A112141 -id:A112141 - cp's OEIS frontend

A339191 Partial products of squarefree semiprimes (A006881).

6, 60, 840, 12600, 264600, 5821200, 151351200, 4994589600, 169816046400, 5943561624000, 225855341712000, 8808358326768000, 405184483031328000, 20664408634597728000, 1136542474902875040000, 64782921069463877280000, 3757409422028904882240000

Offset: 1

Gus Wiseman, Nov 30 2020

nonn

A squarefree semiprime is a product of any two distinct prime numbers.

Do all terms belong to A242031 (weakly decreasing prime signature)?

			The sequence of terms together with their prime indices begins:
          6: {1,2}
         60: {1,1,2,3}
        840: {1,1,1,2,3,4}
      12600: {1,1,1,2,2,3,3,4}
     264600: {1,1,1,2,2,2,3,3,4,4}
    5821200: {1,1,1,1,2,2,2,3,3,4,4,5}
  151351200: {1,1,1,1,1,2,2,2,3,3,4,4,5,6}
The sequence of terms together with their prime signatures begins:
                   6: (1,1)
                  60: (2,1,1)
                 840: (3,1,1,1)
               12600: (3,2,2,1)
              264600: (3,3,2,2)
             5821200: (4,3,2,2,1)
           151351200: (5,3,2,2,1,1)
          4994589600: (5,4,2,2,2,1)
        169816046400: (6,4,2,2,2,1,1)
       5943561624000: (6,4,3,3,2,1,1)
     225855341712000: (7,4,3,3,2,1,1,1)
    8808358326768000: (7,5,3,3,2,2,1,1)
  405184483031328000: (8,5,3,3,2,2,1,1,1)

A000040 lists the primes, with partial products A002110 (primorials).
A001358 lists semiprimes, with partial products A112141.
A002100 counts partitions into squarefree semiprimes (restricted: A338903)
A000142 lists factorial numbers, with partial products A000178.
A005117 lists squarefree numbers, with partial products A111059.
A006881 lists squarefree semiprimes, with partial sums A168472.
A166237 gives first differences of squarefree semiprimes.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A338898/A338912/A338913 give prime indices of semiprimes.
A338899/A270650/A270652 give prime indices of squarefree semiprimes.
A338901 gives first appearances in the list of squarefree semiprimes.
A339113 gives products of primes of squarefree semiprime index.
Cf. A001221, A112798, A167171, A320732, A320891, A320892, A320894, A320911, A338900, A338902, A339003, A339004.

FoldList[Times,Select[Range[20],SquareFreeQ[#]&&PrimeOmega[#]==2&]]

A347413 a(n) = (product of first n semiprimes) mod (sum of first n semiprimes).

Original entry on oeis.org

0, 4, 7, 14, 11, 40, 17, 17, 0, 8, 85, 147, 62, 16, 292, 26, 138, 18, 0, 570, 167, 257, 360, 156, 525, 882, 372, 918, 0, 0, 0, 0, 0, 150, 0, 0, 1070, 2136, 1172, 0, 1265, 1502, 663, 0, 0, 0, 0, 1208, 306, 2995, 0, 1404, 1389, 636, 0, 272, 0, 1944, 5216, 2268, 1548, 1160, 3300, 0, 924, 84, 0, 3723

Offset: 1

J. M. Bergot and Robert Israel, Aug 31 2021

nonn

A surprising number of terms are 0: 3124 of the first 10000 terms.

			The first 3 semiprimes are 4, 6, 9, so a(3) = (4*6*9) mod (4+6+9) = 216 mod 19 = 7.

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

Cf. A001358, A062198, A112141, A347421.

R:= NULL:
s:= 0: p:= 1: count:= 0:
for n from 4 while count < 100 do
  if numtheory:-bigomega(n) = 2 then
    count:= count+1: s:= s+n; p:= p*n;
    R:= R, p mod s;
  fi
od:
R;

from sympy import factorint
def aupton(terms):
    alst, i, n, s, p = [], 1, 0, 0, 1
    while n < terms:
        if sum(factorint(i).values()) == 2:
            n += 1; s += i; p *= i
            alst.append(p%s)
        i += 1
    return alst
print(aupton(68)) # Michael S. Branicky, Sep 01 2021

a(n) = A112141(n) mod A062198(n).

A347421 Numbers k such that the product of the first k semiprimes is divisible by the sum of the first k semiprimes.

Original entry on oeis.org

1, 9, 19, 29, 30, 31, 32, 33, 35, 36, 40, 44, 45, 46, 47, 51, 55, 57, 64, 67, 70, 71, 72, 74, 81, 83, 84, 92, 94, 95, 96, 97, 103, 104, 105, 107, 108, 109, 113, 116, 118, 124, 125, 127, 130, 131, 132, 133, 136, 138, 140, 142, 144, 158, 159, 160, 167, 177, 182, 184, 188, 191, 196, 202, 203, 206

Offset: 1

J. M. Bergot and Robert Israel, Aug 31 2021

nonn

What are the asymptotics of a(n)/n as n -> infinity?

			a(2) = 9 is a term because the first 9 semiprimes are 4, 6, 9, 10, 14, 15, 21, 22, 25, and 4*6*9*10*14*15*21*22*25 = 5239080000 is divisible by 4+6+9+10+14+15+21+22+25 = 126.

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

Cf. A001358, A062198, A112141, A347413.

using Nemo
function A347421List(upto)
    c, s, p = 0, ZZ(0), ZZ(1)
    list = Int32[]
    for n in 4:typemax(Int32)
        if 2 == sum([e for (p, e) in factor(n)])
            s += n; p *= n; c += 1
            if divisible(p, s)
                c > upto && return list
                push!(list, c)
            end
        end
    end
end
A347421List(206) |> println # Peter Luschny, Aug 31 2021

R:= NULL:
s:= 0: p:= 1: zcount:= 0: scount:= 0:
for n from 4 while zcount < 100 do
  if numtheory:-bigomega(n) = 2 then
    s:= s+n; p:= p*n;
    scount:= scount+1;
    if p mod s = 0 then zcount:= zcount+1; R:= R, scount fi
  fi
od:
R;

sp = Select[Range[700], PrimeOmega[#] == 2 &]; Position[Divisible[Rest @ FoldList[Times, 1, sp], Accumulate @ sp], True] // Flatten (* Amiram Eldar, Aug 31 2021 *)

from sympy import factorint
def aupto(limit):
    alst, i, k, s, p = [], 1, 0, 0, 1
    while k < limit:
        if sum(factorint(i).values()) == 2:
            k += 1; s += i; p *= i
            if p%s == 0: alst.append(k)
        i += 1
    return alst
print(aupto(206)) # Michael S. Branicky, Aug 31 2021

A122123 Product of the first n 5-almost primes (A014614).

Original entry on oeis.org

32, 1536, 110592, 8847360, 955514880, 107017666560, 12842119987200, 2080423437926400, 349511137571635200, 61513960212607795200, 11072512838269403136000, 2214502567653880627200000, 460616534072007170457600000, 111929817779497742421196800000

Offset: 1

Jonathan Vos Post, Oct 19 2006

5-almost prime analog of primorial (A002110). The semiprime analog of primorial is A112141. Equivalent for product of what A086047 is for sum. Bigomega(a(n)) = the number of not necessarily distinct prime factors of a(n) = A001222(a(n)) = A008587(n) = 5*n.

			a(10) = 32 * 48 * 72 * 80 * 108 * 112 * 120 * 162 * 168 * 176 = 2^33 * 3^12 * 5^2 * 7^2 * 11 which has 50 prime factors with multiplicity.

Cf. A001222, A002110, A008585, A014613, A008587, A086046, A086047, A112141, A114426.

Rest[FoldList[Times,1,Select[Range[200],PrimeOmega[#]==5&]]] (* Harvey P. Dale, Feb 07 2012 *)

a(n) = Prod[i=1..n] A014614(i).

More terms from Harvey P. Dale, Feb 07 2012

A122609 Product of the first n 5-almost primes (A014614), divided by product of the first n primes, rounded down.

Original entry on oeis.org

16, 256, 3686, 42130, 413642, 3563691, 25155471, 214483497, 1566662070, 9508018081, 55207846924, 298420794188, 1513939638809, 8555519354201, 45872146324653, 228495219428460, 1045656088909905, 4662597642352366, 19485482684457652, 82333025427285855

Offset: 1

Jonathan Vos Post, Oct 20 2006

This is to 5-almost primes as A122093 is to 4-almost primes as A122032 is to 3-almost primes and as A122019 is to 2-almost primes (semiprimes). Note that these can nonmonotonic (look at the graphs). What is the asymptotic value of the ratio A014614(n)/A002110(n)?

It appears that a(n) = 0 for n >= 11839. - Giovanni Resta, Jun 13 2016

			a(1) = floor(32/2) = floor 16 = 16.
a(2) = floor(1536/6) = floor(256) = 256.
a(3) = floor(110592/30) = floor(3686.4) = 3686.
a(4) = floor(8847360/210) = floor(42130.2857) = 42130.
a(5) = floor(955514880/2310) = floor(413642.805) = 413642.
a(6) = floor(107017666560/30030) = floor(3563691.86) = 3563691.
a(7) = floor(12842119987200/510510) floor(61152952320/2431) = floor(25155471.95) = 25155471.
a(8) = floor(2080423437926400/9699690) = floor(214483497.712) = 214483497.
a(9) = floor(349511137571635200/223092870) = floor(1566662070.247) = 1566662070.
a(10) = floor(61513960212607795200/6469693230) = floor(9508018081.501) = 9508018081.

Giovanni Resta, Table of n, a(n) for n = 1..11840

Cf. A001222, A002110, A008585, A014613, A008587, A086046, A086047, A122093, A112141, A114426, A014614.

q = Select[Range[900], PrimeOmega[#] == 5 &]; m = 1; Table[ Floor[ m *= q[[i]] / Prime[i]], {i, Length@ q}] (* Giovanni Resta, Jun 13 2016 *)

a(n) = floor(A122123(n)/A002110(n)) = floor(Prod(i=1..n)5almostprime(i)/Prod(i=1..n)prime(i)) = floor(Prod(i=1..n)A014614(i)/Prod(i=1..n)A000040(i)) = floor(Prod(i=1..n)(A014614(i)/A000040(i))).

a(12) corrected and a(13)-a(20) from Giovanni Resta, Jun 13 2016

A191645 Numerators of the n-th partial "harmonic" sum of 1 + inverse semiprimes.

Original entry on oeis.org

5, 17, 55, 293, 2141, 445, 457, 5153, 131597, 1745411, 1772711, 30586537, 31024117, 597115873, 604577173, 14050770329, 99311504603, 100230122303, 101081931443, 101903852543

Offset: 1

Jonathan Vos Post, Jun 09 2011

Denominators appear to be the same as A140123. The fractions begin: 5/4, 17/12, 55/36, 293/180, 2141/1260, 445/252, 457/252, 5153/2772, 131597/69300, ...

This is the semiprime analog of A024528.

			a(1) =  5 because 1 + 1/4 = 5/4.
a(2) = 17 because 1 + 1/4 + 1/6 = 17/12.
a(3) = 55 because 1 + 1/4 + 1/6 + 1/9 = 55/36.

Cf. A001358, A112141, A001008, A140123, A024528.

A191645 := proc(n) 1+add(1/A001358(i),i=1..n) ; numer(%) ; end proc:
seq(A191645(n),n=1..20) ; # R. J. Mathar, Jun 16 2011

With[{sp=Join[{1},Select[Range[100],PrimeOmega[#]==2&]]},Rest[ Numerator[ Accumulate[1/sp]]]] (* Harvey P. Dale, May 01 2015 *)

s=1; for(k=1,99, bigomega(k)==2 & print1(numerator(s+=1/k)", "))  \\ M. F. Hasler, Jun 17 2011

A226296 The smallest number beginning with n that can be decomposed into divisors consisting exclusively of the first n semiprimes.

Original entry on oeis.org

16, 24, 3456, 466560, 5927040, 6350400, 771573600, 838252800, 990186120000, 102979356480000, 112378266000000, 12379589782560000, 133730136540000000, 140865976625774400000, 1553794408841875200000, 16774637597496979200000, 17868635701681564800000

Offset: 1

Jonathan Vos Post, Christian N. K. Anderson, and Kevin L. Schwartz, Jun 02 2013

This is to A225903 as semiprimes A001358 are to prime A000040, and as Product of the first n semiprimes A112141 is to Primorials A002110.

			a(1) = 16, the smallest multiple of the first semiprime (4) that begins with leftmost digit 1.
a(2) = 24 = 4 * 6, the product of the first 2 semiprimes, and already begins with 2.

Christian N. K. Anderson, Table of n, a(n) for n = 1..375
Christian N. K. Anderson, Decomposition of the first 376 terms, indicating semiprime factors whose exponents are greater than one.

Cf. A001358, A112141, A225903.

A114428 Primes of the form 1 + product of the first n semiprimes.

Original entry on oeis.org

5, 2161, 30241, 453601, 4495130640001, 152834441760001, 911300420785759804800001, 19660095637340203930960075575675174251117567173124497920000000001

Offset: 1

Jonathan Vos Post, Feb 13 2006

nonn

Semiprime analog of primorial primes A005234 (primes p such that 1 + product of primes up to p is prime) as indexed by A014545 (n such that n-th Euclid number (A006862(n)) = 1 + (Product of first n primes) is prime). In that sense, this sequence is indexed by (1, 4, 5, 6, 11, 12, 39, ...).

The next term has 90 digits. - Harvey P. Dale, Sep 21 2011

			a(1) = 5 = 4 + 1 = 1 + A001358(1) = 1 + A112141(1) because 4 is the first semiprime and 5 is prime.
a(2) = 2161 because 2160 + 1 = 1 + A001358(1)*A001358(2)*A001358(3)*A001358(4) = 1 + A112141(4) = 1 + (4*6*9*10) is prime.
a(3) = 1 + A112141(5).
a(4) = 1 + A112141(6).
a(5) = 1 + A112141(11).
a(6) = 1 + A112141(12).
a(7) = (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) + 1 is prime.

Cf. A000040, A001358, A002110, A005234, A014545, A112141.

Select[#+1&/@FoldList[Times,1,Select[Range[200],PrimeOmega[#] == 2&]], PrimeQ] (* Harvey P. Dale, Sep 21 2011 *)

{a(n)} = {1 + A112141} INTERSECTION {A000040}.

a(7) added by Jonathan Vos Post, Dec 12 2010

A114430 Primes of the form 1 + product of the first n 3-almost primes A014612.

Original entry on oeis.org

97, 32920473601, 1448500838401, 65182537728001, 1491301685600774317670400000001, 48235157779343672198731287466250036763794299837586774072944798728192000000000000000001

Offset: 1

Jonathan Vos Post, Feb 13 2006

3-almost prime analog of primorial primes A005234 (primes p such that 1 + product of primes up to p is prime) as indexed by A014545 (n such that n-th Euclid number (A006862(n)) = 1 + (Product of first n primes) is prime). In that sense, this sequence is indexed by (2, 8, 9, 10, 19, ...).

			a(1) = 97 = 96 + 1 = 1 + (8 * 12) = 1 + A014612(1)*A014612(2) = 1 more than the product of the first 2 of the 3-almost primes and is prime.
a(2) = 32920473601 = 1 + (8 * 12 * 18 * 20 * 27 * 28 * 30 * 42) = 1 more than the product of the first 8 of the 3-almost primes and is prime.
a(3) = 1 more than the product of the first 9 of the 3-almost primes and is prime.
a(4) = 1 more than the product of the first 10 of the 3-almost primes and is prime.
a(5) = 1 more than the product of the first 19 of the 3-almost primes and is prime.

Cf. A000040, A001358, A014612, A002110, A005234, A014545, A112141.

Select[Rest[FoldList[Times,1,Select[Range[250],PrimeOmega[#]==3&]]]+1,PrimeQ] (* Harvey P. Dale, Dec 21 2013 *)

{a(n)} = {1 + Prod[from i = 1 to n] A014612(i)} INTERSECTION {A000040}.

One more term (a(6)) from Harvey P. Dale, Dec 21 2013

A226525 Fortunate semiprimes: least m > 1 such that m + sp(n)# is semiprime, where sp# denotes the product of the semiprimes <= sp.

Original entry on oeis.org

2, 2, 2, 5, 2, 3, 2, 7, 3, 19, 11, 3, 23, 5, 61, 29, 31, 3, 29, 31, 13, 19, 5, 7, 23, 47, 3, 53, 47, 19, 13, 7, 41, 53, 2, 43, 7, 103, 2, 61, 59, 71, 17, 59, 79, 43, 167, 71, 97, 7, 151, 37, 103, 83, 127, 103, 11, 53, 29, 7, 67, 83, 151, 107, 37

Offset: 1

Sean A. Irvine, Jonathan Vos Post and Robert G. Wilson v, Jun 09 2013

This is the semiprime analogous to A005235.
Just like the Fortunate primes (A005235) we conjecture that all terms are prime!
If instead of semiprimorials (A112141), we use primorials the sequence would be {2, 3, 3, 3, 5, 1, 5, 1, 1, 3, 4, 4, 11, 1, 4, 7, 4, 1, 1, 23, 1, 29, 1, 9, 32, 1, 71, 31, 4, 32, 23, 5, 125, 1, 97, 1, 11, 7, 27, 1, 29, 61, 11}; not very interesting.
If instead of m > 1, we start with m > 0, we identify the semiprimorials +1 which are semiprimes by index: 2, 3, 8, 9, 15, 16, 19, 21, 23, 27, 29, ....
A112141(66) + 197 has been completed through the 2030 curve.

			a(3) = 3 since sp(3) = 4*6*9*10 = 2160 and the least number greater than the fourth semiprimorial which is the semiprime is 2165 = 5*433. Therefore the difference is a(3) which equals 3.
A112141(47) + a(47) = 24011725937636436154291480954413133199 * 68433092684820794078956407785220072996675433.
A112141(55) + a(55) = 795251036594717254131632161591406578993 * 650971642564884068706166933685477027845256102005635827825839.
A112141(63) + a(63) = 168586841653003537 * 40363128914158968243564625304355041082304983807201816858670871447070744 83558441664261096307889392423.

Dario Alejandro Alpern, Factorization using the Elliptic Curve Method
Cyril Banderier, Fortune's conjecture (Fortunate and unfortunate primes: Nearest primes from a prime factorial)

Cf. A005235.

NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; f[n_] := Block[{k = 1, m = Times @@ NestList[ NextSemiPrime, 2^2, n-1]}, While[ PrimeOmega[m + k] != 2, k++]; k]; Do[ Print@ f[n], {n, 50}]

The difference between the n-th semiprimorial and the next semiprime greater than that semiprimorial plus 1.

cp's OEIS Frontend

A339191 Partial products of squarefree semiprimes (A006881).

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A347413 a(n) = (product of first n semiprimes) mod (sum of first n semiprimes).

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A347421 Numbers k such that the product of the first k semiprimes is divisible by the sum of the first k semiprimes.

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A122123 Product of the first n 5-almost primes (A014614).

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A122609 Product of the first n 5-almost primes (A014614), divided by product of the first n primes, rounded down.

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A191645 Numerators of the n-th partial "harmonic" sum of 1 + inverse semiprimes.

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A226296 The smallest number beginning with n that can be decomposed into divisors consisting exclusively of the first n semiprimes.

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A114428 Primes of the form 1 + product of the first n semiprimes.

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