A339191 -id:A339191 - cp's OEIS frontend

A112141 Product of the first n semiprimes.

4, 24, 216, 2160, 30240, 453600, 9525600, 209563200, 5239080000, 136216080000, 4495130640000, 152834441760000, 5349205461600000, 203269807540800000, 7927522494091200000, 364666034728195200000, 17868635701681564800000, 911300420785759804800000

Offset: 1

Jonathan Vos Post, Nov 28 2005

Semiprime analog of primorial (A002110). Equivalent for product of what A062198 is for sum.

			a(10) = 4*6*9*10*14*15*21*22*25*26 = 136216080000, the product of the first 10 semiprimes.
From _Gus Wiseman_, Dec 06 2020: (Start)
The sequence of terms together with their prime signatures begins:
                        4: (2)
                       24: (3,1)
                      216: (3,3)
                     2160: (4,3,1)
                    30240: (5,3,1,1)
                   453600: (5,4,2,1)
                  9525600: (5,5,2,2)
                209563200: (6,5,2,2,1)
               5239080000: (6,5,4,2,1)
             136216080000: (7,5,4,2,1,1)
            4495130640000: (7,6,4,2,2,1)
          152834441760000: (8,6,4,2,2,1,1)
         5349205461600000: (8,6,5,3,2,1,1)
       203269807540800000: (9,6,5,3,2,1,1,1)
      7927522494091200000: (9,7,5,3,2,2,1,1)
    364666034728195200000: (10,7,5,3,2,2,1,1,1)
  17868635701681564800000: (10,7,5,5,2,2,1,1,1)
(End)

T. D. Noe, Table of n, a(n) for n = 1..100

Partial sums of semiprimes are A062198.
First differences of semiprimes are A065516.
A000040 lists primes, with partial products A002110 (primorials).
A000142 lists factorials, with partial products A000178 (superfactorials).
A001358 lists semiprimes, with partial products A112141 (this sequence).
A005117 lists squarefree numbers, with partial products A111059.
A006881 lists squarefree semiprimes, with partial products A339191.
A101048 counts partitions into semiprimes (restricted: A338902).
A320655 counts factorizations into semiprimes.
A338898/A338912/A338913 give the prime indices of semiprimes, with product/sum/difference A087794/A176504/A176506.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with product/sum/difference A339361/A339362/A338900.
Cf. A001222, A084126, A084127, A115392, A168472, A320732, A320892.

A112141 := proc(n)
    mul(A001358(i),i=1..n) ;
end proc:
seq(A112141(n),n=1..10) ; # R. J. Mathar, Jun 30 2020

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_] := Times @@ NestList[ NextSemiPrime@# &, 2^2, n - 1]; Array[f, 18] (* Robert G. Wilson v, Jun 13 2013 *)
FoldList[Times,Select[Range[30],PrimeOmega[#]==2&]] (* Gus Wiseman, Dec 06 2020 *)

a(n)=my(v=vector(n),i,k=3);while(iCharles R Greathouse IV, Apr 04 2013

from sympy import factorint
def aupton(terms):
    alst, k, p = [], 1, 1
    while len(alst) < terms:
        if sum(factorint(k).values()) == 2:
            p *= k
            alst.append(p)
        k += 1
    return alst
print(aupton(18)) # Michael S. Branicky, Aug 31 2021

a(n) = Product_{i=1..n} A001358(i).

A001222(a(n)) = 2*n.

A347047 Smallest squarefree semiprime whose prime indices sum to n.

Original entry on oeis.org

6, 10, 14, 21, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526

Offset: 3

Gus Wiseman, Aug 22 2021

nonn

Compared to A001747, we have 21 instead of 22 and lack 2 and 4.
Compared to A100484 (shifted) we have 21 instead of 22 and lack 4.
Compared to A161344, we have 21 instead of 22 and lack 4 and 8.
Compared to A339114, we have 11 instead of 9 and lack 4.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

			The initial terms and their prime indices:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   21: {2,4}
   26: {1,6}
   34: {1,7}
   38: {1,8}
   46: {1,9}

The opposite version (greatest instead of smallest) is A332765.
These are the minima of rows of A338905.
The nonsquarefree version is A339114 (opposite: A339115).
A001358 lists semiprimes (squarefree: A006881).
A024697 adds up semiprimes by weight (squarefree: A025129).
A056239 adds up prime indices, row sums of A112798.
A246868 gives the greatest squarefree number whose prime indices sum to n.
A320655 counts factorizations into semiprimes (squarefree: A320656).
A338898, A338912, A338913 give the prime indices of semiprimes.
A338899, A270650, A270652 give the prime indices of squarefree semiprimes.
A339116 groups squarefree semiprimes by greater factor, sums A339194.
A339362 adds up prime indices of squarefree semiprimes.
Cf. A001221, A087112, A089994, A098350, A176504, A338900, A338901, A338904, A338907/A338908, A339005, A339191.

Table[Min@@Select[Table[Times@@Prime/@y,{y,IntegerPartitions[n,{2}]}],SquareFreeQ],{n,3,50}]

from sympy import prime, sieve
def a(n):
    p = [0] + list(sieve.primerange(1, prime(n)+1))
    return min(p[i]*p[n-i] for i in range(1, (n+1)//2))
print([a(n) for n in range(3, 58)]) # Michael S. Branicky, Sep 05 2021

cp's OEIS Frontend

A112141 Product of the first n semiprimes.

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