A113789 -id:A113789 - cp's OEIS frontend

A124057 Numbers n such that n, n+1, n+2 and n+3 are products of exactly 3 primes.

602, 603, 1083, 2012, 2091, 2522, 2523, 2524, 2634, 2763, 3243, 3355, 4202, 4203, 4921, 4922, 4923, 5034, 5035, 5132, 5203, 5282, 5283, 5785, 5882, 5954, 5972, 6092, 6212, 6476, 6962, 6985, 7730, 7731, 7945, 8393, 8825, 8956, 8972, 9188, 9482, 10011

Offset: 1

Jonathan Vos Post, Nov 03 2006

n such that n, n+1, n+2 and n+3 are 3-almost primes. Subset of A113789 Numbers n such that n, n+1 and n+2 are products of exactly 3 primes. A067813 has some runs of up to 7 consecutive 3-almost primes (i.e. starting 211673). But there cannot be 8 consecutive 3-almost primes, as every run of 8 consecutive positive integers contains exactly one multiple of 8 = 2^3 and only 8 of all positive multiples of 8 is a 3-almost prime (i.e. all larger multiples have at least 4 prime factors, with multiplicity).

A subset of A045940. - Zak Seidov, Nov 05 2006

			a(1) = 602 because 602 = 2 * 7 * 43 and 603 = 3^2 * 67 and 604 = 2^2 * 151 and 605 = 5 * 11^2 are all 3-almost primes.
a(2) = 603 because 603 = 3^2 * 67 and 604 = 2^2 * 151 and 605 = 5 * 11^2 and 606 = 2 * 3 * 101 are all 3-almost primes.
a(3) = 1083 because 1083 = 3 * 19^2 and 1084 = 2^2 * 271 and 1085 = 5 * 7 * 31 and 1086 = 2 * 3 * 181 are all 3-almost primes.
a(4) = 2012 because 2012 = 2^2 * 503, 2013 = 3 * 11 * 61, 2014 = 2 * 19 * 53, 2015 = 5 * 13 * 31.
a(5) = 2091 because 2091 = 3 * 17 * 41, 2092 = 2^2 * 523, 2093 = 7 * 13 * 23, 2094 = 2 * 3 * 349.

D. W. Wilson, Table of n, a(n) for n = 1..10000

Cf. A000040, A014612, A056809, A067813, A113789, A045940.

with(numtheory): a:=proc(n) if bigomega(n)=3 and bigomega(n+1)=3 and bigomega(n+2)=3 and bigomega(n+3)=3 then n else fi end: seq(a(n),n=1..15000); # Emeric Deutsch, Nov 07 2006

okQ[{a_,b_,c_,d_}]:=Union[{a,b,c,d}]=={3}; Flatten[Position[Partition[ PrimeOmega[ Range[11000]],4,1],?(okQ)]] (* _Harvey P. Dale, Sep 23 2012 *)

is(n)=if(!isprime((n+3)\4), return(0)); for(k=n,n+3, if(bigomega(k)!=3, return(0))); 1 \\ Charles R Greathouse IV, Feb 05 2017

list(lim)=my(v=List(),u=v,t); forprime(p=2,lim\4, forprime(q=2,min(lim\(2*p),p), t=p*q; forprime(r=2,min(lim\t,q), listput(u,t*r)))); u=Set(u); for(i=4,#u, if(u[i]-u[i-3]==3, listput(v,u[i-3]))); Vec(v) \\ Charles R Greathouse IV, Feb 05 2017

n, n+1, n+2 and n+3 are all elements of A014612. n and n+1 are elements of A113789.

More terms from Zak Seidov, Nov 05 2006

More terms from Emeric Deutsch, Nov 07 2006

A115402 Difference between 3-almostprime(n) and 3-almostprime(n+3).

Original entry on oeis.org

12, 15, 10, 10, 15, 16, 15, 8, 8, 18, 16, 16, 7, 9, 8, 8, 17, 22, 21, 10, 7, 11, 12, 11, 7, 10, 9, 13, 14, 22, 18, 15, 7, 16, 12, 16, 7, 7, 4, 4, 10, 12, 13, 8, 9, 19, 22, 27, 23, 19, 14, 8, 11, 8

Offset: 1

Jonathan Vos Post, Jan 22 2006

			a(1) = A014612(1+3) - A014612(1) = 20 - 8 = 12.
a(2) = A014612(2+3) - A014612(2) = 27 - 12 = 15.
a(3) = A014612(3+3) - A014612(3) = 28 - 18 = 10.
a(39) = A014612(39+3) - A014612(39) = 174 - 170 = 4.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A014612, A056809, A067813, A113789, A115403.

Last[#]-First[#]&/@Partition[Select[Range[300],PrimeOmega[#]==3&],4,1] (* Harvey P. Dale, Nov 09 2012 *)

a(n) = A014612(n+3) - A014612(n).

A268588 Numbers n such that n, n + 1, n + 2, n + 3 and n + 4 are products of exactly three primes.

Original entry on oeis.org

602, 2522, 2523, 4202, 4921, 4922, 5034, 5282, 7730, 18241, 18242, 18571, 19129, 21931, 23161, 23305, 25203, 25553, 25554, 27290, 27291, 29233, 30354, 30793, 32035, 33843, 34561, 35714, 36001, 36835, 40313, 40314, 40394, 45265, 55361, 67609, 69667, 70202, 72721

Offset: 1

K. D. Bajpai, Feb 07 2016

nonn

Subsequence of A045941. - Zak Seidov, Jan 29 2017

			a(1) = 602: 602 = 2 * 7 * 43; 603 = 3 * 3 * 67; 604 = 2 * 2 * 151; 605 = 5 * 11 * 11; 606 = 2 * 3 * 101 are all products of three primes.
a(4) = 4202 : 4202 = 2 * 11 * 191; 4203 = 3 * 3 * 467; 4204 = 2 * 2 * 1051; 4205 = 5 * 29 * 29; 4206 = 2 * 3 * 701 are all products of three primes.

Chai Wah Wu, Table of n, a(n) for n = 1..11023

Cf. A056809, A113789, A124057, A045941.

IsP3:=func< n | &+[k[2]: k in Factorization(n)] eq 3 >; [ n: n in [2..50000] | IsP3(n) and IsP3(n+1) and IsP3(n+2) and IsP3(n+3) and IsP3(n+4)];

with(numtheory): A268588:= proc() if bigomega(n)=3 and bigomega(n+1)=3 and bigomega(n+2)=3 and bigomega(n+3)=3 and bigomega(n+4)=3 then RETURN (n); fi; end: seq(A268588(), n=1..100000);

Select[Range[100000], PrimeOmega[#] == 3 && PrimeOmega[# + 1] == 3 && PrimeOmega[# + 2] == 3 && PrimeOmega[# + 3] == 3 && PrimeOmega[# + 4] == 3 &]
SequencePosition[PrimeOmega[Range[73000]],{3,3,3,3,3}][[All,1]] (* Harvey P. Dale, Sep 03 2021 *)

for(n = 1,50000, bigomega(n)==3 & bigomega(n+1)==3 & bigomega(n+2)==3 & bigomega(n+3)==3 & bigomega(n+4)==3 & print1(n,","))

Comment removed by Zak Seidov, Jan 29 2017

A375239 Numbers k such that k, k+1, ..., k+5 all have 3 prime factors (counted with multiplicity).

Original entry on oeis.org

2522, 4921, 18241, 25553, 27290, 40313, 90834, 95513, 98282, 98705, 117002, 120962, 136073, 136865, 148682, 153794, 181441, 181554, 185825, 211673, 211674, 212401, 215034, 216361, 231002, 231665, 234641, 236041, 236634, 266282, 281402, 285410, 298433, 298434, 330473, 331985, 346505, 381353

Offset: 1

Zak Seidov and Robert Israel, Aug 06 2024

nonn

First differs from A045942 at position 20, where a(20) = 211673 but A045942(20) = 204323.
All terms == 1 or 2 (mod 8).
One of the numbers k, k+1, ..., k+5 is a Zumkeller number (A083207), since it is of the form 2*3*p, where p is prime > 3. - Ivan N. Ianakiev, Aug 08 2024

			a(3) = 18241 is a term because
  18241 = 17 * 29 * 37
  18242 =  2 * 7 * 1303
  18243 =  3^2 * 2027
  18244 =  2^2 * 4561
  18245 =  5 * 41 * 89
  18246 =  2 * 3 * 3041
are all products of 3 primes (counted with multiplicity).

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

Subsequence of A045942 and of A113789. Contains A259756.

R:= NULL: count:= 0: p:= 1:
while count < 100 do
  p:= nextprime(p);
  x:= 4*p;
  if andmap(t -> numtheory:-bigomega(t)=3, [x-2,x-1,x+1,x+2]) then
    if numtheory:-bigomega(x-3) = 3 then R:= R, x-3; count:= count+1;  fi;
    if numtheory:-bigomega(x+3) = 3 then R:= R, x-2; count:= count+1;  fi;
  fi;
od:
R;

s = {}; Do[If[{3, 3, 3, 3, 3, 3} == PrimeOmega[Range[k, k + 5]],
AppendTo[s, k]], {k, 1000000}]; s

A110934 Difference between 3-almostprime(n) and 3-almostprime(n+2).

Original entry on oeis.org

10, 8, 9, 8, 3, 14, 14, 3, 6, 7, 13, 14, 5, 4, 7, 6, 3, 16, 20, 7, 4, 6, 8, 9, 6, 3, 8, 8, 6, 13, 17, 10, 6, 6, 11, 11, 6, 6, 2, 3, 3, 8, 11, 6, 4, 7, 17, 17, 15, 18, 9, 6, 7, 6, 6, 3, 2, 10, 12, 6, 8, 7, 7, 7, 6, 7, 5, 3, 2, 5, 6, 20, 24, 8, 6, 7, 10, 8, 6, 10, 7

Offset: 1

Jonathan Vos Post, Jan 21 2006

This is the 3-almost prime analog of what A113784 is for semiprimes and what A031131 is for primes. The minimum values in the sequence are 2 because we have, for example, the 3 consecutive 3-almost primes 170, 171, 172, so a(39) = A014612(41) - A014612(39) = 172 - 170 = 2. Equivalently, there are 2 consecutive 1 values of A114403 (3-almost prime gaps; first differences of A014612). This happens for elements of A113789 (numbers n such that n, n+1 and n+2 are 3-almost primes).

			a(1) = 10 because the difference between the first and third 3-almost primes is A014612(3) - A014612(1) = 18 - 8 = 10.
a(2) = A014612(4) - A014612(2) = 20 - 12 = 8.
a(3) = A014612(5) - A014612(3) = 27 - 18 = 9.

Cf. A014612, A031131, A067813, A113784, A113789, A114403.

a(n) = A014612(n+2) - A014612(n).

a(28) corrected by R. J. Mathar, Dec 22 2010

A334583 Numbers m such that m, m + 1 and m + 2 each have exactly eight prime factors, not necessarily distinct.

Original entry on oeis.org

40909374, 71410624, 87278750, 126237375, 152439488, 161590624, 166450624, 209140623, 227929624, 243409374, 267308990, 267639470, 290696768, 291513248, 292088510, 295644734, 307885374, 310314158, 319874750, 321890750, 331690624, 336958622, 343030624, 352749248, 354109374, 356269374, 366681248, 391390623, 401375168, 407590623

Offset: 1

Zak Seidov, May 06 2020

nonn

			40909374 = 2 * 3^4 * 11^2 * 2087, 40909375 = 5^5 * 13 * 19 * 53, and 40909376 = 2^6 * 179 * 3571.

Intersection of A045939 and A046310.

Cf. A001222, A113752, A113789.

list(lim)=my(v=List(), k, o); forfactored(n=40909374, lim\1+2, o=bigomega(n); if(o==8, if(k++>2, listput(v, n[1]-2)), k=0)); Vec(v) \\ Charles R Greathouse IV, May 07 2020

A001222(a(n)+i) = 8 for i in {0,1,2}.

cp's OEIS Frontend

A124057 Numbers n such that n, n+1, n+2 and n+3 are products of exactly 3 primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A115402 Difference between 3-almostprime(n) and 3-almostprime(n+3).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A268588 Numbers n such that n, n + 1, n + 2, n + 3 and n + 4 are products of exactly three primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Extensions

A375239 Numbers k such that k, k+1, ..., k+5 all have 3 prime factors (counted with multiplicity).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A110934 Difference between 3-almostprime(n) and 3-almostprime(n+2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A334583 Numbers m such that m, m + 1 and m + 2 each have exactly eight prime factors, not necessarily distinct.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula