A100484 -id:A100484 - cp's OEIS frontend

A360677 Sum of the right half (exclusive) of the prime indices of n.

0, 0, 0, 1, 0, 2, 0, 1, 2, 3, 0, 2, 0, 4, 3, 2, 0, 2, 0, 3, 4, 5, 0, 3, 3, 6, 2, 4, 0, 3, 0, 2, 5, 7, 4, 4, 0, 8, 6, 4, 0, 4, 0, 5, 3, 9, 0, 3, 4, 3, 7, 6, 0, 4, 5, 5, 8, 10, 0, 5, 0, 11, 4, 3, 6, 5, 0, 7, 9, 4, 0, 4, 0, 12, 3, 8, 5, 6, 0, 4, 4, 13, 0, 6, 7

Offset: 1

Gus Wiseman, Mar 05 2023

nonn

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.

			The prime indices of 810 are {1,2,2,2,2,3}, with right half (exclusive) {2,2,3}, so a(810) = 7.
The prime indices of 3675 are {2,3,3,4,4}, with right half (exclusive) {4,4}, so a(3675) = 8.

Positions of 0's are 1 and A000040.
Positions of last appearances are A004171.
Positions of first appearances are A100484.
These partitions are counted by A360672.
The value k > 0 appears A360673(k) times, inclusive A360671.
The left version is A360676.
The inclusive version is A360679.
A112798 lists prime indices, length A001222, sum A056239, median* A360005.
A360616 gives half of bigomega (exclusive), inclusive A360617.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A001248, A026424, A359908, A359912, A360006, A360457.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Take[prix[n],-Floor[Length[prix[n]]/2]]],{n,100}]

Last position of k is 2^(2k+1).
A360676(n) + A360679(n) = A001222(n).
A360677(n) + A360678(n) = A001222(n).

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

A339112 Products of primes of semiprime index (A106349).

Original entry on oeis.org

1, 7, 13, 23, 29, 43, 47, 49, 73, 79, 91, 97, 101, 137, 139, 149, 161, 163, 167, 169, 199, 203, 227, 233, 257, 269, 271, 293, 299, 301, 313, 329, 343, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 511, 529, 553, 559, 577, 607, 611, 631, 637, 647

Offset: 1

Gus Wiseman, Mar 12 2021

nonn

A semiprime (A001358) is a product of any two prime numbers.

Also MM-numbers of labeled multigraphs with loops (without uncovered vertices). 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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with the corresponding multigraphs begins (A..F = 10..15):
     1:            149:   (34)     313:     (36)
     7:   (11)     161: (11)(22)   329:   (11)(23)
    13:   (12)     163:   (18)     343: (11)(11)(11)
    23:   (22)     167:   (26)     347:     (29)
    29:   (13)     169: (12)(12)   373:     (1C)
    43:   (14)     199:   (19)     377:   (12)(13)
    47:   (23)     203: (11)(13)   389:     (45)
    49: (11)(11)   227:   (44)     421:     (1D)
    73:   (24)     233:   (27)     439:     (37)
    79:   (15)     257:   (35)     443:     (1E)
    91: (11)(12)   269:   (28)     449:     (2A)
    97:   (33)     271:   (1A)     467:     (46)
   101:   (16)     293:   (1B)     487:     (2B)
   137:   (25)     299: (12)(22)   491:     (1F)
   139:   (17)     301: (11)(14)   499:     (38)

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

These primes (of semiprime index) are listed by A106349.
The strict (squarefree) case is A340020.
The prime instead of semiprime version:
  primes: A006450
  products: A076610
  strict: A302590
The nonprime instead of semiprime version:
  primes: A007821
  products: A320628
  odd: A320629
  strict: A340104
  odd strict: A340105
The squarefree semiprime instead of semiprime version:
  strict: A309356
  primes: A322551
  products: A339113
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A006881 lists squarefree semiprimes.
A037143 lists primes and semiprimes (and 1).
A056239 gives the sum of prime indices, which are listed by A112798.
A084126 and A084127 give the prime factors of semiprimes.
A101048 counts partitions into semiprimes.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A320892 lists even-omega non-products of distinct semiprimes.
A320911 lists products of squarefree semiprimes (Heinz numbers of A338914).
A320912 lists products of distinct semiprimes (Heinz numbers of A338916).
A338898, A338912, and A338913 give the prime indices of semiprimes.
MM-numbers: A255397 (normal), A302478 (set multisystems), A320630 (set multipartitions), A302494 (sets of sets), A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A328514 (connected sets of sets), A329559 (clutters), A340019 (half-loop graphs).
Cf. A000040, A000720, A001055, A001222, A003963, A005117, A007097, A289509, A320461.

N:= 1000: # for terms up to N
SP:= {}: p:= 1:
for i from 1 do
  p:= nextprime(p);
  if 2*p > N then break fi;
  Q:= map(t -> p*t, select(isprime, {2,seq(i,i=3..min(p,N/p),2)}));
  SP:= SP union Q;
od:
SP:= sort(convert(SP,list)):
PSP:= map(ithprime,SP):
R:= {1}:
for p in PSP do
  Rp:= {}:
  for k from 1 while p^k <= N do
    Rpk:= select(`<=`,R, N/p^k);
    Rp:= Rp union map(`*`,Rpk, p^k);
  od;
  R:= R union Rp;
od:
sort(convert(R,list)); # Robert Israel, Nov 03 2024

semiQ[n_]:=PrimeOmega[n]==2;
Select[Range[100],FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;!semiQ[PrimePi[p]]]&]

A059788 a(n) = largest prime < 2*prime(n).

Original entry on oeis.org

3, 5, 7, 13, 19, 23, 31, 37, 43, 53, 61, 73, 79, 83, 89, 103, 113, 113, 131, 139, 139, 157, 163, 173, 193, 199, 199, 211, 211, 223, 251, 257, 271, 277, 293, 293, 313, 317, 331, 337, 353, 359, 379, 383, 389, 397, 421, 443, 449, 457, 463, 467, 479, 499, 509, 523

Offset: 1

Labos Elemer, Feb 22 2001

nonn

Also, smallest member of the first pair of consecutive primes such that between them is a composite number divisible by the n-th prime. - Amarnath Murthy, Sep 25 2002

Except for its initial term, A006992 is a subsequence based on iteration of n -> A151799(2n). The range of this sequence is a subset of A065091. - M. F. Hasler, May 08 2016

			n=18: p(18)=61, so a(18) is the largest prime below 2*61=122, which is 113.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A005382, A005383, A005384, A005385, A055496, A006992, A052248.

with(numtheory):
A059788 := proc(n)
    prevprime(2*ithprime(n)) ;
end proc:
seq(A059788(n),n=1..50) ; # R. J. Mathar, May 08 2016

a[n_] := Prime[PrimePi[2Prime[n]]]
NextPrime[2*Prime[Range[100]], -1] (* Zak Seidov, May 08 2016 *)

a(n) = precprime(2*prime(n)); \\ Michel Marcus, May 08 2016

a(n) = A007917(A100484(n)). - R. J. Mathar, May 08 2016

A086005 Semiprimes sandwiched between semiprimes.

Original entry on oeis.org

34, 86, 94, 122, 142, 202, 214, 218, 302, 394, 446, 634, 698, 842, 922, 1042, 1138, 1262, 1346, 1402, 1642, 1762, 1838, 1894, 1942, 1982, 2102, 2182, 2218, 2306, 2362, 2434, 2462, 2518, 2642, 2722, 2734, 3098, 3386, 3602, 3694, 3866, 3902, 3958, 4286, 4414

Offset: 1

Reinhard Zumkeller, Jul 07 2003

nonn

These are some of the balanced semiprimes (see A213025). - Alonso del Arte, Jun 04 2012

			94 = 47*2: 94 - 1 = 3*31 and 94 + 1 = 5*19, therefore 94 is in the sequence.

T. D. Noe, Table of n, a(n) for n=1..10000
Eric Weisstein's World of Mathematics, Semiprime

Cf. A001358, A056809, A064911, A100484, A123255.

a086005 n = a086005_list !! (n-1)
a086005_list = filter
   (\x -> a064911 (x - 1) == 1 && a064911 (x + 1) == 1) a100484_list
-- Reinhard Zumkeller, Aug 08 2013, Jun 10 2012

u[n_]:=Plus@@Last/@FactorInteger[n]==2;lst={};Do[If[u[n],sp=n;If[u[sp-1]&&u[sp+1],AppendTo[lst,sp]]],{n,8!}];lst  (* Vladimir Joseph Stephan Orlovsky, Nov 16 2009 *)
(* First run program for A109611 to define semiPrimeQ *) Select[Range[4000], Union[{semiPrimeQ[# - 1], semiPrimeQ[#], semiPrimeQ[# + 1]}] == {True} &] (* Alonso del Arte, Jun 03 2012 *)
Select[Partition[Range@ 4000, 3, 1], Union@ PrimeOmega@ # == {2} &][[All, 2]] (* Michael De Vlieger, Jun 14 2017 *)

from itertools import count, islice
from sympy import factorint, isprime
def agen(): # generator of terms
    nxt = 0
    for k in count(2, 2):
        prv, nxt = nxt, sum(factorint(k+1).values())
        if prv == nxt == 2 and isprime(k//2): yield k
print(list(islice(agen(), 46))) # Michael S. Branicky, Nov 26 2022

a(n) = 2*A086006(n).

a(n) = A056809(n)+1. - Zak Seidov, Sep 30 2012

A138636 a(n) = 6 * prime(n).

Original entry on oeis.org

12, 18, 30, 42, 66, 78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362, 1374

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 14 2008

Column 5 of A272214. - Omar E. Pol, Apr 29 2016

			2*6=12, 3*6=18, ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A001748, A001749, A001750, A100484.

[6*p: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

Mathematica
```
6*Prime[Range[100]]
```

vector(50, n, 6*prime(n)) \\ G. C. Greubel, Feb 02 2019

[6*nth_prime(n) for n in (1..50)] # G. C. Greubel, Feb 02 2019

A338905 Irregular triangle read by rows where row n lists all squarefree semiprimes with prime indices summing to n.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 33, 35, 34, 39, 55, 38, 51, 65, 77, 46, 57, 85, 91, 58, 69, 95, 119, 143, 62, 87, 115, 133, 187, 74, 93, 145, 161, 209, 221, 82, 111, 155, 203, 247, 253, 86, 123, 185, 217, 299, 319, 323, 94, 129, 205, 259, 341, 377, 391, 106, 141

Offset: 3

Gus Wiseman, Nov 28 2020

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.

			Triangle begins:
   6
  10
  14  15
  21  22
  26  33  35
  34  39  55
  38  51  65  77
  46  57  85  91
  58  69  95 119 143
  62  87 115 133 187
  74  93 145 161 209 221
  82 111 155 203 247 253
  86 123 185 217 299 319 323

A004526 (shifted right) gives row lengths.
A025129 (shifted right) gives row sums.
A056239 gives sum of prime indices (Heinz weight).
A339116 is a different triangle whose diagonals are these rows.
A338904 is the not necessarily squarefree version, with row sums A024697.
A338907/A338908 are the union of odd/even rows.
A339114/A332765 are the row minima/maxima.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A087112 groups semiprimes by greater factor.
A168472 gives partial sums of 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.
Cf. A000040, A001221, A014342, A098350, A112798, A320656, A338901, A338906, A339003, A339004, A339005, A339115.

Table[Sort[Table[Prime[k]*Prime[n-k],{k,(n-1)/2}]],{n,3,10}]

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

cp's OEIS Frontend

A360677 Sum of the right half (exclusive) of the prime indices of n.

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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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A339112 Products of primes of semiprime index (A106349).

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A059788 a(n) = largest prime < 2*prime(n).

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A086005 Semiprimes sandwiched between semiprimes.

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