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

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

Author

Gus Wiseman, Nov 20 2020

Keywords

• nonn

Examples

			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}
		

Crossrefs

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.

Programs

Maple

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

Mathematica

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

Python

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

Formula

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