A072000 -id:A072000 - cp's OEIS frontend

A373630 a(n) is the n-th semiprime after the n-th prime.

4, 6, 10, 15, 25, 26, 35, 38, 49, 57, 58, 74, 85, 86, 91, 95, 118, 119, 123, 133, 134, 143, 146, 161, 183, 185, 187, 201, 202, 205, 218, 221, 237, 247, 265, 267, 278, 295, 299, 302, 309, 314, 326, 327, 334, 335, 362, 393, 395, 398, 403, 413, 415, 427, 446, 453, 466, 469, 473, 481, 482, 497, 519

Offset: 1

Zak Seidov and Robert Israel, Jun 11 2024

nonn

			a(5) = 25 because the 5th prime is 11 and the first 5 semiprimes > 11 are 14,15,21,22,25.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A072000.

N:= 10^4: # for terms <= N
P:= select(isprime,[2,seq(i,i=3..N,2)]):
S:= select(t -> numtheory:-bigomega(t)=2, [$1..N]): nS:= nops(S):
f:= proc(n) local j;
  j:= ListTools:-BinaryPlace(S,P[n]);
  if j + n <= nS then S[j+n] else fail fi
end proc:
R:= NULL:
for i from 1 do
  v:= f(i);
  if v = fail then break fi;
  R:= R,v
od:
R;

seq={};Do[i=Prime[n]+1;cnt=0;While[cntJames C. McMahon, Jun 15 2024 *)

from math import isqrt
from sympy import primepi, prime
def A373630(n):
    p = prime(n)
    q = n+int(sum(primepi(p//prime(k))-k+1 for k in range(1,primepi(isqrt(p))+1)))
    def f(x): return int(q+x-sum(primepi(x//prime(k))-k+1 for k in range(1, primepi(isqrt(x))+1)))
    m, k = q, f(q)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Jul 23 2024

a(n) = A001358(n + A072000(A000040(n))).

A101041 Number of numbers not greater than n having no more than two prime factors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, 14, 15, 15, 16, 17, 18, 18, 19, 20, 20, 20, 21, 21, 22, 22, 23, 24, 25, 25, 26, 27, 28, 28, 29, 29, 30, 30, 30, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 38, 39, 39, 40, 41, 41, 41, 42, 42, 43, 43, 44, 44, 45, 45, 46

Offset: 1

Reinhard Zumkeller, Nov 28 2004

nonn

a(n) = 1 + Sum_{k=1..n} A101040(k);
asymptotically: a(n) ~ n*log(log(n))/log(n).
Primes counted with multiplicity. - Harvey P. Dale, Feb 16 2024

Cf. A000720, A072000, A101040.

Accumulate[Table[If[PrimeOmega[n]<3,1,0],{n,80}]] (* Harvey P. Dale, Feb 16 2024 *)

a(n) = A000720(n) + A072000(n) + 1.

A140234 Sum of the semiprimes <= n.

Original entry on oeis.org

0, 0, 0, 0, 4, 4, 10, 10, 10, 19, 29, 29, 29, 29, 43, 58, 58, 58, 58, 58, 58, 79, 101, 101, 101, 126, 152, 152, 152, 152, 152, 152, 152, 185, 219, 254, 254, 254, 292, 331, 331, 331, 331, 331, 331, 331, 377, 377, 377, 426, 426, 477, 477, 477, 477, 532, 532, 589

Offset: 0

Jonathan Vos Post, May 13 2008

This is to semiprimes A001358 as A034387 is to primes A000040. From the prime number theorem A034387(n) has the asymptotic expression: a(n) ~ n^2 / (2 log n), so what is the asymptotic expression for a(n)?

Cf. A001358, A034387, A051352, A062198, A072000, A100959, A140235.

a[n_]:=Total[Select[Range[n],PrimeOmega[#]==2&]];Array[a,58,0] (* James C. McMahon, Jul 06 2025 *)

a(n) = Sum_{j such that j is in A001358 and j<=n} = A062198(A072000(n)).

A140235 Partial sum of non-semiprimes A100959.

Original entry on oeis.org

1, 3, 6, 11, 18, 26, 37, 49, 62, 78, 95, 113, 132, 152, 175, 199, 226, 254, 283, 313, 344, 376, 412, 449, 489, 530, 572, 615, 659, 704, 751, 799, 849, 901, 954, 1008, 1064, 1123, 1183, 1244, 1307, 1371, 1437, 1504, 1572, 1642, 1713, 1785, 1858, 1933, 2009

Offset: 1

Jonathan Vos Post, May 13 2008

This is to semiprimes A001358 as A051352 is to primes A000040. Equivalently, this is to non-semiprimes A100959 as A051349 is to nonprimes A018252.

			a(5) = 18 = 1 + 2 + 3 + 5 + 7.

Cf. A000217, A001358, A018252, A034387, A051349, A051352, A062198, A072000, A100959, A140234.

Accumulate[Select[Range[100],PrimeOmega[#]!=2&]] (* Harvey P. Dale, Aug 22 2021 *)

a(n) = Sum{k=1..n} A100959(k).

Corrected and edited by Giovanni Resta, Jun 20 2016

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.

A186637 Semiprime powers with special exponents: k^(j-1) where both j and k are arbitrary semiprime numbers.

Original entry on oeis.org

64, 216, 729, 1000, 1024, 2744, 3375, 7776, 9261, 10648, 15625, 17576, 35937, 39304, 42875, 54872, 59049, 59319, 65536, 97336, 100000, 117649, 132651, 166375, 185193, 195112, 238328, 262144, 274625, 328509, 405224, 456533, 537824, 551368, 614125, 636056, 658503, 753571, 759375, 804357, 830584, 857375

Offset: 1

Jonathan Vos Post, Feb 24 2011

Semiprime analog of A036454: prime powers with special exponents: q^(p-1) where both p and q are arbitrary prime numbers.

			a(1) = smallest semiprime to power of (smallest semiprime - 1) = 4^(4-1) = 4^3 = 64.

Cf. A001358, A036454, A113877, A186621.

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, factorint
def A186637(n):
    def A072000(n): return int(-((t:=primepi(s:=isqrt(n)))*(t-1)>>1)+sum(primepi(n//p) for p in primerange(s+1)))
    def f(x): return int(n+x-sum(A072000(integer_nthroot(x, p-1)[0]) for p in range(4,x.bit_length()+1) if sum(factorint(p).values())==2))
    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,n,n) # Chai Wah Wu, Sep 12 2024

{a(n)} = {A001358(i) ^ A186621(j)}.

{a(n)} = {a^b where a and b are elements of A001358} = {(p*q)^((r*s)-1) for primes p, q, r, s, not necessarily distinct}.

A192394 Number of semiprimes in the range (prime(n), prime(n)+sqrt(prime(n))).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 3, 2, 2, 2, 2, 2, 4, 4, 4, 1, 2, 2, 2, 4, 6, 3, 4, 5, 5, 4, 4, 5, 2, 3, 4, 3, 4, 4, 6, 6, 7, 7, 3, 2, 2, 3, 4, 4, 5, 4, 4, 2, 3, 4, 4, 6, 8, 5, 6, 7, 7, 5, 4, 5, 5, 6, 5, 5, 5, 8

Offset: 1

Juri-Stepan Gerasimov, Jun 29 2011

nonn

			a(1)=0 because there are no semiprimes in the range (2, 2+sqrt(2));
a(2)=1 because there is one semiprime (4) in the range (3, 3+sqrt(3)).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358.

A192394 := proc(n) local a,p,s; a := 0 ; p := ithprime(n) ; for s from p to floor( p+sqrt(p)) do if isA001358(s) then a := a+1 ;  end if; end do: a; end proc: # R. J. Mathar, Jul 01 2011

(* First run A072000 to define semiPrimePi *) Table[semiPrimePi[Prime[n] + Sqrt[Prime[n]]] - semiPrimePi[Prime[n]], {n, 75}] (* Alonso del Arte, Jul 01 2011 *)
Table[Count[Range[p,p+Sqrt[p]],?(PrimeOmega[#]==2&)],{p,Prime[Range[80]]}] (* _Harvey P. Dale, Feb 10 2025 *)

Corrected by R. J. Mathar, Jul 01 2011

A216593 Number of semiprimes among n-th million natural numbers.

Original entry on oeis.org

210035, 197249, 193162, 190540, 188288, 187308, 185657, 184788, 183856, 183441, 182123, 181556, 181125, 180878, 180234, 179649, 179055, 178710, 178652, 178034, 178015, 177307, 177391, 177003, 176568, 176419, 176021, 175788, 175655, 175189, 174915, 175357

Offset: 1

Zak Seidov, Sep 09 2012

nonn

Let f(m) = number of semiprimes

a(n) = 0 for almost all n. It seems infeasible to find the first such n. - Charles R Greathouse IV, Sep 09 2012

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A001358, A036352, A064911, A066265, A072000.

f[m_] := Sum[ PrimePi[(m - 1)/Prime[i]], {i, PrimePi[ Sqrt[m]]}] - Binomial[ PrimePi[ Sqrt[m]], 2]; ta=Table[f[n*10^6],{n,0,1000}];s=Rest[ta]-Most[ta] (* for first 1000 terms *)
(* using Mmca code by Robert G. Wilson v in A066265 - Zak Seidov, Sep 09 2012 *)

a(n)=sum(k=10^6*(n-1),10^6*n,bigomega(k)==2) \\ Charles R Greathouse IV, Sep 09 2012

a(n) ~ 1000000 n log log n / log n. - Charles R Greathouse IV, Sep 23 2012

A217908 Semiprime powers of distinct semiprimes.

Original entry on oeis.org

1296, 4096, 6561, 10000, 38416, 50625, 194481, 234256, 262144, 390625, 456976, 531441, 1000000, 1048576, 1185921, 1336336, 1500625, 2085136, 2313441, 4477456, 5764801, 6765201, 7529536, 9150625, 10077696, 10556001, 11316496, 11390625, 14776336, 17850625

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Mar 25 2013

nonn

Subset of A113877.

			6561=9^4, and 9 and 4 are both semiprime. 46656 = 6^6 is excluded because the semiprimes are not distinct.

Christian N. K. Anderson, Table of n, a(n) for n = 1..9006, for a(n) < 1.5*10^18

Cf. A113877.

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, factorint
def A217908(n):
    def A072000(n): return int(-((t:=primepi(s:=isqrt(n)))*(t-1)>>1)+sum(primepi(n//p) for p in primerange(s+1)))
    def f(x): return int(n+x-sum(A072000(integer_nthroot(x, p)[0])-(p**p<=x) for p in range(4,x.bit_length()) if sum(factorint(p).values())==2))
    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,n,n) # Chai Wah Wu, Sep 12 2024

A243903 Numbers n such that (number of primes <= n) is greater than or equal to (number of semiprimes <= n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 29, 30, 31, 32, 33

Offset: 1

Harvey P. Dale, Jun 14 2014

nonn

Conjecture: there are no additional terms.

We know from the asymptotic formulas (see Landau) that the sequence is finite. See also the graph of A243906. - N. J. A. Sloane, Jun 14 2014

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1, Teubner, Leipzig, 1909; third edition : Chelsea, New York (1974).

Cf. A000040, A000720, A001358, A072000, A243906.

With[{nn=5000},Flatten[Position[Thread[{Accumulate[Table[ If[ PrimeOmega[n] == 2,1,0],{n,nn}]],PrimePi[Range[nn]]}],_?(Last[#]>=First[#]&),{1}, Heads-> False]]]

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cp's OEIS Frontend

A373630 a(n) is the n-th semiprime after the n-th prime.

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A101041 Number of numbers not greater than n having no more than two prime factors.

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A140234 Sum of the semiprimes <= n.

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A140235 Partial sum of non-semiprimes A100959.

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

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A186637 Semiprime powers with special exponents: k^(j-1) where both j and k are arbitrary semiprime numbers.

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A192394 Number of semiprimes in the range (prime(n), prime(n)+sqrt(prime(n))).

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A216593 Number of semiprimes among n-th million natural numbers.

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