A067813 -id:A067813 - cp's OEIS frontend

A067820 The start of a record-breaking run of consecutive integers with a number of prime factors (counted with multiplicity) equal to 5.

32, 944, 15470, 57967, 632148, 14845324, 69921004, 888781058, 2674685524, 10077383364, 21117216104, 393370860205, 3157222675953, 5509463413255, 24819420480104, 361385490681003, 441826936079342

Offset: 1

Shyam Sunder Gupta, Feb 07 2002

a(16) > 3*10^13. - Brian Trial, May 13 2017
All multiples of 32 greater than 32 are of form 2^5*m and have at least 6 factors. Thus this sequence will be limited to a run of at most 31 integers. - Brian Trial, May 13 2017
a(18) > 2 * 10^15. - Toshitaka Suzuki, Aug 31 2025

			a(3)=15470 because 15470 is the start of a record breaking run of 3 consecutive integers (15470 to 15472) each having 5 prime factors; i.e. bigomega(n)=A001222(n)=5 for n = 15470, ..., 15472.

Cf. A067813, A067814, A067821, A067822.

Subsequence of A014614.

bigomega[n_] := Plus@@Last/@FactorInteger[n]; For[n=1; m=l=0, True, n++, If[bigomega[n]==5, l++, If[l>m, m=l; Print[n-l, " ", l]]; l=0]]
Table[SequencePosition[PrimeOmega[Range[15*10^6]],PadRight[{},n,5],1][[All,1]],{n,6}]//Flatten (* The program generates the first six terms of the sequence. *) (* Harvey P. Dale, Sep 03 2022 *)

Edited by Dean Hickerson, Jul 31 2002
More terms from Jens Kruse Andersen, Aug 23 2003
a(13)-a(14) from Donovan Johnson, Jan 31 2009
a(15) from Brian Trial, May 13 2017
a(16)-a(17) from Toshitaka Suzuki, Aug 31 2025

A113789 Numbers n such that n, n+1 and n+2 are products of exactly 3 primes.

Original entry on oeis.org

170, 244, 284, 428, 434, 506, 602, 603, 604, 637, 962, 1074, 1083, 1084, 1130, 1244, 1309, 1412, 1434, 1490, 1532, 1556, 1586, 1604, 1634, 1675, 1771, 1885, 1946, 2012, 2013, 2035, 2084, 2091, 2092, 2162, 2396, 2404, 2522, 2523, 2524, 2525, 2634, 2635

Offset: 1

Jonathan Vos Post, Jan 21 2006

3-almost prime analog of A056809.
This sequence consists of the least of 3 consecutive 3-almost primes, or 4 or more consecutive 3-almost primes (i.e. n, n+1 and n+2 but not excluding n+3 also 3-almost prime). 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).
Primes counted with multiplicity. - Harvey P. Dale, Sep 04 2019

			a(1) = 170 because 170 = 2 * 5 * 17 and 171 = 3^2 * 19 and 172 = 2^2 * 43 are all 3-almost primes.
a(2) = 244 because 244 = 2^2 * 61 and 245 = 5 * 7^2 and 246 = 2 * 3 * 41 are all 3-almost primes.
a(3) = 284 because 284 = 2^2 * 71 and 285 = 3 * 5 * 19 and 286 = 2 * 11 * 13 are all 3-almost primes.
a(4) = 428 because 428 = 2^2 * 107 and 429 = 3 * 11 * 13 and 430 = 2 * 5 * 43 are all 3-almost primes.
a(5) = 434 because 434 = 2 * 7 * 31 and 435 = 3 * 5 * 29 and 436 = 2^2 * 109 are all 3-almost primes.
a(6) = 506 because 506 = 2 * 11 * 23 and 507 = 3 * 13^2 and 508 = 2^2 * 127 all 3-almost primes.
a(7), a(8), a(9) = 602, 603, 604 because of the record-setting 5 consecutive 3-almost primes: 602 = 2 * 7 * 43; 603 = 3^2 * 67; 604 = 2^2 * 151; 605 = 5 * 11^2; 606 = 2 * 3 * 101.

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

Cf. A014612, A056809, A067813.

Subsequence of A180117.

fQ[n_] := Plus @@ Last /@ FactorInteger@n == 3; Select[ Range@2664, fQ@# && fQ[ # + 1] && fQ[ # + 2] &] (* Robert G. Wilson v, Jan 21 2006 *)
SequencePosition[Table[If[PrimeOmega[n]==3,1,0],{n,3000}],{1,1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 04 2019 *)

is(n)=bigomega(n)==3 && bigomega(n+1)==3 && bigomega(n+2)==3 \\ Charles R Greathouse IV, Feb 05 2017

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

Edited, corrected and extended by Robert G. Wilson v, Jan 21 2006

A067821 The start of a record-breaking run of consecutive integers with a number of prime factors (counted with multiplicity) equal to 6.

Original entry on oeis.org

64, 5264, 33614, 8706123, 101905622, 4843161124, 25189114374, 412352139170, 1122875553872, 28099912628847, 78661670985666

Offset: 1

Shyam Sunder Gupta, Feb 07 2002

a(11) > 3*10^13. - Brian Trial, May 26 2017

a(12) > 2 * 10^15. - Toshitaka Suzuki, Aug 31 2025

			a(3)=33614 because 33614 is the start of a record breaking run of 3 consecutive integers (33614 to 33616) each having 6 prime factors; i.e., bigomega(n)=A001222(n)=6 for n = 33614, ..., 33616.

Cf. A001222, A067813, A067814, A067820, A067822.

bigomega[n_] := Plus@@Last/@FactorInteger[n]; For[n=1; m=l=0, True, n++, If[bigomega[n]==6, l++, If[l>m, m=l; Print[n-l, " ", l]]; l=0]]

Edited by Dean Hickerson, Jul 31 2002
More terms from Jens Kruse Andersen, Aug 23 2003
a(7)-a(9) from Donovan Johnson, Jan 31 2009
a(10) from Brian Trial, May 26 2017
a(11) from Toshitaka Suzuki, Aug 31 2025

A067814 The start of a record-breaking run of consecutive integers with a number of prime factors (counted with multiplicity) equal to 4.

Original entry on oeis.org

16, 135, 1274, 4023, 12122, 204323, 355923, 3405122, 49799889, 202536181, 3195380868, 5208143601, 85843948321, 97524222465

Offset: 1

Shyam Sunder Gupta, Feb 07 2002

3405122 is the first number having 8 and 9 consecutive integers with 4 prime factors. - T. D. Noe, Mar 19 2014

			a(4)=4023 because 4023 is the start of a record breaking run of 4 consecutive integers (4023 to 4026) each having 4 prime factors; i.e. bigomega(n)=A001222(n)=4 for n = 4023, ..., 4026.

Cf. A067813, A067820, A067821, A067822.

bigomega[n_] := Plus@@Last/@FactorInteger[n]; For[n=1; m=l=0, True, n++, If[bigomega[n]==4, l++, If[l>m, m=l; Print[n-l, " ", l]]; l=0]]

Edited by Dean Hickerson, Jul 31 2002

More terms from Don Reble, Aug 11 2002, who remarks that the sequence is now complete.

A067822 The start of a record-breaking run of consecutive integers with a number of prime factors equal to 7.

Original entry on oeis.org

128, 29888, 3145310, 296299374, 15605704374, 242576758750, 1981162639374, 126460514648223

Offset: 1

Shyam Sunder Gupta, Feb 07 2002

			a(3) = 3145310 because 3145310 is the start of a record breaking run of 3 consecutive integers (3145310 to 3145312) each having 7 prime factors; i.e., bigomega(n) = A001222(n) = 7 for n = 3145310, ..., 3145312.

Cf. A001222, A067813, A067814, A067820, A067821.

bigomega[n_] := Plus@@Last/@FactorInteger[n]; For[n=1; m=l=0, True, n++, If[bigomega[n]==7, l++, If[l>m, m=l; Print[n-l, " ", l]]; l=0]]

Edited by Dean Hickerson, Jul 31 2002
More terms from Jens Kruse Andersen, Aug 23 2003
a(7) from Donovan Johnson, Jan 31 2009
a(8) from Brian Trial, Jun 28 2017

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

Original entry on oeis.org

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

A259756 Numbers n such that numbers n through n+6 are the product of exactly three (not necessarily distinct) primes.

Original entry on oeis.org

211673, 298433, 381353, 460801, 506521, 568729, 690593, 705953, 737633, 741305, 921529, 1056529, 1088521, 1105553, 1141985, 1362313, 2016721, 2270633, 2369809, 2535721, 2590985, 2688833, 2956681, 2983025, 3085201, 3112193, 3147553, 3269161

Offset: 1

Zak Seidov, Nov 08 2015

nonn

All terms are == 1 (mod 8). There are no sets of 8 consecutive integers all 3-almost primes.

Note that a(1) = A067813(6). - Michel Marcus, Nov 24 2015

			211673=7*11*2749, 211674=2*3*35279, 211675=5*5*8467, 211676=2*2*52919,
211677=3*37*1907, 211678=2*109*971, 211679=13*19*857.

Zak Seidov and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 4000 from Seidov)

Subsequence of A259504 and A014612. Cf. A067813.

W:= numtheory:-bigomega:
select(t -> isprime((t+3)/4) and W(t) = 3 and W(t+1) = 3 and W(t+2) = 3
and W(t+4) = 3 and W(t+5) = 3 and W(t+6) = 3, [seq(i, i=1..10^7, 8)]); # Robert Israel, Nov 24 2015

SequencePosition[PrimeOmega[Range[327*10^4]],{3,3,3,3,3,3,3}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 12 2019 *)

forcomposite(n=1, 4*10^6, if(bigomega(n)==3 && bigomega(n+1)==3 && bigomega(n+2)==3 && bigomega(n+3)==3 && bigomega(n+4)==3 && bigomega(n+5)==3 && bigomega(n+6)==3, print1(n, ", "))) \\ Altug Alkan, Nov 08 2015

{(bo(n)=bigomega(n));n=211673-8;for(i=1,20000,n=n+8;while((a=bo(n))<>3||!isprime((n+3)/4),n=n+8);if(a==bo(n+1)&&
a==bo(n+2)&&a==bo(n+4)&&a==bo(n+5)&&a==bo(n+6),print1(n",")))}\\ Zak Seidov, Jul 27 2016

list(lim)=my(v=List(), ct=6, is); forfactored(n=211679, lim\1+6, is=vecsum(n[2][, 2])==3; if(is, if(ct++==7, listput(v, n[1]-6)), ct=0)); Vec(v) \\ Charles R Greathouse IV, Jun 26 2019

A117969 Start of least run of maximal length of consecutive n-almost primes.

Original entry on oeis.org

2, 33, 211673, 97524222465

Offset: 1

Rick L. Shepherd, Apr 05 2006

For n>=2 there cannot be more than 2^n - 1 consecutive n-almost primes. Is it known whether there always exists such a run of length 2^n - 1? If not, I conjecture so. This is confirmed to be true for terms through a(4). Terms here equal the last terms of corresponding finite sequences: a(3) = A067813(6). a(4) was computed by Don Reble as A067814(14). a(5) >= A067820(12).

a(4) is smaller than the number 488995430567765317569 found by Forbes. [From T. D. Noe, Oct 29 2008]

			a(2) = 33 because 33, 34, 35 is the least run of three consecutive 2-almost primes (semiprimes).

Tony Forbes, Fifteen consecutive integers with exactly four prime factors, Math. Comp. 71 (2002), 449-452. [From _T. D. Noe_, Oct 29 2008]

Cf. A067813, A067814, A067820, A067821, A067822.

A259504 Numbers n such that n and n+1 are the product of exactly three (not necessarily distinct) primes.

Original entry on oeis.org

27, 44, 75, 98, 116, 124, 147, 153, 164, 170, 171, 174, 230, 244, 245, 284, 285, 332, 356, 369, 387, 425, 428, 429, 434, 435, 474, 506, 507, 530, 548, 555, 574, 595, 602, 603, 604, 605, 609, 627, 637, 638, 645, 651, 657, 710

Offset: 1

Zak Seidov, Nov 08 2015

nonn

Conjecture: this sequence is infinite.
Number of terms < 10^k: 0, 4, 63, 727, 7014, 64556, 585725, 5284711, ... . - Robert G. Wilson v, Nov 09 2015
a(n) = p^3 where p is prime iff p is in intersection of A065508 and A005383. - Altug Alkan, Nov 24 2015
There are 47753279 terms less than 10^9 and 432841730 terms less than 10^10. - Charles R Greathouse IV, Jun 27 2019

			27=3*3*3, 28=2*2*7.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A014612 and A045920.

Cf. A067813.

Select[Range[1000], 3 == PrimeOmega[#] == PrimeOmega[# + 1] &]

forcomposite(n=1, 1e3, if(bigomega(n)==3 && bigomega(n+1)==3, print1(n, ", "))); \\ Altug Alkan, Nov 08 2015

list(lim)=my(v=List(),was=1,is); forfactored(n=28,lim\1+1, is=vecsum(n[2][,2])==3; if(is && was, listput(v,n[1]-1)); was=is); Vec(v) \\ Charles R Greathouse IV, Jun 26 2019

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).

cp's OEIS Frontend

A067820 The start of a record-breaking run of consecutive integers with a number of prime factors (counted with multiplicity) equal to 5.

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A113789 Numbers n such that n, n+1 and n+2 are products of exactly 3 primes.

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A067821 The start of a record-breaking run of consecutive integers with a number of prime factors (counted with multiplicity) equal to 6.

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A067814 The start of a record-breaking run of consecutive integers with a number of prime factors (counted with multiplicity) equal to 4.

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A067822 The start of a record-breaking run of consecutive integers with a number of prime factors equal to 7.

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

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A259756 Numbers n such that numbers n through n+6 are the product of exactly three (not necessarily distinct) primes.

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