A078843 -id:A078843 - cp's OEIS frontend

A116428 The number of n-almost primes less than or equal to 8^n, starting with a(0)=1.

1, 4, 22, 125, 669, 3410, 16677, 78369, 359110, 1612613, 7133274, 31185350, 135062165, 580556958, 2480278767, 10542976739, 44626102826, 188215850830, 791374442571, 3318478309647, 13882441625034, 57952990683107

Offset: 0

Robert G. Wilson v, Feb 14 2006

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}]]]]];
Table[ AlmostPrimePi[n, 8^n], {n, 14}] (* Eric W. Weisstein, Feb 07 2006 *)

almost_prime_count(N, k) = if(k==1, return(primepi(N))); (f(m, p, k, j=0) = my(c=0, s=sqrtnint(N\m, k)); if(k==2, forprime(q=p, s, c += primepi(N\(m*q))-j; j += 1), forprime(q=p, s, c += f(m*q, q, k-1, j); j += 1)); c); f(1, 2, k);
a(n) = if(n == 0, 1, almost_prime_count(8^n, n)); \\ Daniel Suteu, Jul 10 2023

a(15)-a(18) from Donovan Johnson, Oct 01 2010

a(19)-a(21) from Daniel Suteu, Jul 10 2023

A116429 The number of n-almost primes less than or equal to 9^n, starting with a(0)=1.

Original entry on oeis.org

1, 4, 26, 181, 1095, 6416, 35285, 187929, 973404, 4934952, 24628655, 121375817, 592337729, 2868086641, 13798982719, 66043675287, 314715355786, 1494166794434, 7071357084444, 33374079939405

Offset: 0

Robert G. Wilson v, Feb 10 2006

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, 9^n], {n, 13}]

almost_prime_count(N, k) = if(k==1, return(primepi(N))); (f(m, p, k, j=0) = my(c=0, s=sqrtnint(N\m, k)); if(k==2, forprime(q=p, s, c += primepi(N\(m*q))-j; j += 1), forprime(q=p, s, c += f(m*q, q, k-1, j); j += 1)); c); f(1, 2, k);
a(n) = if(n == 0, 1, almost_prime_count(9^n, n)); \\ Daniel Suteu, Jul 10 2023

a(14)-a(16) from Donovan Johnson, Oct 01 2010

a(16) corrected and a(17)-a(19) from Daniel Suteu, Jul 10 2023

A116431 The number of n-almost primes less than or equal to 12^n, starting with a(0)=1.

Original entry on oeis.org

1, 5, 48, 434, 3695, 29165, 218283, 1569995, 10950776, 74621972, 499495257, 3297443264, 21533211312, 139411685398, 896352197825, 5730605551626, 36465861350230

Offset: 0

Robert G. Wilson v, Feb 10 2006

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, 12^n], {n, 12}]

almost_prime_count(N, k) = if(k==1, return(primepi(N))); (f(m, p, k, j=0) = my(c=0, s=sqrtnint(N\m, k)); if(k==2, forprime(q=p, s, c += primepi(N\(m*q))-j; j += 1), forprime(q=p, s, c += f(m*q, q, k-1, j); j += 1)); c); f(1, 2, k);
a(n) = if(n == 0, 1, almost_prime_count(12^n, n)); \\ Daniel Suteu, Jul 10 2023

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A116431(n):
    if n<=1: return 4*n+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)))
    return int(sum(primepi(12**n//prod(c[1] for c in a))-a[-1][0] for a in g(12**n,0,1,1,n))) # Chai Wah Wu, Sep 28 2024

a(13)-a(14) from Donovan Johnson, Oct 01 2010

a(15)-a(16) from Daniel Suteu, Jul 10 2023

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

Original entry on oeis.org

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}]

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)).

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

A209934 a(n) is the first value to occur consecutively in the sequence b_n defined by p_2k(b_n(k)) = p_k(n)^2, k=1,2,3,..., where p_k(n) is the n-th k-almost prime.

Original entry on oeis.org

1, 3, 8, 12, 23, 26, 32, 66, 68, 78, 83, 106, 116, 169, 181, 201, 210, 216, 234, 273, 282, 296, 427, 436, 501, 504, 513, 538, 547, 583, 655, 688, 711, 738, 751, 851, 866, 947, 1065, 1088, 1155, 1274, 1277, 1285, 1350, 1369, 1389, 1456, 1594, 1615, 1702, 1734

Offset: 1

Daniel Tisdale, Mar 15 2012

nonn

A k-almost prime has exactly k prime factors, repetitions included.
Conjecture: Each sequence b_n repeats indefinitely. (Example: for n=3, b_n = 9, 8, 8, 8, 8, 8, ....  It looks like b_3(k) is 8 for all k > 1.)
The conjecture follows from the formula that uses A078843 below (and the strict monotonicity of A078843). However the first repeated value is not for every n the value that repeats indefinitely. For example a(8) = b_8(2) = b_8(3) = 66, but b_8(k) = 64 for k >= 4. - Peter Munn, Aug 05 2019

			for k = 1, 2, 3, 4, 5, 6, ...:
p_k(3) = 5, 9, 18, 36, 72, 144, ... (the 3rd k-almost prime);
p_k(3)^2 = 25, 81, 324, 1296, 5184, 20736, ...;
b_3(k) = 9, 8, 8, 8, 8, 8, ... (index in the 2k-almost primes);
so since b_3(3) = b_3(2) = 8, a(3) = 8.

Cf. A058933, A078840, A078843, A091538.

get_p(m,k) = {local(i,n);i=0;n=1;while(iA209934(n) = {local(m,k,k_old);m=3;k_old=get_k(2,get_p(1,n)^2);k=get_k(4,get_p(2,n)^2);while(kMichael B. Porter, Mar 20 2012

From Peter Munn, Aug 05 2019: (Start)
b_n(k) = A058933(A078840(k,n)^2).
a(n) = b_n(min {k : b_n(k) = b_n(k+1)}).
If n < A078843(k+1) and b_n(k) < A078843(2k+1) then b_n(i) = b_n(k) for i >= k.
(End)

Edited, correcting the subscripting, by Peter Munn, Aug 04 2019

A289173 The largest n-almost prime less than 3^n.

Original entry on oeis.org

2, 6, 20, 60, 208, 624, 2080, 6240, 18720, 58240, 176000, 529408, 1593344, 4780032, 14344192, 43040768, 129138688, 387416064, 1162248192, 3486777344, 10460332032, 31380996096, 94142988288, 282428964864, 847286894592, 2541860683776, 7625582051328

Offset: 1

Zak Seidov, Jun 26 2017

nonn

All terms are even as 3^n is the first odd n-almost prime.

			a(26) = 2541860683776 = 3^26 - 5144553 = 2^18*3^6*47*283 (a 26-almost prime).
From _Michael De Vlieger_, Jun 27 2017: (Start)
Table of prime factors of a(n) for 1 <= n <= 16:
   1:  2
   2:  2   3
   3:  2   2   5
   4:  2   2   3   5
   5:  2   2   2   2  13
   6:  2   2   2   2   3  13
   7:  2   2   2   2   2   5  13
   8:  2   2   2   2   2   3   5  13
   9:  2   2   2   2   2   3   3   5  13
  10:  2   2   2   2   2   2   2   5   7  13
  11:  2   2   2   2   2   2   2   5   5   5  11
  12:  2   2   2   2   2   2   2   2   2   2  11  47
  13:  2   2   2   2   2   2   2   2   2   2   2   2 389
  14:  2   2   2   2   2   2   2   2   2   2   2   2   3 389
  15:  2   2   2   2   2   2   2   2   2   2   2   2   2  17 103
  16:  2   2   2   2   2   2   2   2   2   2   2   2   2   2  37  71(End)

Jon E. Schoenfield, Table of n, a(n) for n = 1..100

Cf. A078843 (where 3^n occurs in n-almost primes).

Table[SelectFirst[Range[3^n - 1, 2^n, -1], PrimeOmega@ # == n &], {n, 18}] (* Michael De Vlieger, Jun 27 2017 *)

for (n = 1,26, m = 3^n-1; while(bigomega(m) <> n, m = m-2); print1 (m ","))

a(n)=my(target=n-1); forstep(k=3^n\2,1,-1, if(bigomega(k)==target, return(2*k))) \\ Charles R Greathouse IV, Jul 05 2017

from math import prod, isqrt
from sympy import primepi, primerange, integer_nthroot
def A289173(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def almostprimepi(n,k):
        if k==0: return int(n>=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)))
        return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n,0,1,1,k)) if k>1 else primepi(n))
    m = almostprimepi(3**n-1,n)
    def f(x): return m+x-almostprimepi(x,n)
    return bisection(f,m,m) # Chai Wah Wu, Mar 29 2025

a(27) from Jon E. Schoenfield, Jul 02 2017

A337219 a(n) is the least positive number k such that 3^n+k has n prime factors counted with multiplicity.

Original entry on oeis.org

2, 1, 1, 3, 9, 7, 21, 63, 157, 471, 5, 15, 45, 135, 405, 1215, 3645, 10935, 32805, 98415, 295245, 885735, 2657205, 4409119, 2741597, 8224791, 16285765, 15302863, 45908589, 137725767, 77632981, 232898943, 161825917, 485477751, 1456433253, 3027122479, 1565174669

Offset: 1

Zak Seidov, Sep 14 2020

nonn

David A. Corneth, Table of n, a(n) for n = 1..56

Cf. A078843.

a[n_] := Module[{k = 1}, While[PrimeOmega[3^n + k] != n, k++]; k]; Array[a, 20] (* Amiram Eldar, Sep 18 2020 *)

a(n) = for(k=1, oo, if(bigomega(3^n+k)==n,return(k))); \\ Daniel Suteu, Oct 17 2020

a(27)-a(37) from Daniel Suteu, Sep 14 2020

Definition edited by Zak Seidov, Oct 17 2020

A376480 a(n) is the least k such that the sum of the first k numbers with n prime factors, counted with multiplicity, is prime.

Original entry on oeis.org

1, 3, 6, 8, 15, 24, 68, 68, 103, 179, 280, 432, 681, 1078, 1705, 2630, 4110, 6414, 10029, 15611, 24297, 37746, 58506, 90631, 140203, 216630, 334543, 516159, 795637, 1225649, 1886573, 2901816, 4460387, 6851543, 10518523, 16138688

Offset: 1

Robert Israel, Sep 24 2024

nonn

For n >=2, a(n) >= A078843(n), as for k < A078843(n) the sum of the first k is even. a(n) = A078843(n) for n = 2, 4, 9, 18, ...

			a(3) = 6 because the sum of the first 6 triprimes is 8 + 12 + 18 + 20 + 27 + 28 = 113 which is prime, and none of the previous partial sums is prime.

Cf. A001222, A078843, A376481.

f:= proc(n)
uses priqueue;
 local pq, t, s, count,v, w, p, i;
 initialize(pq);
 insert([-2^n, [2$n]],pq);
 s:= 0;
 for count from 1 do
   t:= extract(pq);
   v:= -t[1];
   w:= t[2];
   s:= s+v;
   if isprime(s) then return count fi;
   p:= nextprime(w[-1]);
   for i from n to 1 by -1 while w[i] = w[n] do
      insert([t[1]*(p/w[-1])^(n+1-i),[op(w[1..i-1]),p$(n+1-i)]],pq);
 od od;
end proc:
map(f, [$1..36]);

cp's OEIS Frontend

A116428 The number of n-almost primes less than or equal to 8^n, starting with a(0)=1.

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A116429 The number of n-almost primes less than or equal to 9^n, starting with a(0)=1.

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A116431 The number of n-almost primes less than or equal to 12^n, starting with a(0)=1.

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

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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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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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A209934 a(n) is the first value to occur consecutively in the sequence b_n defined by p_2k(b_n(k)) = p_k(n)^2, k=1,2,3,..., where p_k(n) is the n-th k-almost prime.

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A289173 The largest n-almost prime less than 3^n.

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