A114106 -id:A114106 - cp's OEIS frontend

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

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

A126280 Triangle read by rows: T(k,n) is number of numbers <= 10^n that are products of k primes.

Original entry on oeis.org

4, 4, 1, 25, 34, 22, 12, 4, 2, 168, 299, 247, 149, 76, 37, 14, 7, 2, 1229, 2625, 2569, 1712, 963, 485, 231, 105, 47, 22, 7, 3, 1, 9592, 23378, 25556, 18744, 11185, 5933, 2973, 1418, 671, 306, 138, 63, 25, 11, 4, 2, 78498, 210035, 250853, 198062, 124465, 68963

Offset: 1

Jonathan Vos Post & Robert G. Wilson v, Dec 22 2006

The n-th row's sum is 10^n - 1.

			4 4 1
25 34 22 12 4 2
168 299 247 149 76 37 14
7 2
1229 2625 2569 1712 963 485 231
105 47 22 7 3 1
9592 23378 25556 18744 11185 5933 2973
1418 671 306 138 63 25 11 4 2
78498 210035 250853 198062 124465 68963 35585 17572
8491 4016 1878 865 400 179 79 35 14 7 2
664579 1904324 2444359 2050696 1349779 774078 409849 207207
101787 49163 23448 11068 5210 2406 1124 510 233 102 45 21 7 3 1

Robert G. Wilson v, Table of n, a(n) for n = 1..342
Eric Weisstein's World of Mathematics, Semiprime.
Eric Weisstein's World of Mathematics, Almost Prime.

By columns: A006880, A036352, A109251, A114106, A114453, A120047, A120048, A120049, A120050, A120051, A120052, A120053.

The n-th row's sum: A002283 = 10^n -1, A116430, A126279: same array but for powers of two.

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[ PrimePi[ n/Times @@ Prime[ Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[ Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]] ]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[ AlmostPrimePi[m, 10^n], {n, 6}, {m, Floor[Log[2, 10^n]] }] // Flatten

A117526 Least number a(n) which is a product of n primes and such that Pi_n(a(n))/a(n) is maximum.

Original entry on oeis.org

3, 10, 9837, 259441550133

Offset: 1

Martin Raab and Robert G. Wilson v, Mar 25 2006

nonn

Pi_n(a(n))/a(n): 0.66667, 0.40000, 0.25801, 0.2145967653
3=3, 10=2*5, 9837=3*3*1093 & 259441550133=3*89*311*3124409.
3 is the second prime, 10 is the fourth semiprime, 9837 is the 3-almost prime, and 259441550133 is the 4-almost prime.

			a(1)=3 because Pi(2)/2=1/2 < Pi(3)/3=2/3 > Pi(5)/5=3/5.
a(2)=10 because Pi_2(9)/9=1/3 < Pi_2(10)/10=2/5 > Pi_2(14)/14=5/14; Pi_2(10)/10 = Pi_2(15)/15 but 10 < 15.

Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Prime Factor.

Cf. A006880, A066265, A109251, A114106, A114453.

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
fQ[n_] := Plus @@ Last /@ FactorInteger@n == 4; c = r = 0; Do[If[fQ@n, c++ ]; If[c/n > r, Print[n]; r = c/n], {n, 10^6}]

Comment edited and a(4) added by Donovan Johnson, Mar 10 2010

A120045 The (10^n)-th 4-almost prime.

Original entry on oeis.org

16, 88, 693, 5958, 54328, 511725, 4922511, 47997635, 472514554, 4683086217, 46636297326, 466032880556

Offset: 0

Robert G. Wilson v, Feb 15 2006

Cf. A006988, A014613, A114106, A114125, A120044, A120046.

FourAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j*Prime@k)] - k + 1, {i, PrimePi[n^(1/4)]}, {j, i, PrimePi[(n/Prime@i)^(1/3)]}, {k, j, PrimePi@Sqrt[n/(Prime@i*Prime@j)]}];
FourAlmostPrime[n_] := Block[{e = Floor[Log[2, n] +3], a, b}, a = 2^e; Do[b = 2^p; While[FourAlmostPrimePi[a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Do[ Print@FourAlmostPrime[10^n], {n, 0, 11}]

A124033 Number of n-digit numbers having exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

4, 31, 225, 1563, 10222, 63030, 374264, 2160300, 12196405, 67724342, 371233523, 2014305995, 10841722966, 57974736592, 308361428628, 1632877406997

Offset: 1

J. M. Bergot, Apr 08 2011

Essentially the same as A036335.

What would be the ratio between a(n) and all possible numbers with n digits for each n?

			a(1) = A006880(1) = 4.
a(2) = A066265(2) - A066265(1) = 34 - 3 = 31.
a(3) = A109251(3) - A109251(2) = 247 - 22 = 225.
a(4) = A114106(4) - A114106(3) = 1712 - 149 = 1563.
a(5) = A114453(5) - A114453(4) = 11185 - 963 = 10222.
a(6) = A120047(6) - A120047(5) = 68963 - 5933 = 63030.
a(7) = A120048(7) - A120048(6) = 409849 - 35585 = 374264.
a(8) = A120049(8) - A120049(7) = 2367507 - 207207 = 2160300.
a(9) = A120050(9) - A120050(8) = 13377156 - 1180751 = 12196405.
a(10) = A120051(10) - A120051(9) = 74342563 - 6618221 = 67724342.
a(11) = A120052(11) - A120052(10) = 407818620 - 36585097 = 371233523.
a(12) = A120053(12) - A120053(11) = 2214357712 - 200051717 = 2014305995.

Table[Count[Range[10^(n-1),10^n-1],?(PrimeOmega[#]==n&)],{n,8}]  (* _Harvey P. Dale, Apr 22 2011 *)
AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
f[n_] := AlmostPrimePi[n, 10^n - 1] - AlmostPrimePi[n, 10^(n - 1) - 1]; Array[f, 12] (* Robert G. Wilson v, Jul 06 2012 *)

Corrected and extended by Ray Chandler, Apr 11 2011
a(9)-a(12) from Ray Chandler, Apr 12 2011
a(13)-a(16) from Robert G. Wilson v, Jul 06 2012

cp's OEIS Frontend

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

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A126280 Triangle read by rows: T(k,n) is number of numbers <= 10^n that are products of k primes.

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A117526 Least number a(n) which is a product of n primes and such that Pi_n(a(n))/a(n) is maximum.

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A120045 The (10^n)-th 4-almost prime.

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A124033 Number of n-digit numbers having exactly n prime factors (counted with multiplicity).

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