A338910 -id:A338910 - cp's OEIS frontend

A001358 Semiprimes (or biprimes): products of two primes.

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, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 169, 177, 178, 183, 185, 187

Offset: 1

N. J. A. Sloane, R. K. Guy

Numbers of the form p*q where p and q are primes, not necessarily distinct.
These numbers are sometimes called semiprimes or 2-almost primes.
Numbers n such that Omega(n) = 2 where Omega(n) = A001222(n) is the sum of the exponents in the prime decomposition of n.
Complement of A100959; A064911(a(n)) = 1. - Reinhard Zumkeller, Nov 22 2004
The graph of this sequence appears to be a straight line with slope 4. However, the asymptotic formula shows that the linearity is an illusion and in fact a(n)/n ~ log(n)/log(log(n)) goes to infinity. See also the graph of A066265 = number of semiprimes < 10^n.
For numbers between 33 and 15495, semiprimes are more plentiful than any other k-almost prime. See A125149.
Numbers that are divisible by exactly 2 prime powers (not including 1). - Jason Kimberley, Oct 02 2011
The (disjoint) union of A006881 and A001248. - Jason Kimberley, Nov 11 2015
An equivalent definition of this sequence is a'(n) = smallest composite number which is not divided by any smaller composite number a'(1),...,a'(n-1). - Meir-Simchah Panzer, Jun 22 2016
The above characterization can be simplified to "Composite numbers not divisible by a smaller term." This shows that this is the equivalent of primes computed via Eratosthenes's sieve, but starting with the set of composite numbers (i.e., complement of 1 union primes) instead of all positive integers > 1. It's easy to see that iterating the method (using Eratosthenes's sieve each time on the remaining numbers, complement of the previously computed set) yields numbers with bigomega = k for k = 0, 1, 2, 3, ..., i.e., {1}, A000040, this, A014612, etc. - M. F. Hasler, Apr 24 2019
For all n except n = 2, a(n) is a deficient number. - Amrit Awasthi, Sep 10 2024
It is reasonable to assume that the "comforting numbers" which John T. Williams found in Chapter 3 of Milne's book "The House at Pooh Corner" are these semiprimes. Winnie-the-Pooh wonders whether he has 14 or 15 honey pots and concludes: "It's sort of comforting." To arrange a semiprime number of honey pots in a rectangular way, let's say on a shelf, with the larger divisor parallel to the wall, there is only one solution and this is for a simple mind like Winnie-the-Pooh comforting. - Ruediger Jehn, Dec 12 2024

			From _Gus Wiseman_, May 27 2021: (Start)
The sequence of terms together with their prime factors begins:
   4 = 2*2     46 = 2*23     91 = 7*13    141 = 3*47
   6 = 2*3     49 = 7*7      93 = 3*31    142 = 2*71
   9 = 3*3     51 = 3*17     94 = 2*47    143 = 11*13
  10 = 2*5     55 = 5*11     95 = 5*19    145 = 5*29
  14 = 2*7     57 = 3*19    106 = 2*53    146 = 2*73
  15 = 3*5     58 = 2*29    111 = 3*37    155 = 5*31
  21 = 3*7     62 = 2*31    115 = 5*23    158 = 2*79
  22 = 2*11    65 = 5*13    118 = 2*59    159 = 3*53
  25 = 5*5     69 = 3*23    119 = 7*17    161 = 7*23
  26 = 2*13    74 = 2*37    121 = 11*11   166 = 2*83
  33 = 3*11    77 = 7*11    122 = 2*61    169 = 13*13
  34 = 2*17    82 = 2*41    123 = 3*41    177 = 3*59
  35 = 5*7     85 = 5*17    129 = 3*43    178 = 2*89
  38 = 2*19    86 = 2*43    133 = 7*19    183 = 3*61
  39 = 3*13    87 = 3*29    134 = 2*67    185 = 5*37
(End)

Archimedeans Problems Drive, Eureka, 17 (1954), 8.
Raymond Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter II, Problem 60.
Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 1, Teubner, Leipzig; third edition: Chelsea, New York (1974). See p. 211.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
John T. Williams, Pooh and the Philosophers, Dutton Books, 1995.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 10000 terms from T. D. Noe)
Dragos Crisan and Radek Erban, On the counting function of semiprimes, INTEGERS, Vol. 21 (2021), #A122.
Daniel A. Goldston, Sidney W. Graham, János Pintz and Cem Y. Yildirim, Small gaps between primes or almost primes, Transactions of the American Mathematical Society, Vol. 361, No. 10 (2009), pp. 5285-5330, arXiv preprint, arXiv:math/0506067 [math.NT], 2005.
Richard K. Guy, Letters to N. J. A. Sloane, June-August 1968
Sh. T. Ishmukhametov and F. F. Sharifullina, On distribution of semiprime numbers, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 53-59. English translation, Russian Mathematics, Vol. 58, No. 8 (2014), pp. 43-48, alternative link.
Donovan Johnson, Jonathan Vos Post, and Robert G. Wilson v, Selected n and a(n). (2.5 MB)
Dixon Jones, Quickie 593, Mathematics Magazine, Vol. 47, No. 3, May 1974, p. 167.
Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909. See Vol. 1, p. 211.
Xianmeng Meng, On sums of three integers with a fixed number of prime factors, Journal of Number Theory, Vol. 114, No. 1 (2005), pp. 37-65.
Michael Penn, What makes a number "good"?, YouTube video, 2022.
Carlos Rivera, Conjecture 108: On the counting function of semiprimes, The Prime Puzzles & Problems Connection.
Eric Weisstein's World of Mathematics, Semiprime.
Eric Weisstein's World of Mathematics, Almost Prime.
Wikipedia, Almost prime.
Robert G. Wilson v, Subsequences at various powers of 10.
Index to sequences related to sums of cubes
Index entries for "core" sequences

Cf. A064911 (characteristic function).
Cf. A048623, A048639, A000040 (primes), A014612 (products of 3 primes), A014613, A014614, A072000 ("pi" for semiprimes), A065516 (first differences).
Cf. A077554, A077555, A002024, A072966, A100592, A014673, A068318, A061299, A087718, A089994, A089995, A096916, A096932, A106550, A106554, A108541, A108542, A126663, A131284, A138510, A138511, A072931, A088183, A171963, A237040 (semiprimes of form n^3 + 1).
Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r=1), this sequence (r=2), A014612 (r=3), A014613 (r=4), A014614 (r=5), A046306 (r=6), A046308 (r=7), A046310 (r=8), A046312 (r=9), A046314 (r=10), A069272 (r=11), A069273 (r=12), A069274 (r=13), A069275 (r=14), A069276 (r=15), A069277 (r=16), A069278 (r=17), A069279 (r=18), A069280 (r=19), A069281 (r=20).
These are the Heinz numbers of length-2 partitions, counted by A004526.
The squarefree case is A006881 with odd/even terms A046388/A100484 (except 4).
Including primes gives A037143.
The odd/even terms are A046315/A100484.
Partial sums are A062198.
The prime factors are A084126/A084127.
Grouping by greater factor gives A087112.
The product/sum/difference of prime indices is A087794/A176504/A176506.
Positions of even/odd terms are A115392/A289182.
The terms with relatively prime/divisible prime indices are A300912/A318990.
Factorizations using these terms are counted by A320655.
The prime indices are A338898/A338912/A338913.
Grouping by weight (sum of prime indices) gives A338904, with row sums A024697.
The terms with even/odd weight are A338906/A338907.
The terms with odd/even prime indices are A338910/A338911.
The least/greatest term of weight n is A339114/A339115.
Cf. A014342, A098350, A112141, A320732, A332765, A339003/A339004, A339116.

a001358 n = a001358_list !! (n-1)
a001358_list = filter ((== 2) . a001222) [1..]

[n: n in [2..200] | &+[d[2]: d in Factorization(n)] eq 2]; // Bruno Berselli, Sep 09 2015

A001358 := proc(n) option remember; local a; if n = 1 then 4; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 2 then return a; end if; end do: end if; end proc:
seq(A001358(n), n=1..120) ; # R. J. Mathar, Aug 12 2010

Select[Range[200], Plus@@Last/@FactorInteger[#] == 2 &] (* Zak Seidov, Jun 14 2005 *)
Select[Range[200], PrimeOmega[#]==2&] (* Harvey P. Dale, Jul 17 2011 *)

select( isA001358(n)={bigomega(n)==2}, [1..199]) \\ M. F. Hasler, Apr 09 2008; added select() Apr 24 2019

list(lim)=my(v=List(),t);forprime(p=2, sqrt(lim), t=p;forprime(q=p, lim\t, listput(v,t*q))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Sep 11 2011

A1358=List(4); A001358(n)={while(#A1358M. F. Hasler, Apr 24 2019

from sympy import factorint
def ok(n): return sum(factorint(n).values()) == 2
print([k for k in range(1, 190) if ok(k)]) # Michael S. Branicky, Apr 30 2022

from math import isqrt
from sympy import primepi, prime
def A001358(n):
    def f(x): return int(n+x-sum(primepi(x//prime(k))-k+1 for k in range(1, primepi(isqrt(x))+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Jul 23 2024

a(n) ~ n*log(n)/log(log(n)) as n -> infinity [Landau, p. 211], [Ayoub].
Recurrence: a(1) = 4; for n > 1, a(n) = smallest composite number which is not a multiple of any of the previous terms. - Amarnath Murthy, Nov 10 2002
A174956(a(n)) = n. - Reinhard Zumkeller, Apr 03 2010
a(n) = A088707(n) - 1. - Reinhard Zumkeller, Feb 20 2012
Sum_{n>=1} 1/a(n)^s = (1/2)*(P(s)^2 + P(2*s)), where P is the prime zeta function. - Enrique Pérez Herrero, Jun 24 2012
sigma(a(n)) + phi(a(n)) - mu(a(n)) = 2*a(n) + 1. mu(a(n)) = ceiling(sqrt(a(n))) - floor(sqrt(a(n))). - Wesley Ivan Hurt, May 21 2013
mu(a(n)) = -Omega(a(n)) + omega(a(n)) + 1, where mu is the Moebius function (A008683), Omega is the count of prime factors with repetition, and omega is the count of distinct prime factors. - Alonso del Arte, May 09 2014
a(n) = A078840(2,n). - R. J. Mathar, Jan 30 2019
A100484 UNION A046315. - R. J. Mathar, Apr 19 2023
Conjecture: a(n)/n ~ (log(n)/log(log(n)))*(1-(M/log(log(n)))) as n -> oo, where M is the Mertens's constant (A077761). - Alain Rocchelli, Feb 02 2025

More terms from James Sellers, Aug 22 2000

A338899 Concatenated sequence of prime indices of squarefree semiprimes (A006881).

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 2, 3, 2, 4, 1, 5, 1, 6, 2, 5, 1, 7, 3, 4, 1, 8, 2, 6, 1, 9, 2, 7, 3, 5, 2, 8, 1, 10, 1, 11, 3, 6, 2, 9, 1, 12, 4, 5, 1, 13, 3, 7, 1, 14, 2, 10, 4, 6, 2, 11, 1, 15, 3, 8, 1, 16, 2, 12, 3, 9, 1, 17, 4, 7, 1, 18, 2, 13, 2, 14, 4, 8, 1, 19, 2, 15

Offset: 1

Gus Wiseman, Nov 16 2020

This is a triangle with two columns and strictly increasing rows, namely {A270650(n), A270652(n)}.

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:
      6: {1,2}     57: {2,8}     106: {1,16}    155: {3,11}
     10: {1,3}     58: {1,10}    111: {2,12}    158: {1,22}
     14: {1,4}     62: {1,11}    115: {3,9}     159: {2,16}
     15: {2,3}     65: {3,6}     118: {1,17}    161: {4,9}
     21: {2,4}     69: {2,9}     119: {4,7}     166: {1,23}
     22: {1,5}     74: {1,12}    122: {1,18}    177: {2,17}
     26: {1,6}     77: {4,5}     123: {2,13}    178: {1,24}
     33: {2,5}     82: {1,13}    129: {2,14}    183: {2,18}
     34: {1,7}     85: {3,7}     133: {4,8}     185: {3,12}
     35: {3,4}     86: {1,14}    134: {1,19}    187: {5,7}
     38: {1,8}     87: {2,10}    141: {2,15}    194: {1,25}
     39: {2,6}     91: {4,6}     142: {1,20}    201: {2,19}
     46: {1,9}     93: {2,11}    143: {5,6}     202: {1,26}
     51: {2,7}     94: {1,15}    145: {3,10}    203: {4,10}
     55: {3,5}     95: {3,8}     146: {1,21}    205: {3,13}

A270650 is the first column.
A270652 is the second column.
A320656 counts multiset partitions using these rows, or factorizations into squarefree semiprimes.
A338898 is the version including squares, with columns A338912 and A338913.
A338900 gives row differences.
A338901 gives the row numbers for first appearances.
A001221 and A001222 count distinct/all prime indices.
A001358 lists semiprimes.
A004526 counts 2-part partitions, with strict case shifted right once.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A046315 and A100484 list odd and even semiprimes.
A046388 lists odd squarefree semiprimes.
A166237 gives first differences of squarefree semiprimes.
Cf. A030229, A056239, A065516, A112798, A115392, A167171, A176506, A320893, A338904, A338906, A338907, A338910, A338911.

Join@@Cases[Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&],k_:>PrimePi/@First/@FactorInteger[k]]

A338913 Greater prime index of the n-th semiprime.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 5, 3, 6, 5, 7, 4, 8, 6, 9, 4, 7, 5, 8, 10, 11, 6, 9, 12, 5, 13, 7, 14, 10, 6, 11, 15, 8, 16, 12, 9, 17, 7, 5, 18, 13, 14, 8, 19, 15, 20, 6, 10, 21, 11, 22, 16, 9, 23, 6, 17, 24, 18, 12, 7, 25, 19, 26, 10, 13, 27, 8, 20, 28, 14, 11, 29, 21

Offset: 1

Gus Wiseman, Nov 20 2020

nonn

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

After the first three terms, there appear to be no adjacent equal terms.

			The semiprimes are:
  2*2, 2*3, 3*3, 2*5, 2*7, 3*5, 3*7, 2*11, 5*5, 2*13, ...
so the greater prime factors are:
  2, 3, 3, 5, 7, 5, 7, 11, 5, 13, ...
with indices:
  1, 2, 2, 3, 4, 3, 4, 5, 3, 6, ...

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

A115392 lists positions of first appearances of each positive integer.
A270652 is the squarefree case, with lesser part A270650.
A338898 has this as second column.
A338912 is the corresponding lesser prime index.
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.
A087794/A176504/A176506 are product/sum/difference of semiprime indices.
A338910/A338911 list products of pairs of odd/even-indexed primes.
Cf. A037143, A056239, A065516, A084126, A084127, A112798, A128301, A289182, A320732, A320892, A338900, A338904, A338909.

Table[Max[PrimePi/@First/@FactorInteger[n]],{n,Select[Range[100],PrimeOmega[#]==2&]}]

a(n) = A000720(A084127(n)).

A338898 Concatenated sequence of prime indices of semiprimes (A001358).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 1, 3, 1, 4, 2, 3, 2, 4, 1, 5, 3, 3, 1, 6, 2, 5, 1, 7, 3, 4, 1, 8, 2, 6, 1, 9, 4, 4, 2, 7, 3, 5, 2, 8, 1, 10, 1, 11, 3, 6, 2, 9, 1, 12, 4, 5, 1, 13, 3, 7, 1, 14, 2, 10, 4, 6, 2, 11, 1, 15, 3, 8, 1, 16, 2, 12, 3, 9, 1, 17, 4, 7, 5, 5, 1, 18, 2

Offset: 1

Gus Wiseman, Nov 15 2020

This is a triangle with two columns and weakly increasing rows, namely {A338912(n), A338913(n)}.

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 semiprimes together with their prime indices begins:
      4: {1,1}     46: {1,9}      91: {4,6}     141: {2,15}
      6: {1,2}     49: {4,4}      93: {2,11}    142: {1,20}
      9: {2,2}     51: {2,7}      94: {1,15}    143: {5,6}
     10: {1,3}     55: {3,5}      95: {3,8}     145: {3,10}
     14: {1,4}     57: {2,8}     106: {1,16}    146: {1,21}
     15: {2,3}     58: {1,10}    111: {2,12}    155: {3,11}
     21: {2,4}     62: {1,11}    115: {3,9}     158: {1,22}
     22: {1,5}     65: {3,6}     118: {1,17}    159: {2,16}
     25: {3,3}     69: {2,9}     119: {4,7}     161: {4,9}
     26: {1,6}     74: {1,12}    121: {5,5}     166: {1,23}
     33: {2,5}     77: {4,5}     122: {1,18}    169: {6,6}
     34: {1,7}     82: {1,13}    123: {2,13}    177: {2,17}
     35: {3,4}     85: {3,7}     129: {2,14}    178: {1,24}
     38: {1,8}     86: {1,14}    133: {4,8}     183: {2,18}
     39: {2,6}     87: {2,10}    134: {1,19}    185: {3,12}

A112798 restricted to rows of length 2 gives this triangle.
A115392 is the row number for the first appearance of each positive integer.
A176506 gives row differences.
A338899 is the squarefree version.
A338912 is column 1.
A338913 is column 2.
A001221 counts a number's distinct prime indices.
A001222 counts a number's prime indices.
A001358 lists semiprimes.
A004526 counts 2-part partitions.
A006881 lists squarefree semiprimes.
A037143 lists primes and semiprimes.
A046315 and A100484 list odd and even semiprimes.
A046388 and A100484 list odd and even squarefree semiprimes.
A065516 gives first differences of semiprimes.
A084126 and A084127 give the prime factors of semiprimes.
A270650 and A270652 give the prime indices of squarefree semiprimes.
A320655 counts factorizations into semiprimes.
Cf. A056239, A101048, A320892, A320912, A338900, A338901, A338904, A338906, A338907, A338910, A338911.

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

A338912 Lesser prime index of the n-th semiprime.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 4, 2, 3, 2, 1, 1, 3, 2, 1, 4, 1, 3, 1, 2, 4, 2, 1, 3, 1, 2, 3, 1, 4, 5, 1, 2, 2, 4, 1, 2, 1, 5, 3, 1, 3, 1, 2, 4, 1, 6, 2, 1, 2, 3, 5, 1, 2, 1, 4, 3, 1, 5, 2, 1, 3, 4, 1, 2, 6, 1, 3, 2, 6, 2, 5, 1, 4, 1, 3, 2, 1

Offset: 1

Gus Wiseman, Nov 20 2020

nonn

A semiprime is a product of any two prime numbers. 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 semiprimes are:
  2*2, 2*3, 3*3, 2*5, 2*7, 3*5, 3*7, 2*11, 5*5, 2*13, ...
so the lesser prime factors are:
  2, 2, 3, 2, 2, 3, 3, 2, 5, 2, ...
with indices:
  1, 1, 2, 1, 1, 2, 2, 1, 3, 1, ...

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

A084126 is the lesser prime factor (not index).
A084127 is the greater factor, with index A338913.
A115392 lists positions of ones.
A128301 lists positions of first appearances of each positive integer.
A270650 is the squarefree case, with greater part A270652.
A338898 has this as first column.
A001221 counts distinct prime indices.
A001222 counts prime indices.
A001358 lists semiprimes, with odds A046315 and evens A100484.
A006881 lists squarefree semiprimes, with odds A046388 and evens A100484.
A087794/A176504/A176506 are product/sum/difference of semiprime indices.
A338910/A338911 list products of pairs of odd/even-indexed primes.
Cf. A037143, A056239, A065516, A112798, A289182, A320732, A320892, A338900, A338904, A338909.

Table[Min[PrimePi/@First/@FactorInteger[n]],{n,Select[Range[100],PrimeOmega[#]==2&]}]

a(n) = A000720(A084126(n)).

A338904 Irregular triangle read by rows where row n lists all semiprimes whose prime indices sum to n.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 35, 34, 39, 49, 55, 38, 51, 65, 77, 46, 57, 85, 91, 121, 58, 69, 95, 119, 143, 62, 87, 115, 133, 169, 187, 74, 93, 145, 161, 209, 221, 82, 111, 155, 203, 247, 253, 289, 86, 123, 185, 217, 299, 319, 323, 94, 129, 205

Offset: 2

Gus Wiseman, Nov 28 2020

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.

			Triangle begins:
   4
   6
   9  10
  14  15
  21  22  25
  26  33  35
  34  39  49  55
  38  51  65  77
  46  57  85  91 121
  58  69  95 119 143
  62  87 115 133 169 187
  74  93 145 161 209 221
  82 111 155 203 247 253 289
  86 123 185 217 299 319 323
  94 129 205 259 341 361 377 391

A004526 gives row lengths.
A024697 gives row sums.
A087112 is a different triangle of semiprimes.
A098350 has antidiagonals with the same distinct terms as these rows.
A338905 is the squarefree case, with row sums A025129.
A338907/A338906 are the union of odd/even rows.
A339114/A339115 are the row minima/maxima.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A014342 is the self-convolution of primes.
A037143 lists primes and semiprimes.
A056239 gives sum of prime indices (Heinz weight).
A062198 gives partial sums of semiprimes.
A084126 and A084127 give the prime factors of semiprimes.
A289182/A115392 list the positions of odd/even terms in A001358.
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.
Cf. A000040, A001221, A001222, A005117, A112798, A320732, A332877, A338908, A338910, A338911, A339116.

Table[Sort[Table[Prime[k]*Prime[n-k],{k,n/2}]],{n,2,10}]

A338907 Semiprimes whose prime indices sum to an odd number.

Original entry on oeis.org

6, 14, 15, 26, 33, 35, 38, 51, 58, 65, 69, 74, 77, 86, 93, 95, 106, 119, 122, 123, 141, 142, 143, 145, 158, 161, 177, 178, 185, 201, 202, 209, 214, 215, 217, 219, 221, 226, 249, 262, 265, 278, 287, 291, 299, 302, 305, 309, 319, 323, 326, 327, 329, 346, 355

Offset: 1

Gus Wiseman, Nov 28 2020

nonn

All terms are squarefree (A005117).
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 semiprimes in A300063; the semiprimes in A332820. - Peter Munn, Dec 25 2020

			The sequence of terms together with their prime indices begins:
      6: {1,2}      95: {3,8}     202: {1,26}
     14: {1,4}     106: {1,16}    209: {5,8}
     15: {2,3}     119: {4,7}     214: {1,28}
     26: {1,6}     122: {1,18}    215: {3,14}
     33: {2,5}     123: {2,13}    217: {4,11}
     35: {3,4}     141: {2,15}    219: {2,21}
     38: {1,8}     142: {1,20}    221: {6,7}
     51: {2,7}     143: {5,6}     226: {1,30}
     58: {1,10}    145: {3,10}    249: {2,23}
     65: {3,6}     158: {1,22}    262: {1,32}
     69: {2,9}     161: {4,9}     265: {3,16}
     74: {1,12}    177: {2,17}    278: {1,34}
     77: {4,5}     178: {1,24}    287: {4,13}
     86: {1,14}    185: {3,12}    291: {2,25}
     93: {2,11}    201: {2,19}    299: {6,9}

A031368 looks at primes instead of semiprimes.
A098350 has this as union of odd-indexed antidiagonals.
A300063 looks at all numbers (not just semiprimes).
A338904 has this as union of odd-indexed rows.
A338906 is the even version.
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.
A338908 lists squarefree semiprimes of even weight.
A339114/A339115 give the least/greatest semiprime of weight n.
Cf. A000040, A001222, A014342, A024697, A062198, A112798, A300061, A319242, A320655, A338910, A339003.
Subsequence of A332820.

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

from math import isqrt
from sympy import primepi, primerange
def A338907(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)))
    return bisection(f,n,n) # Chai Wah Wu, Apr 03 2025

Complement of A338906 in A001358.

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.

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

cp's OEIS Frontend

A001358 Semiprimes (or biprimes): products of two primes.

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A338899 Concatenated sequence of prime indices of squarefree semiprimes (A006881).

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A338913 Greater prime index of the n-th semiprime.

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A338898 Concatenated sequence of prime indices of semiprimes (A001358).

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A338912 Lesser prime index of the n-th semiprime.

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A338904 Irregular triangle read by rows where row n lists all semiprimes whose prime indices sum to n.

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A338907 Semiprimes whose prime indices sum to an odd number.

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