A072000 -id:A072000 - cp's OEIS frontend

A001358 Semiprimes (or biprimes): products of two primes.

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 169, 177, 178, 183, 185, 187

Offset: 1

N. J. A. Sloane, R. K. Guy

Numbers of the form p*q where p and q are primes, not necessarily distinct.
These numbers are sometimes called semiprimes or 2-almost primes.
Numbers n such that Omega(n) = 2 where Omega(n) = A001222(n) is the sum of the exponents in the prime decomposition of n.
Complement of A100959; A064911(a(n)) = 1. - Reinhard Zumkeller, Nov 22 2004
The graph of this sequence appears to be a straight line with slope 4. However, the asymptotic formula shows that the linearity is an illusion and in fact a(n)/n ~ log(n)/log(log(n)) goes to infinity. See also the graph of A066265 = number of semiprimes < 10^n.
For numbers between 33 and 15495, semiprimes are more plentiful than any other k-almost prime. See A125149.
Numbers that are divisible by exactly 2 prime powers (not including 1). - Jason Kimberley, Oct 02 2011
The (disjoint) union of A006881 and A001248. - Jason Kimberley, Nov 11 2015
An equivalent definition of this sequence is a'(n) = smallest composite number which is not divided by any smaller composite number a'(1),...,a'(n-1). - Meir-Simchah Panzer, Jun 22 2016
The above characterization can be simplified to "Composite numbers not divisible by a smaller term." This shows that this is the equivalent of primes computed via Eratosthenes's sieve, but starting with the set of composite numbers (i.e., complement of 1 union primes) instead of all positive integers > 1. It's easy to see that iterating the method (using Eratosthenes's sieve each time on the remaining numbers, complement of the previously computed set) yields numbers with bigomega = k for k = 0, 1, 2, 3, ..., i.e., {1}, A000040, this, A014612, etc. - M. F. Hasler, Apr 24 2019
For all n except n = 2, a(n) is a deficient number. - Amrit Awasthi, Sep 10 2024
It is reasonable to assume that the "comforting numbers" which John T. Williams found in Chapter 3 of Milne's book "The House at Pooh Corner" are these semiprimes. Winnie-the-Pooh wonders whether he has 14 or 15 honey pots and concludes: "It's sort of comforting." To arrange a semiprime number of honey pots in a rectangular way, let's say on a shelf, with the larger divisor parallel to the wall, there is only one solution and this is for a simple mind like Winnie-the-Pooh comforting. - Ruediger Jehn, Dec 12 2024

			From _Gus Wiseman_, May 27 2021: (Start)
The sequence of terms together with their prime factors begins:
   4 = 2*2     46 = 2*23     91 = 7*13    141 = 3*47
   6 = 2*3     49 = 7*7      93 = 3*31    142 = 2*71
   9 = 3*3     51 = 3*17     94 = 2*47    143 = 11*13
  10 = 2*5     55 = 5*11     95 = 5*19    145 = 5*29
  14 = 2*7     57 = 3*19    106 = 2*53    146 = 2*73
  15 = 3*5     58 = 2*29    111 = 3*37    155 = 5*31
  21 = 3*7     62 = 2*31    115 = 5*23    158 = 2*79
  22 = 2*11    65 = 5*13    118 = 2*59    159 = 3*53
  25 = 5*5     69 = 3*23    119 = 7*17    161 = 7*23
  26 = 2*13    74 = 2*37    121 = 11*11   166 = 2*83
  33 = 3*11    77 = 7*11    122 = 2*61    169 = 13*13
  34 = 2*17    82 = 2*41    123 = 3*41    177 = 3*59
  35 = 5*7     85 = 5*17    129 = 3*43    178 = 2*89
  38 = 2*19    86 = 2*43    133 = 7*19    183 = 3*61
  39 = 3*13    87 = 3*29    134 = 2*67    185 = 5*37
(End)

Archimedeans Problems Drive, Eureka, 17 (1954), 8.
Raymond Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter II, Problem 60.
Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 1, Teubner, Leipzig; third edition: Chelsea, New York (1974). See p. 211.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
John T. Williams, Pooh and the Philosophers, Dutton Books, 1995.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 10000 terms from T. D. Noe)
Dragos Crisan and Radek Erban, On the counting function of semiprimes, INTEGERS, Vol. 21 (2021), #A122.
Daniel A. Goldston, Sidney W. Graham, János Pintz and Cem Y. Yildirim, Small gaps between primes or almost primes, Transactions of the American Mathematical Society, Vol. 361, No. 10 (2009), pp. 5285-5330, arXiv preprint, arXiv:math/0506067 [math.NT], 2005.
Richard K. Guy, Letters to N. J. A. Sloane, June-August 1968
Sh. T. Ishmukhametov and F. F. Sharifullina, On distribution of semiprime numbers, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 53-59. English translation, Russian Mathematics, Vol. 58, No. 8 (2014), pp. 43-48, alternative link.
Donovan Johnson, Jonathan Vos Post, and Robert G. Wilson v, Selected n and a(n). (2.5 MB)
Dixon Jones, Quickie 593, Mathematics Magazine, Vol. 47, No. 3, May 1974, p. 167.
Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909. See Vol. 1, p. 211.
Xianmeng Meng, On sums of three integers with a fixed number of prime factors, Journal of Number Theory, Vol. 114, No. 1 (2005), pp. 37-65.
Michael Penn, What makes a number "good"?, YouTube video, 2022.
Carlos Rivera, Conjecture 108: On the counting function of semiprimes, The Prime Puzzles & Problems Connection.
Eric Weisstein's World of Mathematics, Semiprime.
Eric Weisstein's World of Mathematics, Almost Prime.
Wikipedia, Almost prime.
Robert G. Wilson v, Subsequences at various powers of 10.
Index to sequences related to sums of cubes
Index entries for "core" sequences

Cf. A064911 (characteristic function).
Cf. A048623, A048639, A000040 (primes), A014612 (products of 3 primes), A014613, A014614, A072000 ("pi" for semiprimes), A065516 (first differences).
Cf. A077554, A077555, A002024, A072966, A100592, A014673, A068318, A061299, A087718, A089994, A089995, A096916, A096932, A106550, A106554, A108541, A108542, A126663, A131284, A138510, A138511, A072931, A088183, A171963, A237040 (semiprimes of form n^3 + 1).
Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r=1), this sequence (r=2), A014612 (r=3), A014613 (r=4), A014614 (r=5), A046306 (r=6), A046308 (r=7), A046310 (r=8), A046312 (r=9), A046314 (r=10), A069272 (r=11), A069273 (r=12), A069274 (r=13), A069275 (r=14), A069276 (r=15), A069277 (r=16), A069278 (r=17), A069279 (r=18), A069280 (r=19), A069281 (r=20).
These are the Heinz numbers of length-2 partitions, counted by A004526.
The squarefree case is A006881 with odd/even terms A046388/A100484 (except 4).
Including primes gives A037143.
The odd/even terms are A046315/A100484.
Partial sums are A062198.
The prime factors are A084126/A084127.
Grouping by greater factor gives A087112.
The product/sum/difference of prime indices is A087794/A176504/A176506.
Positions of even/odd terms are A115392/A289182.
The terms with relatively prime/divisible prime indices are A300912/A318990.
Factorizations using these terms are counted by A320655.
The prime indices are A338898/A338912/A338913.
Grouping by weight (sum of prime indices) gives A338904, with row sums A024697.
The terms with even/odd weight are A338906/A338907.
The terms with odd/even prime indices are A338910/A338911.
The least/greatest term of weight n is A339114/A339115.
Cf. A014342, A098350, A112141, A320732, A332765, A339003/A339004, A339116.

a001358 n = a001358_list !! (n-1)
a001358_list = filter ((== 2) . a001222) [1..]

[n: n in [2..200] | &+[d[2]: d in Factorization(n)] eq 2]; // Bruno Berselli, Sep 09 2015

A001358 := proc(n) option remember; local a; if n = 1 then 4; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 2 then return a; end if; end do: end if; end proc:
seq(A001358(n), n=1..120) ; # R. J. Mathar, Aug 12 2010

Select[Range[200], Plus@@Last/@FactorInteger[#] == 2 &] (* Zak Seidov, Jun 14 2005 *)
Select[Range[200], PrimeOmega[#]==2&] (* Harvey P. Dale, Jul 17 2011 *)

select( isA001358(n)={bigomega(n)==2}, [1..199]) \\ M. F. Hasler, Apr 09 2008; added select() Apr 24 2019

list(lim)=my(v=List(),t);forprime(p=2, sqrt(lim), t=p;forprime(q=p, lim\t, listput(v,t*q))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Sep 11 2011

A1358=List(4); A001358(n)={while(#A1358M. F. Hasler, Apr 24 2019

from sympy import factorint
def ok(n): return sum(factorint(n).values()) == 2
print([k for k in range(1, 190) if ok(k)]) # Michael S. Branicky, Apr 30 2022

from math import isqrt
from sympy import primepi, prime
def A001358(n):
    def f(x): return int(n+x-sum(primepi(x//prime(k))-k+1 for k in range(1, primepi(isqrt(x))+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Jul 23 2024

a(n) ~ n*log(n)/log(log(n)) as n -> infinity [Landau, p. 211], [Ayoub].
Recurrence: a(1) = 4; for n > 1, a(n) = smallest composite number which is not a multiple of any of the previous terms. - Amarnath Murthy, Nov 10 2002
A174956(a(n)) = n. - Reinhard Zumkeller, Apr 03 2010
a(n) = A088707(n) - 1. - Reinhard Zumkeller, Feb 20 2012
Sum_{n>=1} 1/a(n)^s = (1/2)*(P(s)^2 + P(2*s)), where P is the prime zeta function. - Enrique Pérez Herrero, Jun 24 2012
sigma(a(n)) + phi(a(n)) - mu(a(n)) = 2*a(n) + 1. mu(a(n)) = ceiling(sqrt(a(n))) - floor(sqrt(a(n))). - Wesley Ivan Hurt, May 21 2013
mu(a(n)) = -Omega(a(n)) + omega(a(n)) + 1, where mu is the Moebius function (A008683), Omega is the count of prime factors with repetition, and omega is the count of distinct prime factors. - Alonso del Arte, May 09 2014
a(n) = A078840(2,n). - R. J. Mathar, Jan 30 2019
A100484 UNION A046315. - R. J. Mathar, Apr 19 2023
Conjecture: a(n)/n ~ (log(n)/log(log(n)))*(1-(M/log(log(n)))) as n -> oo, where M is the Mertens's constant (A077761). - Alain Rocchelli, Feb 02 2025

More terms from James Sellers, Aug 22 2000

A064911 If n is semiprime (or 2-almost prime) then 1 else 0.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Patrick De Geest, Oct 13 2001

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65545 (first 10000 terms from Reinhard Zumkeller)
Eric Weisstein's World of Mathematics, Semiprime
Eric Weisstein's World of Mathematics, Prime zeta function primezeta(s).
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n

Cf. A010051, A064899-A064910, A053409, A046413, A101040, A105700, A280710, A302047, A302048, A302049, A353475, A353476, A353477, A353478, A353479, A353480, A353481.

a064911 = a010051 . a032742 -- Reinhard Zumkeller, Mar 13 2011

with(numtheory):
a:= n-> `if`(bigomega(n)=2, 1, 0):
seq(a(n), n=1..120);  # Alois P. Heinz, Mar 16 2011

Table[If[PrimeOmega[n] == 2, 1, 0], {n, 105}] (* Jayanta Basu, May 25 2013 *)

a(n)=bigomega(n)==2 \\ Charles R Greathouse IV, Mar 13 2011

a(n) = 1 iff n is in A001358 (semiprimes), a(n) = 0 iff n is in A100959 (non-semiprimes). - Reinhard Zumkeller, Nov 24 2004
Dirichlet g.f.: (primezeta(2s) + primezeta(s)^2)/2. - Franklin T. Adams-Watters, Jun 09 2006
a(n) = A057427(A174956(n)); a(n)*A072000(n) = A174956(n). - Reinhard Zumkeller, Apr 03 2010
a(n) = A010051(A032742(n)) (i.e., largest proper divisor is prime). - Reinhard Zumkeller, Mar 13 2011
From Antti Karttunen, Apr 24 2018 & Apr 22 2022: (Start)
a(n) = A280710(n) + A302048(n) = A101040(n) - A010051(n).
a(n) = A353478(n) + A353480(n) = A353477(n) + A353478(n) + A353479(n).
a(n) = A353475(n) + A353476(n).
(End)
a(n) = [Omega(n) = 2], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jul 22 2025

Edited by M. F. Hasler, Oct 18 2017

A066265 a(n) = number of semiprimes < 10^n.

Original entry on oeis.org

0, 3, 34, 299, 2625, 23378, 210035, 1904324, 17427258, 160788536, 1493776443, 13959990342, 131126017178, 1237088048653, 11715902308080, 111329817298881, 1061057292827269, 10139482913717352, 97123037685177087, 932300026230174178, 8966605849641219022, 86389956293761485464, 833671466551239927908, 8056846659984852885191

Offset: 0

Patrick De Geest, Dec 10 2001

nonn

Apart from the first nonzero term the sequence is identical to A036352. - Hugo Pfoertner, Jul 22 2003

			Below 10 there are three semiprimes: 4 (2*2), 6 (2*3) and 9 (3*3).

Eric Weisstein's World of Mathematics, Semiprime
Index entries for sequences related to numbers of primes in various ranges

Cf. A001358, A064911, A072000, A036352 (identical starting from a(2)), A220262, A292785.

f[n_] := Sum[ PrimePi[(10^n - 1)/Prime[i]], {i, PrimePi[ Sqrt[10^n]]}] - Binomial[ PrimePi[ Sqrt[10^n]], 2]; Do[ Print[ f[n]], {n, 0, 14}] (* Robert G. Wilson v, May 16 2005 *)
SemiPrimePi[n_] := Sum[ PrimePi[n/Prime@ i] - i + 1, {i, PrimePi@ Sqrt@ n}]; Array[ SemiPrimePi[10^# - 1] &, 14, 0] (* Robert G. Wilson v, Jan 21 2015 *)

a(n)=my(s);forprime(p=2,sqrt(10^n),s+=primepi((10^n-1)\p)); s-binomial(primepi(sqrt(10^n)),2) \\ Charles R Greathouse IV, Apr 23 2012

use Math::Prime::Util qw/:all/; use integer; sub countsp { my($k,$sum,$pc)=($[0]-1,0,1); prime_precalc(60_000_000); forprimes { $sum += prime_count($k/$) + 1 - $pc++; } int(sqrt($k)); $sum; } foreach my $n (0..16) { say "$n: ", countsp(10**$n); } # Dana Jacobsen, May 11 2014

from math import isqrt
from sympy import primepi, primerange
def A066265(n): return int((-(t:=primepi(s:=isqrt(m:=10**n)))*(t-1)>>1)+sum(primepi(m//k) for k in primerange(1, s+1))) if n>1 else 3*n # Chai Wah Wu, Aug 16 2024

(1/2)*( pi(10^(n/2)) + Sum_{i=1..pi(10^n)} pi( (10^n-1)/P_i) ) = Sum_{i=1..pi(sqrt(10^n))} pi( (10^n-1)/P_i ) - binomial( pi(sqrt(10^n)), 2). - Robert G. Wilson v, May 16 2005

More terms from Hugo Pfoertner, Jul 22 2003
a(14) from Robert G. Wilson v, May 16 2005
a(15)-a(16) from Donovan Johnson, Mar 18 2010
a(17)-a(18) from Dana Jacobsen, May 11 2014
a(19)-a(21) from Henri Lifchitz, Jul 04 2015
a(22)-a(23) from Henri Lifchitz, Nov 09 2024

A128301 Indices of squares (of primes) in the semiprimes.

Original entry on oeis.org

1, 3, 9, 17, 40, 56, 90, 114, 164, 253, 289, 404, 484, 533, 634, 783, 973, 1031, 1233, 1373, 1452, 1683, 1842, 2112, 2483, 2676, 2779, 2995, 3108, 3320, 4124, 4384, 4775, 4926, 5593, 5741, 6172, 6644, 6962, 7448, 7955, 8108, 8978, 9147, 9512, 9697, 10842

Offset: 1

Rick L. Shepherd, Feb 25 2007

nonn

A001358(a(n)) = A001248(n) = A000040(n)^2.

Numbers n with property that tau(semiprime(n)) is not semiprime. - Juri-Stepan Gerasimov, Oct 15 2010

			a(4) = 17 as 49 = 7^2 = prime(4)^2, the fourth square in the semiprimes, is the seventeenth semiprime.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A128302, A001358, A001248, A072000.

With[{sp=Select[Range[50000],PrimeOmega[#]==2&]},Flatten[Table[ Position[ sp,Prime[ n]^2],{n,Floor[Sqrt[Length[sp]]]}]]] (* Harvey P. Dale, Nov 17 2014 *)

a(n)=my(s=0,i=0); n=prime(n)^2; forprime(p=2, sqrt(n), s+=primepi(n\p); i++); s - i * (i-1)/2
\\ Charles R Greathouse IV, Apr 21 2011

-MMath::Pari=factorint,PARI -wle 'my $c = 0; my $s = PARI 1; while (1) { ++$s; my($sp, $si) = @{factorint($s)}; next if @$sp > 2; next if $si->[0] + (@$si > 1 ? $si->[1] : 0) != 2; ++$c; print "$s => $c" if @$sp == 1}' # Hugo van der Sanden, Sep 25 2007

from math import isqrt
from sympy import prime, primepi
def A128301(n):
    m = prime(n)**2
    return int(sum(primepi(m//prime(k))-k+1 for k in range(1,n+1))) # Chai Wah Wu, Jul 23 2024

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

Original entry on oeis.org

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

A100949 Number of partitions of n into a prime and a semiprime.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 3, 2, 2, 1, 2, 2, 5, 1, 2, 2, 3, 2, 4, 2, 3, 3, 5, 5, 4, 1, 2, 4, 5, 2, 4, 3, 5, 6, 4, 5, 6, 3, 4, 5, 6, 5, 4, 3, 4, 4, 8, 7, 6, 4, 3, 7, 8, 6, 4, 4, 3, 10, 7, 6, 7, 4, 6, 10, 7, 6, 5, 6, 4, 7, 8, 9, 7, 5, 6, 9, 8, 9, 4, 5, 7, 8, 9, 11, 8, 4, 4, 11, 12, 10, 6, 10, 7, 13, 9, 9, 6

Offset: 1

Reinhard Zumkeller, Nov 23 2004

Marnell conjectures that a(n) > 0 for n > 10 after analyzing "many thousands of whole numbers". I find no exceptions below 100 million. - Charles R Greathouse IV, May 04 2010

			a(21) = #{7+2*7, 11+2*5, 17+2*2} = 3.

Geoffrey R. Marnell, "Ten Prime Conjectures", Journal of Recreational Mathematics 33:3 (2004-2005), pp. 193-196.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A061358, A000040, A001358.

Cf. A010051.

a100949 n = sum $ map (a010051 . (n -)) $ takeWhile (< n) a001358_list
-- Reinhard Zumkeller, Jun 26 2013

Table[Count[Sort/@(PrimeOmega/@IntegerPartitions[n,{2}]),{1,2}],{n,110}] (* Harvey P. Dale, Mar 25 2018 *)

list(lim)=my(p=primes(primepi(lim)),sp=select(n->bigomega(n)==2, vector(lim\1,i,i)),x=O('x^(lim\1+1))+'x); concat([0,0,0,0,0], Vec(sum(i=1,#p,x^p[i])*sum(i=1,#sp,x^sp[i]))) \\ Charles R Greathouse IV, Jun 14 2013

A100951(n) <= A100950(n) <= a(n) <= min(A000720(n), A072000(n)).

a(n) = Sum_{i=1..floor(n/2)} A010051(i) * A064911(n-i) + A010051(n-i) * A064911(i). - Wesley Ivan Hurt, May 02 2019

A174956 0 unless n is the k-th semiprime when a(n) = k.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 0, 3, 4, 0, 0, 0, 5, 6, 0, 0, 0, 0, 0, 7, 8, 0, 0, 9, 10, 0, 0, 0, 0, 0, 0, 11, 12, 13, 0, 0, 14, 15, 0, 0, 0, 0, 0, 0, 16, 0, 0, 17, 0, 18, 0, 0, 0, 19, 0, 20, 21, 0, 0, 0, 22, 0, 0, 23, 0, 0, 0, 24, 0, 0, 0, 0, 25, 0, 0, 26, 0, 0, 0, 0, 27, 0, 0, 28, 29, 30, 0, 0, 0, 31, 0, 32, 33

Offset: 1

Reinhard Zumkeller, Apr 03 2010

nonn

a(A001358(n)) = n; a(A100959(n)) = 0.

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A049084.

import Data.List (unfoldr)
a174956 n = a174956_list !! (fromInteger n - 1)
a174956_list = unfoldr x (1, 1, a001358_list) where
   x (i, z, ps'@(p:ps)) | i == p = Just (z, (i + 1, z + 1, ps))
                        | i /= p = Just (0, (i + 1, z, ps'))
-- Reinhard Zumkeller, Oct 27 2012

nn = 100; With[{tbl = Table[If[PrimeOmega[n] == 2, 1, 0], {n, nn}]},
Table[If[tbl[[i]] == 0, 0, Total[Take[tbl, i]]], {i, nn}]] (* Harvey P. Dale, Oct 13 2012 *)

first(n)=my(v=List(),u=vector(n)); forprime(p=2,n\2, forprime(q=2,min(p,n\p), listput(v,p*q))); v=Set(v); for(i=1,#v, u[v[i]]=i); u \\ Charles R Greathouse IV, Sep 02 2015

a(n) = A064911(n)*A072000(n).

A125527 Number of semiprimes <= 2^n.

Original entry on oeis.org

0, 1, 2, 6, 10, 22, 42, 82, 157, 304, 589, 1124, 2186, 4192, 8110, 15658, 30253, 58546, 113307, 219759, 426180, 827702, 1608668, 3129211, 6091437, 11868599, 23140878, 45150717, 88157689, 172235073, 336717854, 658662065, 1289149627, 2524532330

Offset: 1

Robert G. Wilson v, Dec 29 2006

nonn

Robert G. Wilson v, Table of n, a(n) for n = 1..63 (using data from A120033, terms n=48, 50..57 from Dana Jacobsen)
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A126279, A066265, A007053, A120033, A127396.

SemiPrimePi[n_] := Sum[ PrimePi[ n/Prime@i] - i + 1, {i, PrimePi@ Sqrt@n}]; Table[ SemiPrimePi[2^n], {n, 47}]

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

use ntheory ":all"; print "$ ",semiprime_count(1 << $),"\n" for 1..48; # Dana Jacobsen, Sep 10 2018

a(n) = A072000(2^n). - R. J. Mathar, Aug 26 2011

A113877 Semiprimes to semiprime powers.

Original entry on oeis.org

256, 1296, 4096, 6561, 10000, 38416, 46656, 50625, 194481, 234256, 262144, 390625, 456976, 531441, 1000000, 1048576, 1185921, 1336336, 1500625, 2085136, 2313441, 4477456, 5764801, 6765201, 7529536, 9150625, 10077696, 10556001, 11316496, 11390625, 14776336

Offset: 1

Jonathan Vos Post, Jan 27 2006

This is the semiprime analog of A053810.

			a(1) = 256 = 4^4 = semiprime(1)^semiprime(1).
a(2) = 1296 = 6^4 = semiprime(2)^semiprime(1).
a(3) = 4096 = 4^6 = semiprime(1)^semiprime(2).
a(4) = 6561 = 9^4 = semiprime(3)^semiprime(1).
a(5) = 10000 = 10^4.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001358, A113854, A053810.

lim = 10^8; s = Select[Range[lim^(1/4)], Total[Transpose[FactorInteger[#]][[2]]] == 2 &]; t = {}; j = 1; While[b = s[[j]]; i = 1; While[a = s[[i]]; e = a^b; If[e <= lim, AppendTo[t, e]]; e < lim && i < Length[s], i++]; i > 1, j++]; t = Union[t] (* T. D. Noe, Jun 05 2013 *)

is(n)=my(b,e=ispower(n,,&b),o); if(e==0,return(0)); o=bigomega(e); (o==2 && bigomega(b)==2) || (e%2==0 && o==3 && isprime(b)) \\ Charles R Greathouse IV, Jun 05 2013

list(lim)=my(v=List());for(e=4,log(lim\=1+.5)\log(4), if(bigomega(e)!=2, next); for(b=4,(lim+.5)^(1/e), if(bigomega(b)==2, listput(v,b^e)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 05 2013

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, factorint
def A113877(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]) 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

{a(n)} = {a^b where a and b are elements of A001358}.
{a(n)} = {(p*q)^(r*s) = (p^(r*s))*(q^r*s) for distinct primes p, q, r, s} UNION {(p*q)^(p*r) = (p^(p*r))*(q^(p*r)) for distinct primes p, q, r} UNION {(p*q)^(r*r) = (p^(r^2))*(q^(r^2)) for distinct primes p, q, r} UNION {(p*q)^(p*q)= (p^(p*q))*(q^(p*q)) for distinct primes p, q} UNION {(p^2)^(p^2) = p^(2*(p^2)) for prime p}.
a(n) ~ (n log n/log log n)^4. - Charles R Greathouse IV, Jun 05 2013

Terms corrected by Charles R Greathouse IV, Jun 05 2013

cp's OEIS Frontend

A175040 Product of the n-th block of identical consecutive values of A072000.

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A001358 Semiprimes (or biprimes): products of two primes.

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A064911 If n is semiprime (or 2-almost prime) then 1 else 0.

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A066265 a(n) = number of semiprimes < 10^n.

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A128301 Indices of squares (of primes) in the semiprimes.

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

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