A125527 -id:A125527 - cp's OEIS frontend

A120033 Number of semiprimes s such that 2^n < s <= 2^(n+1).

0, 1, 1, 4, 4, 12, 20, 40, 75, 147, 285, 535, 1062, 2006, 3918, 7548, 14595, 28293, 54761, 106452, 206421, 401522, 780966, 1520543, 2962226, 5777162, 11272279, 22009839, 43006972, 84077384, 164482781, 321944211, 630487562, 1235382703

Offset: 0

Jonathan Vos Post and Robert G. Wilson v, Mar 20 2006

nonn

The partial sum equals the number of Pi_2(2^n) = A125527(n).

			(2^2, 2^3] there is one semiprime, namely 6. 4 was counted in the previous entry.

Dana Jacobsen, Table of n, a(n) for n = 0..62 (first 48 terms from Charles R Greathouse IV, corrected a(47)-a(48))

Cf. A001358, A072000, A066265, A036378, A120033-A120043.

SemiPrimePi[n_] := Sum[PrimePi[n/Prime[i]] - i + 1, {i, PrimePi[Sqrt[n]]}]; t = Table[SemiPrimePi[2^n], {n, 0, 35}]; Rest@t - Most@t

pi2(n)=my(s,i); forprime(p=2, sqrt(n), s+=primepi(n\p); i++); s - i * (i-1)/2
a(n)=pi2(2^(n+1))-pi2(2^n) \\ Charles R Greathouse IV, May 15 2012

use ntheory ":all"; print "$ ",semiprime_count(1+(1<<$), 1<<($+1)),"\n" for 0..48; # _Dana Jacobsen, Mar 04 2019

use ntheory ":all"; my $l=0; for (0..48) { my $c=semiprime_count(1<<($+1)); print "$ ",$c-$l,"\n"; $l=$c; } # Dana Jacobsen, Mar 04 2019

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

A127396 Number of 3-almostprimes <= 2^n.

Original entry on oeis.org

0, 0, 1, 2, 7, 13, 30, 60, 125, 256, 513, 1049, 2082, 4214, 8401, 16771, 33427, 66550, 132405, 262865, 522296, 1036033, 2055256, 4075039, 8078110, 16008485, 31720903, 62847087, 124501149, 246638355, 488559079, 967785236, 1917099175, 3797688543

Offset: 1

Robert G. Wilson v, Dec 29 2006

Robert G. Wilson v, Table of n, a(n) for n = 1..53
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A126279, A109251, A125527.

ThreeAlmostPrimePi[n_] := Sum[PrimePi[n/(Prime@i*Prime@j)] - j + 1, {i, PrimePi[n^(1/3)]}, {j, i, PrimePi[Sqrt[n/Prime@i]]}]; Table[ ThreeAlmostPrimePi[2^n], {n, 30}]

a(n) = A072114(2^n). - R. J. Mathar, Aug 26 2011

A146168 Number of odd squarefree semiprimes (A046388) < 2^n.

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 20, 46, 96, 197, 404, 798, 1599, 3134, 6169, 12093, 23640, 46199, 90180, 176198, 343927, 671783, 1312304, 2564485, 5012807, 9803883, 19181677, 37545265, 73524262, 144038812, 282313035, 553557959, 1085860455, 2130904274, 4183364732, 8215861037

Offset: 1

Washington Bomfim, Oct 27 2008

nonn

			a(5) = 2. The odd squarefree semiprimes less than 2^5 are 15 and 21. The formula gives 10 - pi(5) - pi(2^4) + 1 = 2.

Amiram Eldar, Table of n, a(n) for n = 1..57 (calculated using the b-files at A060967, A007053 and A125527)

Cf. A046388, A001358 (semiprimes), A000720 (pi(n), the number of primes <= n), A007053 (number of primes <= 2^n), A060967, A125527 (number of semiprimes <= 2^n).

Table[lim=2^n; Sum[PrimePi[lim/p]-PrimePi[p], {p, Prime[Range[2,PrimePi[Sqrt[lim]]]]}], {n,20}]

a(n) = A125527(n) - A060967(n) - A007053(n-1) + 1, for n > 1.

a(34) onwards from Amiram Eldar, Sep 05 2024

A334069 Number of numbers <= 2^n that are the product of exactly four primes, not necessarily distinct.

Original entry on oeis.org

0, 0, 0, 1, 2, 7, 14, 34, 71, 152, 325, 669, 1405, 2866, 5931, 12139, 24782, 50444, 102458, 207945, 420511, 850518, 1716168, 3460304, 6968639, 14022029, 28189833, 56631732, 113697179, 228115641, 457456902, 916899721, 1836996851, 3678943569, 7365141297, 14740076678, 29490954290

Offset: 1

Robert G. Wilson v, Apr 13 2020

nonn

			a(6) = 7 because
  16 = 2 * 2 * 2 * 2,
  24 = 2 * 2 * 2 * 3,
  36 = 2 * 2 * 3 * 3,
  40 = 2 * 2 * 2 * 5,
  54 = 2 * 3 * 3 * 3,
  56 = 2 * 2 * 2 * 7, and
  60 = 2 * 2 * 3 * 5
are the seven numbers less than 2^6 = 64 that are each the product of four primes.

Robert G. Wilson v, Table of n, a(n) for n = 1..53
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A007053, A014613, A082996, A125527, A127396, A126279.

Partial sums of A120035.

FourAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j*Prime@k)] - k + 1, {i, PrimePi[n^(1/4)]}, {j, i, PrimePi[(n/Prime@i)^(1/3)]}, {k, j, PrimePi@Sqrt[n/(Prime@i*Prime@j)]}]; Array[FourAlmostPrimePi[2^#] &, 37]

a(n) = A082996(2^n).

A175613 Number of semiprimes <= 2^prime(n).

Original entry on oeis.org

1, 2, 10, 42, 589, 2186, 30253, 113307, 1608668, 88157689, 336717854, 19015826478, 282528883551, 1091574618496, 16360940729894

Offset: 1

Juri-Stepan Gerasimov, Dec 04 2010

			a(2)=2 because first 2 semiprimes are 4, 6 both <2^prime(2)=8.

Cf. A001358, A007053, a proper subset of A125527.

(* First run program given in A072000 to define the SemiPrimePi function *) Table[SemiPrimePi[2^Prime[n]], {n, 10}](* Alonso del Arte, Dec 10 2010 *)

a(n)=my(N=2^prime(n),s,i); forprime(p=2, sqrtint(N), s+=primepi(N\p); i++); s - i * (i-1)/2 \\ Charles R Greathouse IV, Apr 25 2016

from math import isqrt
from sympy import prime, primepi
def A175613(n):
    m = 1<Chai Wah Wu, Jul 23 2024

a(n) = A072000(A034785(n)) = A125527(A000040(n)). - R. J. Mathar, Dec 10 2010

a(14) & a(15) from Robert G. Wilson v, Oct 19 2011.

A259172 Numbers in A259145 that are neither prime nor semiprime.

Original entry on oeis.org

561, 595, 1105, 1235, 1245, 1495, 1547, 1885, 2405, 2555, 2717, 2849, 3115, 3495, 3655, 3657, 3689, 3815, 4521, 4795, 4945, 5035, 5385, 5395, 5453, 5457, 5709, 5865, 6083, 6141, 6251, 6285, 6365, 6391, 6501, 6695, 6755, 6969, 7021, 7887, 8113, 8255, 8355

Offset: 1

Carlos Eduardo Olivieri, Jun 19 2015

Regarding the distribution: Let K be the union of primes and semiprimes in A259145. Let S be the set of other terms. The growth rate of the cardinality of S with respect to the cardinality of K is significantly slower. For instance, if we take the first 50000 terms of A259145, about 32.5 percent are contained in S. If we take the first 350000 terms, about 38.2 percent are contained in S.
a(n) that are in A002997 (Carmichael numbers) for a(n) <= 10^6 are 561, 1105, 8911, 10585, 29341, 825265.
a(n) that are in A051015 (Zeisel numbers) for a(n) <= 3*10^6 are 1885, 353977, 2953711.

Carlos Eduardo Olivieri, Table of n, a(n) for n = 1..8906
Eric Naslund, Integers with a predetermined prime factorization, arXiv:1203.2363 [math.NT], 2012.

Cf. A001221, A001358, A002997, A007053, A051015, A125527, A259145.

Subsequence of A000469, A033942, A050384 (conjuctered).

Select[Range[25000], PrimeQ[#^2 - EulerPhi[#]] && PrimeNu[#] > 2 &]

A001221(a(n)) > 2.

A000005(a(n)) = 2^k, k >= 3.

A362042 Number of odd semiprimes less than 2^n.

Original entry on oeis.org

0, 0, 0, 0, 2, 4, 11, 24, 51, 103, 207, 417, 815, 1622, 3164, 6210, 12146, 23711, 46295, 90307, 176369, 344155, 672091, 1312721, 2565048, 5013566, 9804910, 19183069, 37547164, 73526846, 144042323, 282317826, 553564500, 1085869406, 2130916524, 4183381508, 8215884036

Offset: 0

Sidney Cadot, Apr 15 2023

nonn

Odd numbers with two prime factors are used as the modulus in the RSA algorithm. This sequence shows the growth of the number of 'candidate' RSA moduli for keys up to a given number of bits.

			For n=5, there are four integers less than 32 (i.e., 2^5) that are the product of two odd primes: {3*3, 3*5, 3*7, 5*5} = {9, 15, 21, 25}; hence, a(5)=4.

Cf. A046315, A007053, A085770, A125527.

a[n_]:=Length@Select[Range[1, 2^n - 1, 2], Total[Last /@ FactorInteger[#]] == 2 &]
Table[a[n],{n,0,24}]

a(n) = A125527(n) - A007053(n-1) for n > 0. - Jinyuan Wang, Apr 16 2023

More terms from Jinyuan Wang, Apr 16 2023

A146169 Percentage (rounded) of semiprimes <= 2^n which are odd and squarefree.

Original entry on oeis.org

0, 0, 17, 20, 36, 48, 56, 61, 65, 69, 71, 73, 75, 76, 77, 78, 79, 80, 80, 81, 81, 82, 82, 82, 83, 83, 83, 83, 84, 84, 84, 84

Offset: 2

Washington Bomfim, Oct 27 2008

nonn

More than 84% of the semiprimes in the interval [4, 2^32] are odd and squarefree. This percentage appears to rise indefinitely as n grows.

a(n) = 100 for all n > N. What is the least such N? - Charles R Greathouse IV, May 12 2013

			a(5)= 20 since the interval [4, 2^5] contains 10 semiprimes, namely 4,6,9,10,14,15,21,22,25 and 26; and two of those semiprimes, (15 and 21), are odd and squarefree.

Cf. A001358(semiprimes), A125527(Number of semiprimes <= 2^n), A146168(Number of odd squarefree semiprimes < 2^n).

a(n)=my(s,i,N=2^n); forprime(p=2, sqrtint(N), s+=primepi(N\p); i++); s-=i*(i-1)/2; i=primepi(sqrtint(N))+primepi(N/2)-1; round(100*(s-i)/s) \\ Charles R Greathouse IV, May 12 2013

a(n) = round(A146168(n)/A125527(n)*100)

cp's OEIS Frontend

A120033 Number of semiprimes s such that 2^n < s <= 2^(n+1).

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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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A127396 Number of 3-almostprimes <= 2^n.

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A146168 Number of odd squarefree semiprimes (A046388) < 2^n.

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A334069 Number of numbers <= 2^n that are the product of exactly four primes, not necessarily distinct.

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A175613 Number of semiprimes <= 2^prime(n).

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A259172 Numbers in A259145 that are neither prime nor semiprime.

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A362042 Number of odd semiprimes less than 2^n.

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