A092097 -id:A092097 - cp's OEIS frontend

A037088 Triangle read by rows: T(n,k) is number of numbers x, 2^n <= x < 2^(n+1), with k prime factors (counted with multiplicity).

2, 2, 2, 2, 4, 2, 5, 4, 5, 2, 7, 12, 6, 5, 2, 13, 20, 17, 7, 5, 2, 23, 40, 30, 20, 8, 5, 2, 43, 75, 65, 37, 21, 8, 5, 2, 75, 147, 131, 81, 41, 22, 8, 5, 2, 137, 285, 257, 173, 91, 44, 22, 8, 5, 2, 255, 535, 536, 344, 199, 96, 46, 22, 8, 5, 2, 464, 1062, 1033, 736, 403, 215, 99, 47

Offset: 1

Alford Arnold

Sequence A092097 gives the limiting behavior at the end of the rows. - T. D. Noe, Feb 22 2008

			The triangular array begins 2; 2,2; 2,4,2; 5,4,5,2; 7,12,6,5,2; ...
a(7) = 5 because the 3-almost primes between 16 and 32 are (18,20,27,28,30).

T. D. Noe, Rows n=1..24 of triangle, flattened

A001222 counts factors of n. A000040, A001358, A014612-A014614 are special cases. A036378 and A025488 are applications of binary order A029837. Leading diagonal is essentially A036378 and has partial sums A007053.

t[n_, k_] := Count[Range[2^n, 2^(n+1)-1], x_ /; Total[FactorInteger[x][[All, 2]]] == k]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 07 2013 *)

More terms from Naohiro Nomoto, Jun 18 2001

A096352 Triangle read by rows: each row represents all possible values for the size of the subset S{n - x} of {2^n...2^(n+1) - 1}, where S{n - x} represents all the members of that set with n - x factors.

Original entry on oeis.org

2, 4, 5, 2, 4, 6, 7, 8, 5, 12, 17, 20, 21, 22, 7, 20, 30, 37, 41, 44, 46, 47, 13, 40, 65, 81, 91, 96, 99, 101, 102, 103, 23, 75, 131, 173, 199, 215, 224, 229, 232, 233, 234, 43, 147, 257, 344, 403, 439, 461, 473, 482, 487, 490, 492, 493

Offset: 1

Andrew S. Plewe, Jun 29 2004

The number of members in the n-th row appears to be equal to 2 + ( (n) * ((1 + sqrt(5))/2) ), or the n-th member of the lower Wythoff sequence (A000201) plus two. For the four rows show above, these values are 3, 5, 6, 8.

The first member of each row n is the number of primes in the set {2^n...2^(n + 1) - 1} (sequence A036378). The last member of each row follows sequence A092097, which is also equivalent to taking the difference of successive members of A052130 (the number of products of half-odd primes less than 2^n).

			Let x = 1. In set {2^2..2^(3) - 1}, or {4, 5, 6, 7}, S{n - 1} = S{2 - 1} = S{1} = subset of all numbers with one factor (the primes). The size of this subset is 2, or {5, 7}. For the set {2^3...2^(4) - 1}, the size of subset S{3 - 1} is 4. For {2^4..2^(5) - 1}, the size of subset S{4 - 1} is 5. For all subsequent sets, the size of subset S{n - 1} will be 5.
The triangle begins:
  2,4,5
  2,4,6,7,8
  5,12,17,20,21,22
  7,20,30,37,41,44,46,47
  ...

Cf. A000201, A036378, A092097, A052130.

A125288 a(n) = least integer k such that for all integers m greater than k, 2Pi(n,m) is greater than Pi(n,2m).

Original entry on oeis.org

10, 297, 49650, 180701087317

Offset: 1

Jonathan Vos Post and Robert G. Wilson v, Jan 22 2007

Pi(n, m) is the number of integers <= m that have n prime factors counting multiplicity, also known as n-almost-primes (A078840).

			a(1) = 10 since the first term relates to 1-almost-primes, which are the primes themselves; and there are 4 primes <= 10, and 2*4 = 8 primes <= 2*10 = 20; but for m = 11 and all larger integers, the number of primes <= 2*m is less than twice the number of primes <= m. - _Peter Munn_, Dec 23 2022

Cf. A078840, A092097, A117526, A126279, A125149.

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 *)

a(4) from Donovan Johnson, Nov 13 2010

Edited by Peter Munn, Jan 05 2023

cp's OEIS Frontend

A037088 Triangle read by rows: T(n,k) is number of numbers x, 2^n <= x < 2^(n+1), with k prime factors (counted with multiplicity).

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A096352 Triangle read by rows: each row represents all possible values for the size of the subset S{n - x} of {2^n...2^(n+1) - 1}, where S{n - x} represents all the members of that set with n - x factors.

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A125288 a(n) = least integer k such that for all integers m greater than k, 2Pi(n,m) is greater than Pi(n,2m).

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cp's OEIS Frontend

A037088 Triangle read by rows: T(n,k) is number of numbers x, 2^n <= x < 2^(n+1), with k prime factors (counted with multiplicity).

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Mathematica

Extensions

A096352 Triangle read by rows: each row represents all possible values for the size of the subset S{n - x} of {2^n...2^(n+1) - 1}, where S{n - x} represents all the members of that set with n - x factors.

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A125288 a(n) = least integer k such that for all integers m greater than k, 2*Pi(n,m) is greater than Pi(n,2*m).

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A125288 a(n) = least integer k such that for all integers m greater than k, 2Pi(n,m) is greater than Pi(n,2m).