A101695 -id:A101695 - cp's OEIS frontend

A101696 a(n) = sum(i=1,n)(i-th i-almost prime). Cumulative sums of A101695.

2, 8, 26, 66, 174, 398, 878, 2174, 4862, 10494, 22014, 45054, 98302, 222718, 480766, 1021438, 2127358, 4355582, 8943102, 18773502, 38696446, 79590910, 175142398, 368080382, 764442110, 1586525694, 3247470078, 6644856318, 13489960446

Offset: 1

Jonathan Vos Post, Dec 12 2004, Sep 28 2006

nonn

It seems that this sum can never be a prime after a(1) = 2, since the n-th n-almost prime is always even. The number of prime factors (with multiplicity) of a(n) is 1, 3, 2, 3, 3, 2, 2, 2, 4, 5, 4, 4, 3, 3, 5, 4, 3, 4, 7, 4, 2, 5, 5, 2, 3, 7, 4, 3, 4.

This is the diagonalization of the set of sequences {j-almost prime(k)}. The cumulative sums of this sequence are in A101696. a(1)=2 is prime. a(2)=8 is a 3-almost prime. a(3)=26 is a semiprime. a(4)=66 is a 3-almost prime. a(5)= 174 is a 3-almost prime. a(6)=398 is a semiprime. a(7)=878 is a semiprime. a(8)=2174 is a semiprime. a(9)=4862 is a 4-almost prime. a(10)=10494 is a 5-almost prime. a(11)=22014 is a 4-almost prime. a(12)=45054 is a 3-almost prime. a(13)=98302 is a 3-almost prime. a(14)=222718 is a 3-almost prime. a(15)=480766 is a 5-almost prime. a(16)=1021438 is a 4-almost prime. a(17)=2127358 is a 3-almost prime. a(18)=4355582 is a 4-almost prime. a(19)=8943102 is a 7-almost prime. a(20)=18773502 is a 4-almost prime. 21-almost numbers are not yet listed in the OEIS.

			a(1) = first 1-almost prime = first prime = A000040(1) = 2.
a(2) = 2 + 2nd 2-almost prime = 2 + A001358(2) = 2+ 6 = 8.
a(3) = a(2) + 3rd 3-almost prime = 8+A014612(3) = 8+18 = 26.
a(4) = a(3) + 4th 4-almost prime = 26+A014613(4) = 26+40 = 66.
a(5) = a(4) + 5th 5-almost prime = 66+A014614(5) = 66+108=174.
...
a(12) = a(11) + 12th 12-almost prime = 22014 + 23040 = 45054 (the first nontrivial palindrome in the sequence).

Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A000040, A001358, A014612, A014613, A046314, A046306, A046308, A046310, A046312, A046314, A069272, A069273, A069274, A069275, A069276, A069277, A069278, A069279, A069280, A069281, A101637, A101638, A101605, A101606, A101695.

a(1) = first 1-almost prime = first prime = A000040(1). a(2) = a(1) + 2nd 2-almost prime = a(1) + 2nd semiprime = A000040(1)+A001358(2). a(3) = a(2) + 3rd 3-almost prime = a(2) + A014612(3). a(4) = a(3) + 4th 4-almost prime = a(3) + A014613(4)... a(n) = a(n-1) + n-th n-almost prime.

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A122942 Partial product of n-th n-almost prime (A101695) divided by product of the first n primes, rounded down.

Original entry on oeis.org

1, 2, 7, 41, 403, 6960, 196527, 13405218, 1566662070, 304256578608, 113065670502087, 78229220671714544, 101598769325059903837, 293965406612712860369329, 1613982664799943153033715558

Offset: 1

Jonathan Vos Post, Oct 24 2006

			a(1) = floor(2/2) = floor(1) = 1.
a(2) = floor(12/6) = floor(2) = 2.
a(3) = floor(216/30) = floor(7.2) = 7.
a(4) = floor(8640/210) = floor(41.1428571) = 41.
a(5) = floor(933120/2310) = floor(403.948052) = 403.
a(6) = floor(209018880/30030) = floor(6960.33566) = 6960.

Cf. A101695, A000040.

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 *)
AlmostPrime[k_, n_] := Block[{e = Floor[Log[2, n]+k], a, b}, a = 2^e; Do[b = 2^p; While[ AlmostPrimePi[k, a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; AlmostPrime[1, 1] = 2;
t = Table[ AlmostPrime[n, n], {n, 20}]; Floor[Rest@ FoldList[Times, 1, t]/Rest@ FoldList[Times, 1, Prime@ Range@ 20]] (* Robert G. Wilson v, Aug 31 2007 *)

a(n) = floor( Product_{i=1..n} A101695(i) / A000040(i) ).

More terms from Robert G. Wilson v, Aug 31 2007

A123118 Partial products of A101695.

Original entry on oeis.org

2, 12, 216, 8640, 933120, 209018880, 100329062400, 130026464870400, 349511137571635200, 1968446726803449446400, 22676506292775737622528000, 522466704985552994823045120000, 27820307107070725868337506549760000

Offset: 1

Jonathan Vos Post, Sep 28 2006

nonn

The number of prime factors (with multiplicity) of a(n) is T(n) = A000217(n) = n*(n+1)/2.

			a(1) = 2 = prime(1).
a(2) = 12 = 2 * 6 = prime(1) * semiprime(2) = 2^2 * 3.
a(3) = 216 = 2 * 6 * 18 = prime(1) * semiprime(2) * 3-almostprime(3) = 2^3 * 3^3.
a(4) = 8640 = 2 * 6 * 18 * 40 = prime(1) * semiprime(2) * 3-almostprime(3) * 4-almostprime(4) = 2^6 * 3^3 * 5.
a(15) = 893179304874387947794472921245209518407680000 = 2 * 6 * 18 * 40 * 108 * 224 * 480 * 1296 * 2688 * 5632 * 11520 * 23040 * 53248 * 124416 * 258048 = 2^88 * 3^23 * 5^4 * 7^3 * 11 * 13.

Cf. A000040, A001358, A014612, A014613, A046314, A046306, A046308, A046310, A046312, A046314, A069272, A069273, A069274, A069275, A069276, A069277, A069278, A069279, A069280, A069281, A101637, A101638, A101605, A101606, A101695.

a(n) = Prod[i=1..n] i-th i-almost prime = Prod[i=1..n] A101695(i).

A078840 Table of n-almost-primes T(n,k) (n >= 0, k > 0), read by antidiagonals, starting at T(0,1)=1 followed by T(1,1)=2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 9, 12, 16, 11, 10, 18, 24, 32, 13, 14, 20, 36, 48, 64, 17, 15, 27, 40, 72, 96, 128, 19, 21, 28, 54, 80, 144, 192, 256, 23, 22, 30, 56, 108, 160, 288, 384, 512, 29, 25, 42, 60, 112, 216, 320, 576, 768, 1024, 31, 26, 44, 81, 120, 224, 432, 640, 1152

Offset: 0

Benoit Cloitre and Paul D. Hanna, Dec 10 2002

An n-almost-prime is a positive integer that has exactly n prime factors.
This sequence is a rearrangement of the natural numbers. - Robert G. Wilson v, Feb 11 2006
Each antidiagonal begins with the n-th prime and ends with 2^n.
From Eric Desbiaux, Jun 27 2009: (Start)
(A001222 gives this sequence)
A001221 gives another table:
  1
  -    2    3    4    5    7    8    9   11 ... A000961
  -    6   10   12   14   15   18   20   21 ... A007774
  -   30   42   60   66   70   78   84   90 ... A033992
  -  210  330  390  420  462  510  546  570 ... A033993
  - 2310 2730 3570 3990 4290 4620 4830 5460 ... A051270
Antidiagonals begin with A000961 and end with A002110.
Diagonal is A073329 which is last term in n-th row of A048692. (End)

			Table begins:
  1
  -  2  3   5   7  11  13  17  19  23  29 ...
  -  4  6   9  10  14  15  21  22  25  26 ...
  -  8 12  18  20  27  28  30  42  44  45 ...
  - 16 24  36  40  54  56  60  81  84  88 ...
  - 32 48  72  80 108 112 120 162 168 176 ...
  - 64 96 144 160 216 224 240 324 336 352 ...

Robert G. Wilson v, Table of n, a(n) for n = 0..10011 (corrected by Ivan Neretin).
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A078840, A078841, A078842, A078843, A078844, A078445, A078846, A109636.
T(1, k)=A000040(k), T(2, k)=A001358(k), T(3, k)=A014612(k), T(4, k)=A014613(k), T(5, k)=A014614(k), T(6, k)=A046306(k), T(7, k)=A046308(k), T(8, k)=A046310(k), T(9, k)=A046312(k), T(10, k)=A046314(k).
T(11, k)=A069272(k), T(12, k)=A069273(k), T(13, k)=A069274(k), T(14, k)=A069275(k), T(15, k)=A069276(k), T(16, k)=A069277(k), T(17, k)=A069278(k), T(18, k)=A069279(k), T(19, k)=A069280(k), T(20, k)=A069281(k).
T(k, 1)=A000079(k), T(k, 2)=A007283(k), T(k, 3)=A116453(k), T(k, k)=A101695(k), T(k, k+1)=A078841(k).
A091538 is this sequence with zeros inserted, making a square array.

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 *)
AlmostPrime[k_, n_] := Block[{e = Floor[Log[2, n]+k], a, b}, a = 2^e; Do[b = 2^p; While[ AlmostPrimePi[k, a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Table[ AlmostPrime[k, n - k + 1], {n, 11}, {k, n}] // Flatten (* Robert G. Wilson v *)
mx = 11; arr = NestList[Take[Union@Flatten@Outer[Times, #, primes], mx] &, primes = Prime@Range@mx, mx]; Prepend[Flatten@Table[arr[[k, n - k + 1]], {n, mx}, {k, n}], 1] (* Ivan Neretin, Apr 30 2016 *)
(* The next code skips the initial 1. *)
width = 15; (seq = Table[
  Rest[NestList[1 + NestWhile[# + 1 &, #, ! PrimeOmega[#] == z &] &,
  2^z, width - z + 1]] - 1, {z, width}]) // TableForm
Flatten[Map[Reverse[Diagonal[Reverse[seq], -width + #]] &, Range[width]]]
(* Peter J. C. Moses, Jun 05 2019 *)
Grid[Table[Select[Range[200], PrimeOmega[#] == n &], {n, 0, 7}]]
(* Clark Kimberling, Nov 17 2024 *)

T(n,k)=if(k<0,0,s=1; while(sum(i=1,s,if(bigomega(i)-n,0,1))

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi, prime
def A078840_T(n,k):
    if n == 1: return prime(k)
    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)))
    def f(x): return int(k-1+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

Edited by Robert G. Wilson v, Feb 11 2006

A078841 Main diagonal of the table of k-almost primes (A078840): a(n) = (n+1)-st integer that is an n-almost prime.

Original entry on oeis.org

1, 3, 9, 20, 54, 112, 240, 648, 1344, 2816, 5760, 12800, 26624, 62208, 129024, 270336, 552960, 1114112, 2293760, 4915200, 9961472, 20447232, 47775744, 96468992, 198180864, 411041792, 830472192, 1698693120, 3422552064, 7046430720

Offset: 0

Benoit Cloitre and Paul D. Hanna, Dec 10 2002

nonn

A k-almost prime is a positive integer that has exactly k prime factors counted with multiplicity.

			a(0) = 1 since one is the multiplicative identity,
a(1) = 2nd 1-almost prime is the second prime number = A000040(2) = 3,
a(2) = 3rd 2-almost prime = 3rd semiprime = A001358(3) = 9 = {3*3}.
a(3) = 4th 3-almost prime = A014612(4) = 20 = {2*2*5}.
a(4) = 5th 4-almost prime = A014613(5) = 54 = {2*3*3*3},
a(5) = 6th 5-almost prime = A014614(6) = 112 = {2*2*2*2*7}, ....

Chai Wah Wu, Table of n, a(n) for n = 0..1000 (terms 0..228 from Robert G. Wilson v)
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A078840, A078842, A078843, A078844, A078445, A078846, A101695.

f[n_] := Plus @@ Last /@ FactorInteger@n; t = Table[{}, {40}]; Do[a = f[n]; AppendTo[ t[[a]], n]; t[[a]] = Take[t[[a]], 10], {n, 2, 148*10^8}]; Table[ t[[n, n + 1]], {n, 30}] (* Robert G. Wilson v, Feb 11 2006 *)
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 *)
AlmostPrime[k_, n_] := Block[{e = Floor[ Log[2, n] + k], a, b}, a = 2^e; Do[b = 2^p; While[ AlmostPrimePi[k, a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; AlmostPrime[1, 1] = 2; lst = {}; Do[ AppendTo[lst, AlmostPrime[n-1, n]], {n, 30}]; lst (* Robert G. Wilson v, Nov 13 2007 *)

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A078841(n):
    if n <= 1: return (n<<1)+1
    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)))
    def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

Conjecture: Lim as n->inf. of a(n+1)/a(n) = 2. - Robert G. Wilson v, Nov 13 2007

a(14)-a(29) from Robert G. Wilson v, Feb 11 2006

A122943 Odd numbers n ordered by n/2^BigOmega(n), where BigOmega(n) is the number of prime divisors of n with repetition.

Original entry on oeis.org

1, 3, 9, 5, 27, 7, 15, 81, 21, 11, 45, 25, 13, 243, 63, 33, 135, 17, 35, 75, 19, 39, 729, 23, 189, 49, 99, 405, 51, 105, 55, 225, 57, 29, 117, 31, 125, 65, 2187, 69, 567, 147, 37, 297, 1215, 153, 77, 315, 41, 165, 675, 85, 171, 43, 87, 175, 351, 91, 93, 375, 47, 95, 195

Offset: 1

Franklin T. Adams-Watters, Oct 24 2006

nonn

This is the limit of the sequence of largest odd factors of the k-almost primes as k -> infinity.
The location of 3^k in this sequence is A078843(k).
Removing 1 and prime numbers from this sequence gives A374074. - Friedjof Tellkamp, Nov 27 2024

T. D. Noe, Table of n, a(n) for n = 1..1065
Almost primes, sequences related to

Cf. A001222, A101695, A078841, A078840, A000265, A007814, A374074.

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}]]]]] (* from Eric Weisstein, Feb 07 2006 *); AlmostPrime[k_, n_] := Block[{e = Floor[ Log[2, n] + k], a, b}, a = 2^e; Do[b = 2^p; While[ AlmostPrimePi[k, a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; f[n_] := Block[{ kap = AlmostPrime[20, n]}, kap / 2^IntegerExponent[ kap, 2]]; Array[f, 64] (* or *)
f[n_] := n/2^PrimeOmega[n]; Take[2 Ordering[ Table[ f[ 2n - 1], {n, 1100}]] - 1, 63] (* Robert G. Wilson v, Feb 08 2011 *)
f[n_] := n/2^PrimeOmega[n]; nn=9; t = Select[Table[{f[2 n - 1], 2 n - 1}, {n, 3^nn/2 + 1}], #[[1]] <= f[3^nn] &]; Transpose[Sort[t]][[2]]

A101695(n) = a(n) * 2^(n - BigOmega(a(n))). a(n) = A101695(n) / 2^A007814(A101695(n)) = A000265(A101695(n)).

A215405 Largest prime factor of the n-th n-almost prime.

Original entry on oeis.org

2, 3, 3, 5, 3, 7, 5, 3, 7, 11, 5, 5, 13, 3, 7, 11, 5, 17, 7, 5, 19, 13, 3, 23, 7, 7, 11, 5, 17, 7, 11, 5, 19, 29, 13, 31, 5, 13, 3, 23, 7, 7, 37, 11, 5, 17, 11, 7, 41, 11, 5, 17, 19, 43, 29, 7, 13, 13, 31, 5, 47, 19, 13, 3, 23, 53, 7, 7, 37

Offset: 1

Juri-Stepan Gerasimov, Aug 09 2012

nonn

Technically, the prime numbers are "1-almost prime."
Prime(m) (m>=1) occurs first at index n = 1, 2, 4, 6, 10, 13, 18, 21, 24, 34, 36, 43, 49, 54, 61, 66, 75, 79, 91, 97, 101, 107, 113, 124, 138, 144, 148, 157, 162, 167, 187, 194, 202, 207, 224, 229,... in the sequence. - R. J. Mathar, Aug 09 2012
n <= a(n) at 1, 2, 3, 4, 6, 10, 13,...
n < 2*a(n) at n = 1, 2, 3, 4, 6, 7, 9, 10, 13, 16, 18, 21, 22, 24, 29, 33, 34, 36, 40, 43, 49, 54, 55, 59, 61, 66, 69,...
Also largest prime factor of A122943(n) for n>1. - Eric Desbiaux, Mar 20 2016

			a(2) = 3 because the 2nd 2-almost prime (semiprime, A001358) is 6 = 2 * 3, the largest prime factor there being 3.
a(3) = 3 because the 3rd 3-almost prime (A014612) is 18 = 2 * 3^2, the largest prime factor there being 3.
a(4) = 5 because the 4th 4-almost prime (A014613) is 40 = 2^3 * 5, the largest prime factor there being 5.

Cf. A078841, A101695.

A215405 := proc(n)
        A006530(A101695(n)) ;
end proc: # R. J. Mathar, Aug 09 2012

Corrected by R. J. Mathar, Aug 09 2012

A330394 Irregular triangle T(n,k) read by rows in which n-th row lists in increasing order all integers m such that Omega(m) = n and each prime factor p of m has index pi(p) <= n.

Original entry on oeis.org

1, 2, 4, 6, 9, 8, 12, 18, 20, 27, 30, 45, 50, 75, 125, 16, 24, 36, 40, 54, 56, 60, 81, 84, 90, 100, 126, 135, 140, 150, 189, 196, 210, 225, 250, 294, 315, 350, 375, 441, 490, 525, 625, 686, 735, 875, 1029, 1225, 1715, 2401, 32, 48, 72, 80, 108, 112, 120, 162

Offset: 0

Robert Price, Mar 03 2020

Positive integers not in T are: 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, ... .
Row n has exactly one squarefree member: primorial(n) = A002110(n).
Sorting all terms (except 1) gives A324521.

			Triangle T(n,k) begins:
  1;
  2;
  4,  6,  9;
  8, 12, 18, 20, 27, 30, 45, 50, 75, 125;
  ...

Robert Price, Table of n, a(n) for n = 0..8788

Column k=1 gives A000079.
Last elements of rows give A307539.
Row lengths give A088218.
Row sums give A332967(n) = A124960(2n,n).
T(n,n) gives A101695(n) for n > 0.
Cf. A000720, A001222, A005117, A027748, A324521.

b:= proc(n, i) option remember; `if`(n=0, [1], [seq(
      map(x-> x*ithprime(j), b(n-1, j))[], j=1..i)])
    end:
T:= n-> sort(b(n$2))[]:
seq(T(n), n=0..5);  # Alois P. Heinz, Mar 03 2020

t = Table[Union[Apply[Times, Tuples[Prime[Range[n]], {n}], {1}]], {n, 0, 5}];
t // TableForm
Flatten[t]

A340467 a(n) is the n-th squarefree number having n prime factors.

Original entry on oeis.org

2, 10, 66, 462, 4290, 53130, 903210, 17687670, 406816410, 11125544430, 338431883790, 11833068917670, 457077357006270, 20384767656323070, 955041577211912190, 49230430891074322890, 2740956243836856315270, 168909608387276001835590, 11054926927790884163355330

Offset: 1

Alois P. Heinz, Jan 08 2021

nonn

a(n) is the n-th product of n distinct primes.
All terms are even.
This sequence differs from A073329 which has also nonsquarefree terms.

			a(1) = A000040(1) = 2.
a(2) = A006881(2) = 10.
a(3) = A007304(3) = 66.
a(4) = A046386(4) = 462.
a(5) = A046387(5) = 4290.
a(6) = A067885(6) = 53130.
a(7) = A123321(7) = 903210.
a(8) = A123322(8) = 17687670.
a(9) = A115343(9) = 406816410.
a(10) = A281222(10) = 11125544430.

Alois P. Heinz, Table of n, a(n) for n = 1..350

Main diagonal of A340316.
Cf. A001221, A001222, A005117, A070826, A072047, A073329, A101695, A340313.
Cf. A000040, A006881, A007304, A046386, A046387, A067885, A123321, A123322, A115343, A281222.

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A340467(n):
    if n == 1: return 2
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f) # Chai Wah Wu, Aug 31 2024

a(n) = A340316(n,n).
a(n) = A005117(m) <=> A072047(m) = n = A340313(m).
A001221(a(n)) = A001222(a(n)) = n.
a(n) < A070826(n+1), the least odd number with exactly n distinct prime divisors.

A376739 Main diagonal of A376738: a(n) is the n-th number which is the product of n (possibly non-distinct) primes having the same number of decimal digits.

Original entry on oeis.org

2, 6, 18, 40, 108, 224, 480, 1296, 2688, 5760, 12800, 31104, 64512, 138240, 286720, 614400, 1492992, 3096576, 6422528

Offset: 1

Paolo Xausa, Oct 03 2024

See A376738 for more information.

First differs from A101695 at n = 10.

Cf. A101695, A376738.

A376739[n_] := Module[{m = 2^n - 1}, Do[While[Total[(f = FactorInteger[++m])[[All, 2]]] != n || Length[Union[IntegerLength[f[[All, 1]]]]] > 1], n]; m];
Array[A376739, 15]

a(n) = A376738(n,n).

cp's OEIS Frontend

A101696 a(n) = sum(i=1,n)(i-th i-almost prime). Cumulative sums of A101695.

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A122942 Partial product of n-th n-almost prime (A101695) divided by product of the first n primes, rounded down.

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A123118 Partial products of A101695.

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A078840 Table of n-almost-primes T(n,k) (n >= 0, k > 0), read by antidiagonals, starting at T(0,1)=1 followed by T(1,1)=2.

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A078841 Main diagonal of the table of k-almost primes (A078840): a(n) = (n+1)-st integer that is an n-almost prime.

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A122943 Odd numbers n ordered by n/2^BigOmega(n), where BigOmega(n) is the number of prime divisors of n with repetition.

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A215405 Largest prime factor of the n-th n-almost prime.

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A330394 Irregular triangle T(n,k) read by rows in which n-th row lists in increasing order all integers m such that Omega(m) = n and each prime factor p of m has index pi(p) <= n.

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