A115392 -id:A115392 - 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.

A338901 Position of the first appearance of prime(n) as a factor in the list of squarefree semiprimes.

Original entry on oeis.org

1, 1, 2, 3, 6, 7, 9, 11, 13, 17, 18, 21, 23, 25, 29, 31, 34, 36, 40, 42, 45, 47, 50, 52, 56, 58, 61, 64, 67, 70, 76, 78, 81, 82, 86, 89, 93, 97, 100, 104, 106, 107, 112, 113, 116, 118, 125, 129, 133, 134, 135, 139, 141, 147, 150, 154, 159, 160, 165, 167, 169

Offset: 1

Gus Wiseman, Nov 16 2020

nonn

The a(n)-th squarefree semiprime is the first divisible by prime(n).

After a(1) = 1, these are the positions of even terms in the list of all squarefree semiprimes A006881.

Amiram Eldar, Table of n, a(n) for n = 1..10000

A001358 lists semiprimes, with odds A046315 and evens A100484.
A004526 counts 2-part partitions, with strict case A140106 (shifted left).
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odds A046388 and evens A100484.
A115392 is the not necessarily squarefree version.
A166237 gives the first differences of squarefree semiprimes.
A270650 and A270652 give the prime indices of squarefree semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A338898 gives prime indices of semiprimes, with differences A176506.
A338899 gives prime indices of squarefree semiprimes, differences A338900.
A338912 and A338913 give the prime indices of semiprimes.
Cf. A001221, A001222, A002100, A056239, A065516, A112798, A167171, A320891, A320911, A338903, A338905.

rs=First/@FactorInteger[#]&/@Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&];
Table[Position[rs,i][[1,1]],{i,Union@@rs}]

A006881(a(n)) = A100484(n).

A338906 Semiprimes whose prime indices sum to an even number.

Original entry on oeis.org

4, 9, 10, 21, 22, 25, 34, 39, 46, 49, 55, 57, 62, 82, 85, 87, 91, 94, 111, 115, 118, 121, 129, 133, 134, 146, 155, 159, 166, 169, 183, 187, 194, 203, 205, 206, 213, 218, 235, 237, 247, 253, 254, 259, 267, 274, 289, 295, 298, 301, 303, 314, 321, 334, 335, 339

Offset: 1

Gus Wiseman, Nov 28 2020

nonn

A semiprime is a product of any two prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
      4: {1,1}      87: {2,10}    183: {2,18}    274: {1,33}
      9: {2,2}      91: {4,6}     187: {5,7}     289: {7,7}
     10: {1,3}      94: {1,15}    194: {1,25}    295: {3,17}
     21: {2,4}     111: {2,12}    203: {4,10}    298: {1,35}
     22: {1,5}     115: {3,9}     205: {3,13}    301: {4,14}
     25: {3,3}     118: {1,17}    206: {1,27}    303: {2,26}
     34: {1,7}     121: {5,5}     213: {2,20}    314: {1,37}
     39: {2,6}     129: {2,14}    218: {1,29}    321: {2,28}
     46: {1,9}     133: {4,8}     235: {3,15}    334: {1,39}
     49: {4,4}     134: {1,19}    237: {2,22}    335: {3,19}
     55: {3,5}     146: {1,21}    247: {6,8}     339: {2,30}
     57: {2,8}     155: {3,11}    253: {5,9}     341: {5,11}
     62: {1,11}    159: {2,16}    254: {1,31}    358: {1,41}
     82: {1,13}    166: {1,23}    259: {4,12}    361: {8,8}
     85: {3,7}     169: {6,6}     267: {2,24}    365: {3,21}

A031215 looks at primes instead of semiprimes.
A098350 has this as union of even-indexed antidiagonals.
A300061 looks at all numbers (not just semiprimes).
A338904 has this as union of even-indexed rows.
A338907 is the odd version.
A338908 is the squarefree case.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A056239 gives the sum of prime indices (Heinz weight).
A084126 and A084127 give the prime factors of semiprimes.
A087112 groups semiprimes by greater factor.
A289182/A115392 list the positions of odd/even terms in A001358.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338911 lists products of pairs of primes both of even index.
A339114/A339115 give the least/greatest semiprime of weight n.
Cf. A000040, A001222, A024697, A037143, A112798, A300063, A319242, A320655, A332765, A338910, A339004.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],PrimeOmega[#]==2&&EvenQ[Total[primeMS[#]]]&]

from math import isqrt
from sympy import primepi, primerange
def A338906(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((primepi(x//p)-a>>1) for a,p in enumerate(primerange(isqrt(x)+1),-1))
    return bisection(f,n,n) # Chai Wah Wu, Apr 03 2025

A338906 \/ A338907 = A001358.

A338910 Numbers of the form prime(x) * prime(y) where x and y are both odd.

Original entry on oeis.org

4, 10, 22, 25, 34, 46, 55, 62, 82, 85, 94, 115, 118, 121, 134, 146, 155, 166, 187, 194, 205, 206, 218, 235, 253, 254, 274, 289, 295, 298, 314, 334, 335, 341, 358, 365, 382, 391, 394, 415, 422, 451, 454, 466, 482, 485, 514, 515, 517, 527, 529, 538, 545, 554

Offset: 1

Gus Wiseman, Nov 20 2020

nonn

			The sequence of terms together with their prime indices begins:
      4: {1,1}     146: {1,21}    314: {1,37}
     10: {1,3}     155: {3,11}    334: {1,39}
     22: {1,5}     166: {1,23}    335: {3,19}
     25: {3,3}     187: {5,7}     341: {5,11}
     34: {1,7}     194: {1,25}    358: {1,41}
     46: {1,9}     205: {3,13}    365: {3,21}
     55: {3,5}     206: {1,27}    382: {1,43}
     62: {1,11}    218: {1,29}    391: {7,9}
     82: {1,13}    235: {3,15}    394: {1,45}
     85: {3,7}     253: {5,9}     415: {3,23}
     94: {1,15}    254: {1,31}    422: {1,47}
    115: {3,9}     274: {1,33}    451: {5,13}
    118: {1,17}    289: {7,7}     454: {1,49}
    121: {5,5}     295: {3,17}    466: {1,51}
    134: {1,19}    298: {1,35}    482: {1,53}

A338911 is the even instead of odd version.
A339003 is the squarefree case.
A001221 counts distinct prime indices.
A001222 counts prime indices.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A289182/A115392 list the positions of odd/even terms of A001358.
A300912 lists semiprimes with relatively prime indices.
A318990 lists semiprimes with divisible indices.
A338904 groups semiprimes by weight.
A338906/A338907 are semiprimes of even/odd weight.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give prime indices of squarefree semiprimes.
A338909 lists semiprimes with non-relatively prime indices.
Cf. A005117, A037143, A055684, A056239, A065516, A112798, A195017, A320655, A320732, A320892, A339004.

q:= n-> (l-> add(i[2], i=l)=2 and andmap(i->
    numtheory[pi](i[1])::odd, l))(ifactors(n)[2]):
select(q, [$1..1000])[];  # Alois P. Heinz, Nov 23 2020

Select[Range[100],PrimeOmega[#]==2&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]

from math import isqrt
from sympy import primepi, primerange
def A338910(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p)-a>>1 for a,p in enumerate(primerange(isqrt(x)+1),-1) if a&1)
    return bisection(f,n,n) # Chai Wah Wu, Apr 03 2025

Numbers m such that A001222(m) = A195017(m) = 2. - Peter Munn, Jan 17 2021

A338911 Numbers of the form prime(x) * prime(y) where x and y are both even.

Original entry on oeis.org

9, 21, 39, 49, 57, 87, 91, 111, 129, 133, 159, 169, 183, 203, 213, 237, 247, 259, 267, 301, 303, 321, 339, 361, 371, 377, 393, 417, 427, 453, 481, 489, 497, 519, 543, 551, 553, 559, 579, 597, 623, 669, 687, 689, 703, 707, 717, 749, 753, 789, 791, 793, 813, 817

Offset: 1

Gus Wiseman, Nov 20 2020

nonn

			The sequence of terms together with their prime indices begins:
      9: {2,2}     237: {2,22}    481: {6,12}
     21: {2,4}     247: {6,8}     489: {2,38}
     39: {2,6}     259: {4,12}    497: {4,20}
     49: {4,4}     267: {2,24}    519: {2,40}
     57: {2,8}     301: {4,14}    543: {2,42}
     87: {2,10}    303: {2,26}    551: {8,10}
     91: {4,6}     321: {2,28}    553: {4,22}
    111: {2,12}    339: {2,30}    559: {6,14}
    129: {2,14}    361: {8,8}     579: {2,44}
    133: {4,8}     371: {4,16}    597: {2,46}
    159: {2,16}    377: {6,10}    623: {4,24}
    169: {6,6}     393: {2,32}    669: {2,48}
    183: {2,18}    417: {2,34}    687: {2,50}
    203: {4,10}    427: {4,18}    689: {6,16}
    213: {2,20}    453: {2,36}    703: {8,12}

A338910 is the odd instead of even version.
A339004 is the squarefree case.
A001221 counts distinct prime indices.
A001222 counts prime indices.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A338899, A270650, A270652 list prime indices of squarefree semiprimes.
A289182/A115392 list the positions of odd/even terms of A001358.
A300912 lists semiprimes with relatively prime indices.
A318990 lists semiprimes with divisible indices.
A338904 groups semiprimes by weight.
A338906/A338907 list semiprimes of even/odd weight.
A338909 lists semiprimes with non-relatively prime indices.
A338912 and A338913 list prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
Cf. A005117, A037143, A055684, A056239, A065516, A112798, A128301, A195017, A320655, A320732, A320892, A338898, A339002, A339003.

q:= n-> (l-> add(i[2], i=l)=2 and andmap(i->
    numtheory[pi](i[1])::even, l))(ifactors(n)[2]):
select(q, [$1..1000])[];  # Alois P. Heinz, Nov 23 2020

Select[Range[100],PrimeOmega[#]==2&&OddQ[Times@@(1+PrimePi/@First/@FactorInteger[#])]&]

from math import isqrt
from sympy import primerange, primepi
def A338911(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p)-a>>1 for a,p in enumerate(primerange(isqrt(x)+1),-1) if a&1^1)
    return bisection(f,n,n) # Chai Wah Wu, Apr 03 2025

Numbers m such that A001222(m) = 2 and A195017(m) = -2. - Peter Munn, Jan 17 2021

A339003 Numbers of the form prime(x) * prime(y) where x and y are distinct and both odd.

Original entry on oeis.org

10, 22, 34, 46, 55, 62, 82, 85, 94, 115, 118, 134, 146, 155, 166, 187, 194, 205, 206, 218, 235, 253, 254, 274, 295, 298, 314, 334, 335, 341, 358, 365, 382, 391, 394, 415, 422, 451, 454, 466, 482, 485, 514, 515, 517, 527, 538, 545, 554, 566, 614, 626, 635, 649

Offset: 1

Gus Wiseman, Nov 21 2020

nonn

The squarefree semiprimes in A332822. - Peter Munn, Dec 25 2020

			The sequence of terms together with their prime indices begins:
     10: {1,3}     187: {5,7}     358: {1,41}    527: {7,11}
     22: {1,5}     194: {1,25}    365: {3,21}    538: {1,57}
     34: {1,7}     205: {3,13}    382: {1,43}    545: {3,29}
     46: {1,9}     206: {1,27}    391: {7,9}     554: {1,59}
     55: {3,5}     218: {1,29}    394: {1,45}    566: {1,61}
     62: {1,11}    235: {3,15}    415: {3,23}    614: {1,63}
     82: {1,13}    253: {5,9}     422: {1,47}    626: {1,65}
     85: {3,7}     254: {1,31}    451: {5,13}    635: {3,31}
     94: {1,15}    274: {1,33}    454: {1,49}    649: {5,17}
    115: {3,9}     295: {3,17}    466: {1,51}    662: {1,67}
    118: {1,17}    298: {1,35}    482: {1,53}    685: {3,33}
    134: {1,19}    314: {1,37}    485: {3,25}    694: {1,69}
    146: {1,21}    334: {1,39}    514: {1,55}    697: {7,13}
    155: {3,11}    335: {3,19}    515: {3,27}    706: {1,71}
    166: {1,23}    341: {5,11}    517: {5,15}    713: {9,11}

A338910 is the not necessarily squarefree version.
A339004 is the even instead of odd version.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd and even terms A046388 and A100484.
A289182/A115392 list the positions of odd/even terms of A001358.
A300912 lists products of two primes of relatively prime index.
A320656 counts factorizations into squarefree semiprimes.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338904 groups semiprimes by weight.
A338906/A338907 list semiprimes of even/odd weight.
A339002 lists products of two distinct primes of non-relatively prime index.
A339005 lists products of two distinct primes of divisible index.
Cf. A001221, A001222, A056239, A112798, A166237, A195017, A318990, A320911, A338901, A338903, A338911.
Subsequence of A332822.

Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]

from math import isqrt
from sympy import primepi, primerange
def A339003(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p)-a>>1 for a,p in enumerate(primerange(isqrt(x)+1),1) if a&1)
    return bisection(f,n,n) # Chai Wah Wu, Apr 03 2025

Numbers m such that A001221(m) = A001222(m) = A195017(m) = 2. - Peter Munn, Dec 31 2020

A289182 Positions of odd semiprimes in A001358.

Original entry on oeis.org

3, 6, 7, 9, 11, 13, 15, 17, 18, 19, 20, 23, 24, 26, 28, 30, 31, 32, 34, 36, 37, 39, 40, 42, 43, 44, 46, 48, 49, 51, 53, 54, 56, 57, 59, 60, 61, 63, 65, 66, 68, 69, 71, 72, 74, 75, 77, 78, 79, 80, 81, 83, 85, 86, 89, 90, 91, 92, 94, 95, 97, 98

Offset: 1

Zak Seidov, Jun 27 2017

nonn

Complement to A115392.

Cf. A001358, A115392.

sp=Select[Range[4,1000],2==PrimeOmega[#]&];Flatten[Position[Mod[sp, 2],1]]

lista(nn) = vsp = select(x->(bigomega(x)==2), vector(nn, k, k)); select(x->(x%2), vsp, 1); \\ Michel Marcus, Jul 02 2017

from math import isqrt
from sympy import primepi, primerange
def A289182(n):
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    def f(x): return int(-((t:=primepi(s:=isqrt(x)))*(t-1)>>1)+sum(primepi(x//p) for p in primerange(3,s+1)))
    return f(m:=iterfun(lambda x:int(n+x-f(x)),n))+primepi(m>>1) # Chai Wah Wu, Apr 03 2025

a(n) ~ n. - Charles R Greathouse IV, Jul 02 2017

A339004 Numbers of the form prime(x) * prime(y) where x and y are distinct and both even.

Original entry on oeis.org

21, 39, 57, 87, 91, 111, 129, 133, 159, 183, 203, 213, 237, 247, 259, 267, 301, 303, 321, 339, 371, 377, 393, 417, 427, 453, 481, 489, 497, 519, 543, 551, 553, 559, 579, 597, 623, 669, 687, 689, 703, 707, 717, 749, 753, 789, 791, 793, 813, 817, 843, 879, 917

Offset: 1

Gus Wiseman, Nov 22 2020

nonn

The squarefree semiprimes in A332821. - Peter Munn, Dec 25 2020

			The sequence of terms together with their prime indices begins:
     21: {2,4}     267: {2,24}    543: {2,42}
     39: {2,6}     301: {4,14}    551: {8,10}
     57: {2,8}     303: {2,26}    553: {4,22}
     87: {2,10}    321: {2,28}    559: {6,14}
     91: {4,6}     339: {2,30}    579: {2,44}
    111: {2,12}    371: {4,16}    597: {2,46}
    129: {2,14}    377: {6,10}    623: {4,24}
    133: {4,8}     393: {2,32}    669: {2,48}
    159: {2,16}    417: {2,34}    687: {2,50}
    183: {2,18}    427: {4,18}    689: {6,16}
    203: {4,10}    453: {2,36}    703: {8,12}
    213: {2,20}    481: {6,12}    707: {4,26}
    237: {2,22}    489: {2,38}    717: {2,52}
    247: {6,8}     497: {4,20}    749: {4,28}
    259: {4,12}    519: {2,40}    753: {2,54}

A338911 is the not necessarily squarefree version.
A339003 is the odd instead of even version, with not necessarily squarefree version A338910.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A289182/A115392 list the positions of odd/even terms in A001358.
A300912 lists products of pairs of primes with relatively prime indices.
A318990 lists products of pairs of primes with divisible indices.
A320656 counts factorizations into squarefree semiprimes.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338904 groups semiprimes by weight.
A338906/A338907 list semiprimes of even/odd weight.
Cf. A000040, A001221, A001222, A056239, A112798, A166237, A195017, A320911, A338901, A338903, A339002.
Subsequence of A332821.

Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&&OddQ[Times@@(1+ PrimePi/@First/@FactorInteger[#])]&]

from math import isqrt
from sympy import primepi, primerange
def A339004(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p)-a>>1 for a,p in enumerate(primerange(isqrt(x)+1),1) if a&1^1)
    return bisection(f,n,n) # Chai Wah Wu, Apr 03 2025

Numbers m such that A001221(m) = A001222(m) = 2 and A195017(m) = -2. - Peter Munn, Dec 31 2020

A338908 Squarefree semiprimes whose prime indices sum to an even number.

Original entry on oeis.org

10, 21, 22, 34, 39, 46, 55, 57, 62, 82, 85, 87, 91, 94, 111, 115, 118, 129, 133, 134, 146, 155, 159, 166, 183, 187, 194, 203, 205, 206, 213, 218, 235, 237, 247, 253, 254, 259, 267, 274, 295, 298, 301, 303, 314, 321, 334, 335, 339, 341, 358, 365, 371, 377, 382

Offset: 1

Gus Wiseman, Nov 28 2020

nonn

A squarefree semiprime is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
     10: {1,3}     115: {3,9}     213: {2,20}
     21: {2,4}     118: {1,17}    218: {1,29}
     22: {1,5}     129: {2,14}    235: {3,15}
     34: {1,7}     133: {4,8}     237: {2,22}
     39: {2,6}     134: {1,19}    247: {6,8}
     46: {1,9}     146: {1,21}    253: {5,9}
     55: {3,5}     155: {3,11}    254: {1,31}
     57: {2,8}     159: {2,16}    259: {4,12}
     62: {1,11}    166: {1,23}    267: {2,24}
     82: {1,13}    183: {2,18}    274: {1,33}
     85: {3,7}     187: {5,7}     295: {3,17}
     87: {2,10}    194: {1,25}    298: {1,35}
     91: {4,6}     203: {4,10}    301: {4,14}
     94: {1,15}    205: {3,13}    303: {2,26}
    111: {2,12}    206: {1,27}    314: {1,37}

A031215 looks at primes instead of semiprimes.
A300061 and A319241 (squarefree) look all numbers (not just semiprimes).
A338905 has this as union of even-indexed rows.
A338906 is the nonsquarefree version.
A338907 is the odd version.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A024697 is the sum of semiprimes of weight n.
A025129 is the sum of squarefree semiprimes of weight n.
A056239 gives the sum of prime indices of n.
A289182/A115392 list the positions of odd/even terms in A001358.
A320656 counts factorizations into squarefree semiprimes.
A332765 gives the greatest squarefree semiprime of weight n.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338904 groups semiprimes by weight.
A338911 lists products of pairs of primes both of even index.
A339114/A339115 give the least/greatest semiprime of weight n.
A339116 groups squarefree semiprimes by greater prime factor.
Cf. A000040, A001221, A001222, A087112, A098350, A112798, A168472, A338901, A338904, A339004, A339005.

Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&& EvenQ[Total[PrimePi/@First/@FactorInteger[#]]]&]

A338903 Number of integer partitions of the n-th squarefree semiprime into squarefree semiprimes.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 5, 4, 6, 5, 12, 14, 19, 22, 27, 36, 38, 51, 77, 86, 128, 141, 163, 163, 207, 233, 259, 260, 514, 657, 813, 983, 1010, 1215, 1255, 1720, 2112, 2256, 3171, 3370, 3499, 3864, 4103, 6292, 7313, 7620, 8374, 10650, 17579, 18462, 23034, 25180

Offset: 1

Gus Wiseman, Nov 24 2020

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

			The a(n) partitions for n = 1, 5, 7, 9, 10, 11, 13:
  6  21    26       34          35        38           46
     15,6  14,6,6   22,6,6      21,14     26,6,6       34,6,6
           10,10,6  14,14,6     15,14,6   22,10,6      26,14,6
                    14,10,10    15,10,10  14,14,10     21,15,10
                    10,6,6,6,6            14,6,6,6,6   22,14,10
                                          10,10,6,6,6  26,10,10
                                                       15,15,10,6
                                                       22,6,6,6,6
                                                       14,14,6,6,6
                                                       14,10,10,6,6
                                                       10,10,10,10,6
                                                       10,6,6,6,6,6,6

A002100 counts partitions into squarefree semiprimes.
A056768 uses primes instead of squarefree semiprimes.
A101048 counts partitions into semiprimes.
A338902 is the not necessarily squarefree version.
A339113 includes the Heinz numbers of these partitions.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A006881 lists squarefree semiprimes, with odd and even terms A046388 and A100484.
A320656 counts factorizations into squarefree semiprimes.
A338898/A338912/A338913 give prime indices of semiprimes, with sum/difference/product A176504/A176506/A087794.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes.
Cf. A000041, A000607, A065516, A115392, A128301, A320655, A320732, A320892, A320912, A338915, A338916.

nn=100;
sqs=Select[Range[nn],SquareFreeQ[#]&&PrimeOmega[#]==2&];
Table[Length[IntegerPartitions[n,All,sqs]],{n,sqs}]

a(n) = A002100(A006881(n)).

cp's OEIS Frontend

A112141 Product of the first n semiprimes.

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A338901 Position of the first appearance of prime(n) as a factor in the list of squarefree semiprimes.

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A338906 Semiprimes whose prime indices sum to an even number.

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A338910 Numbers of the form prime(x) * prime(y) where x and y are both odd.

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A338911 Numbers of the form prime(x) * prime(y) where x and y are both even.

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A339003 Numbers of the form prime(x) * prime(y) where x and y are distinct and both odd.

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A289182 Positions of odd semiprimes in A001358.

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