_]:=Min[First/@FactorInteger[n]];f2[n_]:=First/@FactorInteger[n][[2,1]];f3[n_]:=First/@FactorInteger[n][[3,1]];f4[n_]:=Max[First/@FactorInteger[n]];lst={};Do[If[f0[n],AppendTo[lst,f3[n]]],{n,0,2*7!}];lst

cp's OEIS Frontend

A006881 Squarefree semiprimes: Numbers that are the product of two distinct primes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

Python

Sage

Formula

Extensions

A270650 Min(i, j), where p(i)*p(j) is the n-th term of A006881.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A270652 Max(i,j), where p(i)*p(j) is the n-th term of A006881.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

A096916 Lesser prime factor of n-th product of two distinct primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

A176170 Smallest prime-factor of n-th product of 4 distinct primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A176881 a(n)=p-q for n-th product of 2 distinct primes p and q (q

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A195759 Greater prime factor of n-th Blum number.

Views

Author

Keywords

Comments