A078845 -id:A078845 - cp's OEIS frontend

A116432 The number of n-almost primes less than or equal to e^n, starting with a(0)=1.

1, 1, 2, 4, 5, 7, 12, 18, 24, 37, 54, 74, 107, 159, 218, 315, 450, 634, 888, 1269, 1782, 2496, 3520, 4933, 6899, 9681, 13555, 18888, 26407, 36855, 51352, 71526, 99654, 138608, 192708, 267833, 372107, 516420, 716816, 994191, 1378195, 1909694

Offset: 0

Robert G. Wilson v, Feb 10 2006

nonn

Cf. A078840, A078841, A078842, A116432, A078843, A116426, A078844, A116427, A078845, A116428, A116429, A116430, A078846, A116431.

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, E^n], {n, 42}]

A116433 Consider the array T(r,c) where is the number of c-almost primes less than or equal to r^c, r >= 1, c >= 0. Read the array by antidiagonals.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 3, 1, 0, 1, 3, 6, 5, 1, 0, 1, 3, 9, 13, 8, 1, 0, 1, 4, 13, 30, 34, 14, 1, 0, 1, 4, 17, 50, 90, 77, 23, 1, 0, 1, 4, 22, 82, 200, 269, 177, 39, 1, 0, 1, 4, 26, 125, 385, 726, 788, 406, 64, 1, 0, 1, 5, 34, 181, 669, 1688, 2613, 2249, 887, 103, 1, 0, 1, 5

Offset: 0

Paul D. Hanna and Robert G. Wilson v, Feb 15 2006

			The array begins:
  0 0 0 0 0 0 0 0 0 0 0
  1 1 1 1 1 1 1 1 1 1 1
  1 2 3 5 8 14 23 39 64 103 169
  1 2 6 13 34 77 177 406 887 1962 4225
  1 3 9 30 90 269 788 2249 6340 17526 47911
T(3,2)=3 because there are 3 2-almost primes <= 3^2 = 9, namely 4, 6, and 9 (see A001358).

Cf. The rows are: A000004, A000012, A078843, A116426, A078844, A116427, A078845, A116428, A116429, A116430, A078846, A116431.

The columns are: A057427, A000720.

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[ If[k == 0, 1, AlmostPrimePi[n - k + 1, k^(n - k + 1)]], {n, 0, 7}, {k, n, 0, -1}] // Flatten

NAME corrected by R. J. Mathar, Jun 20 2021

A376479 Array read by antidiagonals: T(n,k) is the index of prime(k)^n in the numbers with n prime factors, counted with multiplicity.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 9, 5, 1, 5, 17, 30, 8, 1, 6, 40, 82, 90, 14, 1, 7, 56, 328, 385, 269, 23, 1, 8, 90, 551, 2556, 1688, 788, 39, 1, 9, 114, 1243, 5138, 18452, 7089, 2249, 64, 1, 10, 164, 1763, 15590, 44329, 126096, 28893, 6340, 103, 1, 11, 253, 3112, 24646, 179313, 361249, 827901, 115180, 17526

Offset: 1

Robert Israel, Sep 24 2024

T(n,k) is the number of numbers j with n prime factors, counted with multiplicity, such that j <= prime(k)^n.

			T(2,3) = 9 because the third prime is 5 and 5^2 = 25 is the 9th semiprime.

Cf. A001222, A078843 (second column), A078844 (third column), A078845 (fourth column), A078846 (fifth column), A128301 (second row), A128302 (third row), A128304 (fourth row).

T:= Matrix(12,12):
with(priqueue);
for m from 1 to 12 do
  initialize(pq);
  insert([-2^m, [2$m]],pq);
  k:= 0:
  for count from 1 do
    t:= extract(pq);
    w:= t[2];
    if nops(convert(w,set))=1 then
      k:= k+1;
      T[m,k]:= count;
      if m+k = 13 then break fi;
    fi;
    p:= nextprime(w[-1]);
    for i from m to 1 by -1 while w[i] = w[m] do
      insert([t[1]*(p/w[-1])^(m+1-i),[op(w[1..i-1]),p$(m+1-i)]],pq);
od od od:
seq(seq(T[i,s-i],i=1..s-1),s=2..13)

cp's OEIS Frontend

A116432 The number of n-almost primes less than or equal to e^n, starting with a(0)=1.

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A116433 Consider the array T(r,c) where is the number of c-almost primes less than or equal to r^c, r >= 1, c >= 0. Read the array by antidiagonals.

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A376479 Array read by antidiagonals: T(n,k) is the index of prime(k)^n in the numbers with n prime factors, counted with multiplicity.

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