A052130 -id:A052130 - cp's OEIS frontend

A126281 a(n) is the least m to satisfy the requirements of A052130.

1, 2, 5, 8, 10, 13, 16, 18, 21, 24, 27, 29, 32, 35, 37, 40, 43, 46, 48, 51, 54, 56, 59, 62, 65, 67, 70, 73

Offset: 1

Robert G. Wilson v, Dec 24 2006

A052130: For m very large, a(n) = number of numbers between 1 and 2^m with m-n prime factors (counted with multiplicity).
In observing the triangular array of A126279, the array T(k,n) defined as the k-th almost prime count of n-th power of two, it is noticed that the k-th term from the right converges to a fixed value beginning with the n-th power of two.
Will this sequence continue to match A117630: floor(n*log(3)/log(3/2)) ?

J. H. Smith, Perfect Numbers.

Cf. A052130, A126279.

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}]] ]]]; f[n_] := Block[{a = 0, m = n}, While[ b = AlmostPrimePi[m-n+1, 2^m]; b > a, m++; a = b]; m--; m]; Array[f, 24] (* Eric W. Weisstein, Feb 07 2006 *)

a(25)-a(28) from Robert G. Wilson v, Sep 07 2012

Expression in comment corrected by L. Edson Jeffery, Apr 03 2015

A126279 Triangle read by rows: T(k,n) is number of numbers <= 2^n that are products of k primes.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 6, 6, 2, 1, 11, 10, 7, 2, 1, 18, 22, 13, 7, 2, 1, 31, 42, 30, 14, 7, 2, 1, 54, 82, 60, 34, 15, 7, 2, 1, 97, 157, 125, 71, 36, 15, 7, 2, 1, 172, 304, 256, 152, 77, 37, 15, 7, 2, 1, 309, 589, 513, 325, 168, 81, 37, 15, 7, 2, 1, 564, 1124, 1049, 669, 367, 177, 83, 37

Offset: 1

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

			Triangle begins:
  1
  2 1
  4 2 1
  6 6 2 1
  11 10 7 2 1
  18 22 13 7 2 1
  31 42 30 14 7 2 1
  54 82 60 34 15 7 2 1
  97 157 125 71 36 15 7 2 1
  172 304 256 152 77 37 15 7 2 1

Adolf Hildebrand, On the number of prime factors of an integer. Ramanujan revisited (Urbana-Champaign, Ill., 1987), 167 - 185, Academic Press, Boston, MA, 1988.
Edmund Georg Hermann Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, pp. 205 - 211.

Jonathan Vos Post & Robert G. Wilson v, Table of n, a(n) for n = 1..1330
Jonathan Vos Post & Robert G. Wilson v, Regular Triangle of A126279 for the first 52 rows, with some holes.
Eric Weisstein's World of Mathematics, Semiprime.
Eric Weisstein's World of Mathematics, Almost Prime.

First column: A007053, second column: A125527, third column: A127396, 4th column: A334069. The last row reversed: A052130; the n-th row's sum: A000225 = 2^n -1.

Cf. A126280: same array but for powers of ten.

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, 2^n], {n, 16}, {m, n}] // Flatten

A092097 Limit number of (m-n)-almost-primes in range [2^m..2^{m+1}-1].

Original entry on oeis.org

2, 5, 8, 22, 47, 103, 234, 493, 1087, 2282, 4901, 10427, 21993, 46389, 97394, 204567, 427099, 892587, 1858338, 3865692, 8027140, 16642918, 34463760, 71273199, 147235636, 303814862, 626313383, 1289883519, 2654196000

Offset: 0

Andrew S. Plewe, Feb 19 2004

Also number of odd numbers k for which floor(log_2(k)) - bigomega(k) = n, where bigomega is A001222. - Franklin T. Adams-Watters, Jun 20 2006

The value of m at which the number of (m-n)-almost-primes reaches its limit is floor(n/(log_2(3)-1))+n-1: 1,4,7,9,12,15,17,20,23,26,28; not A026356: 2,4,7,9,12,15,17,20,22,25,28 as originally conjectured. - Franklin T. Adams-Watters, Jun 20 2006

			a(0) = 2: m-almost primes in [2^m..2^{m+1}-1] are 2^m and 3*2^{m-1}.
a(1) = 5; (m-1)-almost-primes in [2^m..2^{m-1}] are 5*2^{m-2}, 7*2^{m-2}, 9*2^{m-3}, 15*2^{m-3} and 27*2^{m-4}.

Cf. A052130, A001222, A026356, A120033-A120043.

For n>0, a(n) = A052130(n+1)-A052130(n).

Edited and extended by Franklin T. Adams-Watters, Jun 20 2006

A373943 a(n) is the cardinality of the set containing all rational numbers of the form 2 <= m/2^(bigomega(m) - 1) <= n.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 6, 7, 7, 7, 10, 11, 13, 13, 13, 15, 18, 19, 21, 22, 22, 22, 24, 25, 27, 29, 30, 31, 34, 35, 36, 37, 38, 38, 40, 41, 43, 45, 47, 48, 49, 50, 54, 57, 57, 58, 61, 62, 63, 63, 63, 65, 66, 67, 67, 70, 71, 74, 75, 77, 79, 82, 82, 84, 86, 89, 91

Offset: 1

Friedjof Tellkamp, Jun 23 2024

nonn

			a(10) = 7 = card{2, 3, 9/2, 5, 27/4, 7, 15/2}.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000720, A001222, A052130, A374022.

z = 100;
k[n_] := Max[1, Floor[Log[3/2, n/2]]];
m[n_] := n 2^(k[n] - 1);
PrimePiK = Table[0, Floor[Log[2, m[z]]], m[z]];
For[i = 2, i <= m[z], i++, PrimePiK[[PrimeOmega[i], i]] = 1]
PrimePiK = Accumulate /@ PrimePiK;
a = Table[PrimePiK[[k[n], m[n]]], {n, z}] (*sequence*)
x = Union@Select[Table[i/2^(PrimeOmega[i] - 1), {i, 1, m[z], 2}], # <= z &] (*set*)

nap(n, k) = sum(i=1, n, bigomega(i)==k);
a(n) = my(k=max(1, floor(log(n/2)/(log(3)-log(2))))); nap(n*2^(k-1), k); \\ Michel Marcus, Jun 27 2024

from math import isqrt, prod
from sympy import primepi, primerange, integer_nthroot
def A373943(n):
    if n<=4: return primepi(n)
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    k = 1
    while 3**k<(r:=n<Chai Wah Wu, Dec 03 2024

a(n) = card{x | x = m/2^(bigomega(m)-1), x<=n}.
a(n) = pi_k(n * 2^(k - 1)), with pi_k(n) as the counting function for k-almost primes and k sufficiently large.
k needs to be at least max(1, floor(log(n/2)/(log(3)-log(2)))) and m = n * 2^(k - 1).
a(n) = A374022(n) + A000720(n).
a(2^n) = A052130(n-1).

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.

A114791 Consider the array T where the element at T(r,c) is the number of n-almost primes less than or equal to 2^m. Sequence read by successive columns beginning at r=c.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 6, 6, 1, 2, 7, 10, 11, 1, 2, 7, 13, 22, 18, 1, 2, 7, 14, 30, 42, 31, 1, 2, 7, 15, 34, 60, 82, 54, 1, 2, 7, 15, 36, 71, 125, 157, 97, 1, 2, 7, 15, 37, 77, 152, 256, 304, 172, 1, 2, 7, 15, 37, 81, 168, 325, 513, 589, 309, 1, 2, 7, 15, 37, 83, 177, 367, 669, 1049

Offset: 1

Robert G. Wilson v, Feb 21 2006

Eventually each column reads 1,2,7,15,37,84,187,421,914,... = A052130.

			1 2 4 6 11 18 31 54 97 172 309 564 1028 1900 3512 6542
..1 2 6 10 22 42 82 157 304 589 1124 2186 4192 8110 15658
....1 2 7 13 30 60 125 256 513 1049 2082 4214 8401 16771
......1 2 7 14 34 71 152 325 669 1405 2866 5931 12139
........1 2 7 15 36 77 168 367 770 1643 3410 7150
..........1 2 7 15 37 81 177 392 831 1790 3757
............1 2 7 15 37 83 182 406 867 1880
..............1 2 7 15 37 84 185 414 887
................1 2 7 15 37 84 186 418

First row is A007053, Cf. A052130.

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[n, 2^k], {k, 12}, {n, k, 1, -1}] // Flatten

Each column sums to 2^n-1.

A197871 Irregular triangle T(n,k) of the number of numbers with k prime factors (repetitions allowed) less than n^2.

Original entry on oeis.org

0, 2, 1, 4, 3, 1, 6, 6, 2, 1, 9, 9, 4, 2, 11, 13, 7, 3, 1, 15, 17, 10, 4, 2, 18, 22, 13, 7, 2, 1, 22, 26, 19, 8, 4, 1, 25, 34, 22, 12, 4, 2, 30, 40, 28, 13, 7, 2, 34, 48, 32, 18, 7, 3, 1, 39, 56, 38, 21, 9, 4, 1, 44, 62, 48, 24, 11, 4, 2, 48, 75, 51, 29, 13, 6, 2

Offset: 1

Daniel Tisdale, Oct 18 2011

			In the third row, reading from the left, 6 is the number of primes <= 16, 6 is the number of semiprimes <= 16, 2 is the number of numbers with three prime divisors (repetitions allowed) <= 16, and 1 is the number of numbers with four divisors <= 16.
The triangle begins:
   0
   2  1
   4  3  1
   6  6  2  1
   9  9  4  2
  11 13  7  3  1
  15 17 10  4  2
  ...

G. J. O. Jameson, The Prime Number Theorem, Cambridge, 2004, p.145.

Similar to A052130.

Join[{0}, Flatten[Table[Transpose[Tally[Table[Plus @@ Last /@ FactorInteger[i], {i, 2, n^2}]]][[2]], {n, 2, 15}]]]

T(n,k) = #select(x->(bigomega(x) == k), [1..n^2]);
row(n) = my(v = vector(n, k, T(n,k))); my(pos); for (k=1, n, if (v[k], pos=k)); Vec(v, pos); \\ Michel Marcus, Aug 16 2022

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A126281 a(n) is the least m to satisfy the requirements of A052130.

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A126279 Triangle read by rows: T(k,n) is number of numbers <= 2^n that are products of k primes.

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A092097 Limit number of (m-n)-almost-primes in range [2^m..2^{m+1}-1].

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A373943 a(n) is the cardinality of the set containing all rational numbers of the form 2 <= m/2^(bigomega(m) - 1) <= n.

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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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A114791 Consider the array T where the element at T(r,c) is the number of n-almost primes less than or equal to 2^m. Sequence read by successive columns beginning at r=c.

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A197871 Irregular triangle T(n,k) of the number of numbers with k prime factors (repetitions allowed) less than n^2.

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