A180117 -id:A180117 - cp's OEIS frontend

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

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

A124941 Numbers k such that k and k+4 are 4-almost primes.

Original entry on oeis.org

36, 56, 84, 100, 132, 136, 152, 228, 340, 344, 372, 376, 472, 484, 488, 532, 546, 564, 568, 580, 621, 632, 686, 708, 770, 804, 808, 820, 846, 852, 856, 868, 872, 950, 1012, 1192, 1204, 1206, 1208, 1274, 1304, 1326, 1336, 1444, 1524, 1550, 1572, 1576, 1690

Offset: 1

Zak Seidov, Nov 13 2006

nonn

			36=2^2*3^2, 40=2^3*5; 56=2^3*7, 60=2^2*3*5.

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

Cf. A014613, A124940, A124942, A157483, A180117.

Select[Range[1690], PrimeOmega[#]==PrimeOmega[#+4]==4 &] (* James C. McMahon, Dec 07 2024 *)

isok(n) = (bigomega(n) == 4) && (bigomega(n+4) == 4); \\ Michel Marcus, Oct 11 2013

A180150 Numbers n such that n and n+2 are both divisible by exactly 4 primes (counted with multiplicity).

Original entry on oeis.org

54, 88, 150, 196, 232, 248, 294, 306, 328, 340, 342, 348, 460, 488, 490, 568, 570, 664, 712, 738, 774, 850, 856, 858, 868, 870, 948, 1012, 1014, 1060, 1096, 1110, 1148, 1190, 1204, 1206, 1208, 1210, 1218, 1254, 1274, 1276, 1290, 1302, 1314, 1420, 1430, 1448

Offset: 1

Jonathan Vos Post, Aug 12 2010

"Quadruprimes" or "4-almost primes" that keep that property when incremented by 2. This sequence is to 4 as 3 is to A180117, as A092207 is to 2, and as A001359 is to 1. That is, this sequence is the 4th row of the infinite array A[k,n] = n-th natural number m such that m and m+2 are both divisible by exactly k primes (counted with multiplicity). The first row is the lesser of twin primes. The second row is the sequence such that m and m+2 are both semiprimes.

			a(1) = 54 because 54 = 2 * 3^3 is divisible by exactly 4 primes (counted with multiplicity), and so is 54+2 = 56 = 2^3 * 7.

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

Cf. A000040, A001222, A001358, A014614, A033987, A101637, A114106 (number of 4-almost primes <= 10^n).

SequencePosition[PrimeOmega[Range[1500]],{4,,4}][[;;,1]] (* _Harvey P. Dale, Jan 14 2024 *)

is(n)=bigomega(n)==4 && bigomega(n+2)==4 \\ Charles R Greathouse IV, Jan 31 2017

{m in A014613 and m+2 in A014613} = {m such that bigomega(m) = bigomega(m+2) = 4} = {m such that A001222(m) = A001222(m+2) = 4}.

More terms from R. J. Mathar, Aug 13 2010

A180151 Numbers k such that k and k + 2 are both divisible by exactly five primes (counted with multiplicity).

Original entry on oeis.org

270, 592, 700, 750, 918, 1168, 1240, 1638, 1648, 1672, 1710, 1750, 2070, 2310, 2392, 2548, 2550, 2608, 2728, 2860, 2862, 2896, 2898, 3184, 3330, 3568, 3630, 3822, 3848, 3850, 3942, 3976, 4230, 4264, 4648, 4662, 5070, 5080, 5236, 5238, 5390, 5550, 5560

Offset: 1

Jonathan Vos Post, Aug 12 2010

"5-almost primes" that keep that property when incremented by 2. This sequence is to 5 as 4 is to A180150, as 3 is to A180117, as A092207 is to 2, and as A001359 is to 1. That is, this sequence is the 5th row of the infinite array A[k,n] = n-th natural number m such that m and m+2 are both divisible by exactly k primes (counted with multiplicity). The first row is the lesser of twin primes. The second row is the sequence such that m and m+2 are both semiprimes.

			a(1) = 270 because 270 = 2 * 3^3 * 5 is divisible by exactly 5 primes (counted with multiplicity), and so is 270+2 = 272 = 2^4 * 17.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A001358, A001359, A014614, A033987, A101637, A180117, A180150.

f[n_] := Plus @@ (Last@# & /@ FactorInteger@n); fQ[n_] := f[n] == 5 == f[n + 2]; Select[ Range@ 10000, fQ] (* Robert G. Wilson v, Aug 15 2010 *)

for(x=2,10^4,if(bigomega(x)==5&&bigomega(x+2)==5,print1(x", "))) \\ Zak Seidov, Aug 12 2010

{m in A014614 and m+2 in A014614} = {m such that bigomega(m) = bigomega(m+2) = 5} = {m such that A001222(m) = A001222(m+2) = 5}.

Corrected and extended by Zak Seidov and R. J. Mathar, Aug 12 2010

A180245 n-th natural number m such that m and m+2 are both divisible by exactly n primes (counted with multiplicity).

Original entry on oeis.org

3, 33, 42, 196, 918, 6640, 24750, 246078, 781248, 6565374, 25227774, 165009150, 673932798, 5268548608, 25737162750, 179511912448, 818179991550, 4228689854464, 26455088693248, 104384041582590, 820632501420030

Offset: 1

Jonathan Vos Post, Aug 19 2010

nonn

Main diagonal A[n,n] of A[k,n] = n-th natural number m such that m and m+2 are both divisible by exactly k primes (counted with multiplicity).
This is the main diagonal of the array mentioned in A180117, A180150, and A180151.
Row 1 = A001359 = the lesser of twin primes.
Row 2 = A092207 = Numbers n such that n and n+2 are semiprimes.
Row 3 = A180117 = m and m+2 are both divisible by exactly 3 primes (counted with multiplicity).
Row 4 = A180150 = m and m+2 are both divisible by exactly 4 primes (counted with multiplicity).
Row 5 = A180151 = m and m+2 are both divisible by exactly 5 primes (counted with multiplicity).

			a(1) = 3 because 3 is the first natural number m such that m and m+2 are both divisible by exactly 1 prime (i.e., the first of the lesser of twin primes).
a(2) = 33 because that is the 2nd natural number m such that m and m+2 are both divisible by exactly 2 primes (i.e. 33 = 3 * 11 is semiprime and when 2 is added becomes 35 = 5 * 7 which is also semiprimes) the 1st such being 4.

Cf. A001359, A092207, A180117, A180150, A180151.

Corrected and extended by Jack Brennen, D. S. McNeil and Ray Chandler, Aug 19 2010

a(16)-a(21) from Donovan Johnson, Aug 27 2010

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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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A124941 Numbers k such that k and k+4 are 4-almost primes.

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A180150 Numbers n such that n and n+2 are both divisible by exactly 4 primes (counted with multiplicity).

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A180151 Numbers k such that k and k + 2 are both divisible by exactly five primes (counted with multiplicity).

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A180245 n-th natural number m such that m and m+2 are both divisible by exactly n primes (counted with multiplicity).

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