A112141 -id:A112141 - cp's OEIS frontend

A166483 Number of integers in [1..A112141(n)] that are multiples of the first n semiprimes, where A112141(n) is the product of the first n semiprimes.

1, 8, 84, 912, 13344, 210240, 4536000, 101969280, 2619086400, 69184886400, 2314236355200, 79546238956800, 2824874214624000, 108310887381312000, 4266362489957568000, 197608471322790528000, 9780059370744254592000, 502210378611975825792000, 27829868062695415516800000, 1595350685432248426281600000

Offset: 1

Daniel Tisdale, Oct 14 2009

nonn

Previous name: Numerator of sequence of ratios of semiprimes and multiples thereof in intervals of length 4*6, 4*6*9, 4*6*9*10, ... [A112141] to the length of the interval. See example below. The first few ratios are 1/4, 8/24, 84/216, 912/2160, ...
Conjecture: sequences for 2-ads, 3-ads (composites of 3 primes), 4-ads, etc., converge to 1/2, 1/4, 1/8, ..., respectively.
The first few analogous ratios for 3-ads are 1/8, 16/96, 336/1728, 7296/34560, ...

			For n=2, the product of first two semiprimes is 4*6 = 24, and there are 8 multiples of the semiprimes 4,6 in the interval [1..24] including those two semiprimes, with repetitions (such as 4*3, 2*6) counted only once. So a(2) = 8.

Cf. A112141 (product of the first n semiprimes).

f:= proc(n)
  local N, V, T, S;
  N:= convert(SP[1..n],`*`);
  T:= 0:
  for S in combinat:-powerset(convert(SP[1..n],set)) minus {{}} do
    if nops(S)::odd then T:= T + N/ilcm(op(S)) else T:= T - N/ilcm(op(S)) fi
  od;
  T
end proc:
map(f, [$1..20]); # Robert Israel, Mar 24 2024

ismul(x, v) = for (k=1, #v, if ((x % v[k]) == 0, return(1)););
vsp(n) = my(v=vector(n), i, k=3); while(iMichel Marcus, Aug 16 2022

a(2) and a(6) corrected, and new name from Michel Marcus, Aug 15 2022

More terms from Robert Israel, Mar 24 2024

A317490 a(n) = min {i - j} where i*j is a factorization of the n-th partial product of the semiprimes (A112141).

Original entry on oeis.org

0, 2, 6, 3, 12, 3, 126, 153, 765, 1050, 5348, 14850, 85050, 501200, 91280, 3661983, 25633881, 66271296, 215467945, 254861640, 5311480020, 75327142968, 122703152000, 2187956957004, 3449084839200, 19305922856220, 11327171375520, 58038845751810, 2222926571960640

Offset: 1

Robert G. Wilson v, Jul 29 2018

nonn

			a(1) =   0 since the first semiprime is    4 =    2 *    2;
a(2) =   2 since 4*6               =      24 =    4 *    6;
a(3) =   6 since 4*6*9             =     216 =   12 *   18;
a(4) =   3 since 4*6*9*10          =    2160 =   45 *   48;
a(5) =  12 since 4*6*9*10*14       =   30240 =  168 *  180;
a(6) =   3 since 4*6*9*10*14*15    =  453600 =  172 *  175;
a(7) = 126 since 4*6*9*10*14*15*21 = 9525600 = 3024 * 3150; etc.

Inspired by A003681, and analogous to A061057 and A061060.

Cf. A112141.

SemiPrimePi[n_] := Sum[PrimePi[n/Prime[i]] - i + 1, {i, PrimePi[Sqrt[n]]}]; SemiPrime[n_] := Block[{e = Floor[Log[2, n] + 1], a, b}, a = 2^e; Do[b = 2^p; While[SemiPrimePi@a < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; a[n_] := Block[{sp = Times @@ Array[SemiPrime@# &, n], d}, d = DivisorSigma[0, sp]/2; -Subtract @@  Take[ Divisors@ sp, {d, d + 1}]]; a[1] = 0; Array[a, 29]

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

Original entry on oeis.org

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

A176506 Difference between the prime indices of the two factors of the n-th semiprime.

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, Apr 19 2010

Are there no adjacent equal terms? I have verified this up to n = 10^6. - Gus Wiseman, Dec 04 2020

			From _Gus Wiseman_, Dec 04 2020: (Start)
The sequence of semiprimes together with the corresponding differences begins:
   4: 1 - 1 = 0
   6: 2 - 1 = 1
   9: 2 - 2 = 0
  10: 3 - 1 = 2
  14: 4 - 1 = 3
  15: 3 - 2 = 1
  21: 4 - 2 = 2
  22: 5 - 1 = 4
  25: 3 - 3 = 0
  26: 6 - 1 = 5
  33: 5 - 2 = 3
(End)

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

Cf. A109313.
Cf. A001358, A049084, A084126, A084127.
A087794 is product of the same indices.
A176504 is the sum of the same indices.
A115392 lists positions of first appearances.
A128301 lists positions of 0's.
A172348 lists positions of 1's.
A338898 has this sequence as row differences.
A338900 is the squarefree case.
A338912/A338913 give the two prime indices of semiprimes.
A006881 lists squarefree semiprimes.
A024697 is the sum of semiprimes of weight n.
A056239 gives sum of prime indices (Heinz weight).
A087112 groups semiprimes by greater factor.
A270650/A270652/A338899 give the prime indices of squarefree semiprimes.
A338904 groups semiprimes by weight.
A338907/A338906 list semiprimes of odd/even weight.
A339114/A339115 give the least/greatest semiprime of weight n.
Cf. A000040, A001222, A062198, A112141, A112798, A325699, A320655, A339005.

isA001358 := proc(n) numtheory[bigomega](n) = 2 ; end proc:
A001358 := proc(n) option remember ; if n = 1 then return 4 ; else for a from procname(n-1)+1 do if isA001358(a) then return a; end if; end do; end if; end proc:
A084126 := proc(n) min(op(numtheory[factorset](A001358(n)))) ; end proc:
A084127 := proc(n) max(op(numtheory[factorset](A001358(n)))) ; end proc:
A176506 := proc(n) numtheory[pi](A084127(n)) - numtheory[pi](A084126(n)) ; end proc: seq(A176506(n),n=1..120) ; # R. J. Mathar, Apr 22 2010
# Alternative:
N:= 500: # to use the first N semiprimes
Primes:= select(isprime, [2,seq(i,i=3..N/2,2)]):
SP:= NULL:
for i from 1 to nops(Primes) do
  for j from 1 to i do
    sp:= Primes[i]*Primes[j];
    if sp > N then break fi;
    SP:= SP, [sp, i-j]
od od:
SP:= sort([SP],(s,t) -> s[1] t[2], SP); # Robert Israel, Jan 17 2019

M = 500; (* to use the first M semiprimes *)
primes = Select[Join[{2}, Range[3, M/2, 2]], PrimeQ];
SP = {};
For[i = 1, i <= Length[primes], i++,
  For[j = 1, j <= i, j++,
    sp = primes[[i]] primes[[j]];
    If[sp > M, Break []];
    AppendTo[SP, {sp, i - j}]
]];
SortBy[SP, First][[All, 2]] (* Jean-François Alcover, Jul 18 2020, after Robert Israel *)
Table[If[!SquareFreeQ[n],0,-Subtract@@PrimePi/@First/@FactorInteger[n]],{n,Select[Range[100],PrimeOmega[#]==2&]}] (* Gus Wiseman, Dec 04 2020 *)

lista(nn) = {my(vsp = select(x->(bigomega(x)==2), [1..nn])); vector(#vsp, k, my(f=factor(vsp[k])[,1]); primepi(vecmax(f)) - primepi(vecmin(f)));} \\ Michel Marcus, Jul 18 2020

a(n) = A049084(A084127(n)) - A049084(A084126(n)). [corrected by R. J. Mathar, Apr 22 2010]

a(n) = A338913(n) - A338912(n). - Gus Wiseman, Dec 04 2020

a(51) and a(69) corrected by R. J. Mathar, Apr 22 2010

A114426 Product of the first n 4-almost primes (A014613).

Original entry on oeis.org

16, 384, 13824, 552960, 29859840, 1672151040, 100329062400, 8126654054400, 682638940569600, 60072226770124800, 5406500409311232000, 540650040931123200000, 56227604256836812800000

Offset: 1

Jonathan Vos Post, Feb 13 2006

4-almost prime analog of primorial (A002110). The semiprime analog of primorial is A112141. Equivalent for product of what A086046 is for sum. Bigomega(a(n)) = the number of not necessarily distinct prime factors of a(n) = A001222(a(n)) = A008586(n) = 4*n.

			a(5) = 29859840 = 16 * 24 * 36 * 40 * 54 = the product of the first 5 values of the 4-almost primes = 2^13 * 3^6 * 5, which has 4*5 = 20 prime factors (with multiplicity).

Cf. A001222, A002110, A008585, A014613, A008586, A086046, A112141.

FoldList[Times,Select[Range[200],PrimeOmega[#]==4&]] (* Harvey P. Dale, Dec 02 2018 *)

a(n) = Prod[from i = 1 to n] A014613(i).

A122032 Product of the first n 3-almost primes, divided by product of the first n primes, rounded down.

Original entry on oeis.org

4, 16, 57, 164, 403, 870, 1535, 3393, 6492, 10075, 16250, 22837, 35092, 53862, 77929, 102925, 130837, 163010, 189773, 245903, 330117, 413691, 508391, 599788, 680172, 767719, 864615, 945420, 1075524, 1189739, 1217843, 1282919, 1376563, 1465693, 1505040

Offset: 1

Jonathan Vos Post, Oct 14 2006

Note that this is nonmonotonic. What is the asymptotic value of the ratio A112141(n)/A002110(n)?

Probably it can be easily proved that a(n) = 0 for n >= 116. - Giovanni Resta, Jun 13 2016

			a(1) = floor(8/2) = floor(4) = 4.
a(2) = floor(96/6) = floor(4) = 16.
a(3) = floor(1728/30) = floor(57.6) = 57.
a(4) = floor(34560/210) = floor(164.571429) = 164.
a(5) = floor(933120/2310) = floor(403.948052) = 403.
a(6) = floor(26127360/30030) = floor(870.041958) = 870.
a(7) = floor(783820800/510510) = floor(1535.36816) = 1535.
a(8) = floor(32920473600/9699690) = floor(3393.97172) = 3393.
a(9) = floor(1448500838400/223092870) = floor(6492.81547) = 6492.
a(10) = floor(65182537728000/6469693230) = floor(10075.058) = 10075.
a(11) = floor(3259126886400000/200560490130) = floor(16250.0943) = 16250.
a(12) = floor(169474598092800000/7420738134810) = floor(22837.9704) = 22837.

Giovanni Resta, Table of n, a(n) for n = 1..250

Cf. A000040, A014612, A002110, A112141, A123389.

tp = Select[Range[1000], PrimeOmega[#] == 3 &]; m = 1; Table[ Floor[m *= tp[[i]] / Prime[i]], {i, Length@ tp}] (* Giovanni Resta, Jun 13 2016 *)
Floor[#[[1]]/#[[2]]]&/@Module[{nn=200,tap,len},tap=FoldList[ Times, Select[ Range[ nn],PrimeOmega[#]==3&]];len=Length[tap];Thread[{tap, FoldList[Times, Prime[ Range[len]]]}]] (* Harvey P. Dale, Sep 08 2019 *)

a(n) = floor(A114425(n)/A002110(n)) = floor(Prod(i=1..n)3almostprime(i)/Prod(i=1..n)prime(i)) = floor(Prod(i=1..n)A014612(i)/Prod(i=1..n)A000040(i)) = floor(Prod(i=1..n)(A014612(i)/A000040(i))).

a(10) corrected by and a(13)-a(35) from Giovanni Resta, Jun 13 2016

cp's OEIS Frontend

A224081 Numbers k such that A112141(k) + 1 is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A224082 Numbers k such that A112141(k) - 1 is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A166483 Number of integers in [1..A112141(n)] that are multiples of the first n semiprimes, where A112141(n) is the product of the first n semiprimes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

PARI

Extensions

A317490 a(n) = min {i - j} where i*j is a factorization of the n-th partial product of the semiprimes (A112141).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

Python

Formula

Extensions

A176506 Difference between the prime indices of the two factors of the n-th semiprime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A114426 Product of the first n 4-almost primes (A014613).

Views

Author

Keywords