A338907 -id:A338907 - 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]]

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&]

A176506 Difference between the prime indices of the two factors of the n-th semiprime.

Original entry on oeis.org

0, 1, 0, 2, 3, 1, 2, 4, 0, 5, 3, 6, 1, 7, 4, 8, 0, 5, 2, 6, 9, 10, 3, 7, 11, 1, 12, 4, 13, 8, 2, 9, 14, 5, 15, 10, 6, 16, 3, 0, 17, 11, 12, 4, 18, 13, 19, 1, 7, 20, 8, 21, 14, 5, 22, 0, 15, 23, 16, 9, 2, 24, 17, 25, 6, 10, 26, 3, 18, 27, 11, 7, 28, 19, 1, 29, 12, 20, 2, 21, 4, 30, 8, 31, 13, 22

Offset: 1

Juri-Stepan Gerasimov, Apr 19 2010

Are there no adjacent equal terms? I have verified this up to n = 10^6. - Gus Wiseman, Dec 04 2020

			From _Gus Wiseman_, Dec 04 2020: (Start)
The sequence of semiprimes together with the corresponding differences begins:
   4: 1 - 1 = 0
   6: 2 - 1 = 1
   9: 2 - 2 = 0
  10: 3 - 1 = 2
  14: 4 - 1 = 3
  15: 3 - 2 = 1
  21: 4 - 2 = 2
  22: 5 - 1 = 4
  25: 3 - 3 = 0
  26: 6 - 1 = 5
  33: 5 - 2 = 3
(End)

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

Cf. A109313.
Cf. A001358, A049084, A084126, A084127.
A087794 is product of the same indices.
A176504 is the sum of the same indices.
A115392 lists positions of first appearances.
A128301 lists positions of 0's.
A172348 lists positions of 1's.
A338898 has this sequence as row differences.
A338900 is the squarefree case.
A338912/A338913 give the two prime indices of semiprimes.
A006881 lists squarefree semiprimes.
A024697 is the sum of semiprimes of weight n.
A056239 gives sum of prime indices (Heinz weight).
A087112 groups semiprimes by greater factor.
A270650/A270652/A338899 give the prime indices of squarefree semiprimes.
A338904 groups semiprimes by weight.
A338907/A338906 list semiprimes of odd/even weight.
A339114/A339115 give the least/greatest semiprime of weight n.
Cf. A000040, A001222, A062198, A112141, A112798, A325699, A320655, A339005.

isA001358 := proc(n) numtheory[bigomega](n) = 2 ; end proc:
A001358 := proc(n) option remember ; if n = 1 then return 4 ; else for a from procname(n-1)+1 do if isA001358(a) then return a; end if; end do; end if; end proc:
A084126 := proc(n) min(op(numtheory[factorset](A001358(n)))) ; end proc:
A084127 := proc(n) max(op(numtheory[factorset](A001358(n)))) ; end proc:
A176506 := proc(n) numtheory[pi](A084127(n)) - numtheory[pi](A084126(n)) ; end proc: seq(A176506(n),n=1..120) ; # R. J. Mathar, Apr 22 2010
# Alternative:
N:= 500: # to use the first N semiprimes
Primes:= select(isprime, [2,seq(i,i=3..N/2,2)]):
SP:= NULL:
for i from 1 to nops(Primes) do
  for j from 1 to i do
    sp:= Primes[i]*Primes[j];
    if sp > N then break fi;
    SP:= SP, [sp, i-j]
od od:
SP:= sort([SP],(s,t) -> s[1] t[2], SP); # Robert Israel, Jan 17 2019

M = 500; (* to use the first M semiprimes *)
primes = Select[Join[{2}, Range[3, M/2, 2]], PrimeQ];
SP = {};
For[i = 1, i <= Length[primes], i++,
  For[j = 1, j <= i, j++,
    sp = primes[[i]] primes[[j]];
    If[sp > M, Break []];
    AppendTo[SP, {sp, i - j}]
]];
SortBy[SP, First][[All, 2]] (* Jean-François Alcover, Jul 18 2020, after Robert Israel *)
Table[If[!SquareFreeQ[n],0,-Subtract@@PrimePi/@First/@FactorInteger[n]],{n,Select[Range[100],PrimeOmega[#]==2&]}] (* Gus Wiseman, Dec 04 2020 *)

lista(nn) = {my(vsp = select(x->(bigomega(x)==2), [1..nn])); vector(#vsp, k, my(f=factor(vsp[k])[,1]); primepi(vecmax(f)) - primepi(vecmin(f)));} \\ Michel Marcus, Jul 18 2020

a(n) = A049084(A084127(n)) - A049084(A084126(n)). [corrected by R. J. Mathar, Apr 22 2010]

a(n) = A338913(n) - A338912(n). - Gus Wiseman, Dec 04 2020

a(51) and a(69) corrected by R. J. Mathar, Apr 22 2010

A176504 a(n) = m + k where prime(m)*prime(k) = semiprime(n).

Original entry on oeis.org

2, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 7, 9, 8, 10, 8, 9, 8, 10, 11, 12, 9, 11, 13, 9, 14, 10, 15, 12, 10, 13, 16, 11, 17, 14, 12, 18, 11, 10, 19, 15, 16, 12, 20, 17, 21, 11, 13, 22, 14, 23, 18, 13, 24, 12, 19, 25, 20, 15, 12, 26, 21, 27, 14, 16, 28, 13, 22, 29, 17, 15, 30, 23, 13, 31

Offset: 1

Juri-Stepan Gerasimov, Apr 19 2010

nonn

			From _Gus Wiseman_, Dec 04 2020: (Start)
A semiprime (A001358) is a product of any two prime numbers. The sequence of all semiprimes together with their prime indices and weights begins:
   4: 1 + 1 = 2
   6: 1 + 2 = 3
   9: 2 + 2 = 4
  10: 1 + 3 = 4
  14: 1 + 4 = 5
  15: 2 + 3 = 5
  21: 2 + 4 = 6
  22: 1 + 5 = 6
  25: 3 + 3 = 6
  26: 1 + 6 = 7
(End)

A056239 is the version for not just semiprimes.
A087794 gives the product of the same two indices.
A176506 gives the difference of the same two indices.
A338904 puts the n-th semiprime in row a(n).
A001358 lists semiprimes.
A006881 lists squarefree semiprimes.
A338898/A338912/A338913 give the prime indices of semiprimes.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with product/sum/difference A339361/A339362/A338900.
Cf. A001222, A046315, A065516, A084126, A084127, A100484, A112798, A115392, A128301, A338906/A338907.

From R. J. Mathar, Apr 20 2010: (Start)
isA001358 := proc(n) numtheory[bigomega](n) = 2 ; end proc:
A001358 := proc(n) option remember ; if n = 1 then return 4 ; else for a from procname(n-1)+1 do if isA001358(a) then return a; end if; end do; end if; end proc:
A084126 := proc(n) min(op(numtheory[factorset](A001358(n)))) ; end proc:
A084127 := proc(n) max(op(numtheory[factorset](A001358(n)))) ; end proc:
A176504 := proc(n) numtheory[pi](A084126(n)) + numtheory[pi](A084127(n)) ; end proc: seq(A176504(n),n=1..80) ; (End)

Table[If[SquareFreeQ[n],Total[PrimePi/@First/@FactorInteger[n]],2*PrimePi[Sqrt[n]]],{n,Select[Range[100],PrimeOmega[#]==2&]}] (* Gus Wiseman, Dec 04 2020 *)

a(n) = A056239(A001358(n)) = A338912(n) + A338913(n). - Gus Wiseman, Dec 04 2020

sqrt(n/(log n log log n)) << a(n) << n/log log n. - Charles R Greathouse IV, Apr 17 2024

Entries checked by R. J. Mathar, Apr 20 2010

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}]

A332820 Integers in the multiplicative subgroup of positive rationals generated by the products of two consecutive primes and the cubes of primes. Numbers k for which A048675(k) is a multiple of three.

Original entry on oeis.org

1, 6, 8, 14, 15, 20, 26, 27, 33, 35, 36, 38, 44, 48, 50, 51, 58, 63, 64, 65, 68, 69, 74, 77, 84, 86, 90, 92, 93, 95, 106, 110, 112, 117, 119, 120, 122, 123, 124, 125, 141, 142, 143, 145, 147, 156, 158, 160, 161, 162, 164, 170, 171, 177, 178, 185, 188, 196, 198, 201, 202, 208, 209, 210, 214, 215, 216, 217, 219, 221, 225

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between this sequence, A332821 and A332822, which list the integers in respective cosets of the subgroup.
As the sequence lists the integers in a multiplicative subgroup of the positive rationals, the sequence is closed under multiplication and, provided the result is an integer, under division.
It follows that for any n in this sequence, all powers n^k are present (k >= 0), as are all cubes.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, the resulting numbers are a permutation of the full sequence; and if we take the square root of each square term we get the full sequence.
There are no primes in the sequence, therefore if k is present and p is a prime, k*p and k/p are absent (noting that k/p might not be an integer). This property extends from primes to all terms of A050376 (often called Fermi-Dirac primes), therefore to squares of primes, 4th powers of primes etc.
The terms are the even numbers in A332821 halved. The terms are also the numbers m such that 5m is in A332821, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332822, and so on for alternate primes: 7, 13, 19 etc.
The numbers that are half of the even terms of this sequence are in A332822, which consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332821, which consists exactly of those numbers. These properties extend in a pattern of alternating primes as described in the previous paragraph.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.
If m and n are in this sequence then so is m*n (the definition of "multiplicative semigroup"), while if n is in this sequence, and x is in the complement A359830, then n*x is in A359830. This essentially follows from the fact that A048675 is totally additive sequence. Compare to A329609. - Antti Karttunen, Jan 17 2023

Positions of zeros in A332823; equivalently, numbers in row 3k of A277905 for some k >= 0.
Cf. A048675, A195017, A332821, A332822, A353350 (characteristic function), A353348 (its Dirichlet inverse), A359830 (complement).
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: A000578\{0}, A006094, A090090, A099788, A245630 (A191002 in ascending order), A244726\{0}, A325698, A338471, A338556, A338907.
Subsequence of {1} U A268388.

Select[Range@ 225, Or[Mod[Total@ #, 3] == 0 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]], # == 1] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332820(n) =  { my(f = factor(n)); !((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); };

{a(n) : n >= 1} = {1} U {2 * A332822(k) : k >= 1} U {A003961(a(k)) : k >= 1}.
{a(n) : n >= 1} = {1} U {a(k)^2 : k >= 1} U {A331590(2, A332822(k)) : k >= 1}.
From Peter Munn, Mar 17 2021: (Start)
{a(n) : n >= 1} = {k : k >= 1, 3|A048675(k)}.
{a(n) : n >= 1} = {k : k >= 1, 3|A195017(k)}.
{a(n) : n >= 1} = {A332821(k)/2 : k >= 1, 2|A332821(k)}.
{a(n) : n >= 1} = {A332822(k)/3 : k >= 1, 3|A332822(k)}.
(End)

New name from Peter Munn, Mar 08 2021

A339114 Least semiprime whose prime indices sum to n.

Original entry on oeis.org

4, 6, 9, 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: 2

Gus Wiseman, Nov 28 2020

nonn

Converges to A100484.
After a(4) = 9, also the least squarefree semiprime whose prime indices sum to 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 terms together with their prime indices begins:
      4: {1,1}     106: {1,16}    254: {1,31}
      6: {1,2}     118: {1,17}    262: {1,32}
      9: {2,2}     122: {1,18}    274: {1,33}
     14: {1,4}     134: {1,19}    278: {1,34}
     21: {2,4}     142: {1,20}    298: {1,35}
     26: {1,6}     146: {1,21}    302: {1,36}
     34: {1,7}     158: {1,22}    314: {1,37}
     38: {1,8}     166: {1,23}    326: {1,38}
     46: {1,9}     178: {1,24}    334: {1,39}
     58: {1,10}    194: {1,25}    346: {1,40}
     62: {1,11}    202: {1,26}    358: {1,41}
     74: {1,12}    206: {1,27}    362: {1,42}
     82: {1,13}    214: {1,28}    382: {1,43}
     86: {1,14}    218: {1,29}    386: {1,44}
     94: {1,15}    226: {1,30}    394: {1,45}

A024697 is the sum of the same semiprimes.
A098350 has this sequence as antidiagonal minima.
A338904 has this sequence as row minima.
A339114 (this sequence) is the squarefree case for n > 4.
A339115 is the greatest among the same semiprimes.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A037143 lists primes and semiprimes.
A056239 gives the sum of prime indices of n.
A084126 and A084127 give the prime factors of semiprimes.
A087112 groups semiprimes by greater factor.
A320655 counts factorizations into semiprimes.
A332765/A332877 is 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.
A338907/A338906 list semiprimes of odd/even weight.
A338907/A338908 list squarefree semiprimes of odd/even weight.
Cf. A000040, A001221, A001222, A014342, A025129, A062198, A112798, A338905, A339116.

Table[Min@@Table[Prime[k]*Prime[n-k],{k,n-1}],{n,2,30}]
Take[DeleteDuplicates[SortBy[{Times@@#,Total[PrimePi[#]]}&/@Tuples[ Prime[ Range[ 200]],2],{Last,First}],GreaterEqual[#1[[2]],#2[[2]]]&][[All,1]],60] (* Harvey P. Dale, Sep 06 2022 *)

a(n) = vecmin(vector(n-1, k, prime(k)*prime(n-k))); \\ Michel Marcus, Dec 03 2020

A339116 Triangle of all squarefree semiprimes grouped by greater prime factor, read by rows.

Original entry on oeis.org

6, 10, 15, 14, 21, 35, 22, 33, 55, 77, 26, 39, 65, 91, 143, 34, 51, 85, 119, 187, 221, 38, 57, 95, 133, 209, 247, 323, 46, 69, 115, 161, 253, 299, 391, 437, 58, 87, 145, 203, 319, 377, 493, 551, 667, 62, 93, 155, 217, 341, 403, 527, 589, 713, 899

Offset: 2

Gus Wiseman, Dec 01 2020

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

			Triangle begins:
   6
  10  15
  14  21  35
  22  33  55  77
  26  39  65  91 143
  34  51  85 119 187 221
  38  57  95 133 209 247 323
  46  69 115 161 253 299 391 437
  58  87 145 203 319 377 493 551 667
  62  93 155 217 341 403 527 589 713 899

Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 rows)

A339194 gives row sums.
A100484 is column k = 1.
A001748 is column k = 2.
A001750 is column k = 3.
A006094 is column k = n - 1.
A090076 is column k = n - 2.
A319613 is the central column k = 2*n.
A087112 is the not necessarily squarefree version.
A338905 is a different triangle of squarefree semiprimes.
A339195 is the generalization to all squarefree numbers, row sums A339360.
A001358 lists semiprimes.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd terms A046388.
A024697 is the sum of semiprimes of weight n.
A025129 is the sum of squarefree semiprimes of weight n.
A332765 gives the greatest squarefree semiprime of weight n.
A338898/A338912/A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338904 groups semiprimes by weight.
A338907/A338908 list squarefree semiprimes of odd/even weight.
Cf. A000040, A001221, A014342, A098350, A112798, A168472, A320656, A338901, A339003, A339114/A339115.
Subsequence of A019565.

Table[Prime[i]*Prime[j],{i,2,10},{j,i-1}]

row(n) = {prime(n)*primes(n-1)}
{ for(n=2, 10, print(row(n))) } \\ Andrew Howroyd, Jan 19 2023

T(n,k) = prime(n) * prime(k) for k < n.

Offset corrected by Andrew Howroyd, Jan 19 2023

A332765 Consider all permutations p_i of the first n primes; a(n) is the minimum over p_i of the maximal product of two adjacent primes in the permutation.

Original entry on oeis.org

6, 10, 15, 22, 35, 55, 77, 91, 143, 187, 221, 253, 323, 391, 493, 551, 667, 713, 899, 1073, 1189, 1271, 1517, 1591, 1763, 1961, 2183, 2419, 2537, 2773, 3127, 3233, 3599, 3953, 4189, 4331, 4757, 4897, 5293, 5723, 5963, 6499, 6887, 7171, 7663, 8051, 8633, 8989, 9797, 9991, 10403, 10807

Offset: 2

Bobby Jacobs, Apr 23 2020

nonn

The optimal permutation of n primes is {p_n, p_1, p_n-1, p_2, …, p_ceiling(n/2)}. - Ivan N. Ianakiev, Apr 28 2020

Also the greatest squarefree semiprime whose prime indices sum to n + 1. A squarefree semiprime (A006881) 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. - Gus Wiseman, Dec 06 2020

			Here are the ways (up to reversal) to order the first four primes:
  2, 3, 5, 7: Products: 6, 15, 35;  Largest product: 35
  2, 3, 7, 5: Products: 6, 21, 35;  Largest product: 35
  2, 5, 3, 7: Products: 10, 15, 21; Largest product: 21
  2, 5, 7, 3: Products: 10, 35, 21; Largest product: 35
  2, 7, 3, 5: Products: 14, 21, 15; Largest product: 21
  2, 7, 5, 3: Products: 14, 35, 15; Largest product: 35
  3, 2, 5, 7: Products: 6, 10, 35;  Largest product: 35
  3, 2, 7, 5: Products: 6, 14, 35;  Largest product: 35
  3, 5, 2, 7: Products: 15, 10, 14; Largest product: 15
  3, 7, 2, 5: Products: 21, 14, 10; Largest product: 21
  5, 2, 3, 7: Products: 10, 6, 21;  Largest product: 21
  5, 3, 2, 7: Products: 15, 6, 14;  Largest product: 15
The minimum largest product is 15, so a(4) = 15.
From _Gus Wiseman_, Dec 06 2020: (Start)
The sequence of terms together with their prime indices begins:
      6: {1,2}     551: {8,10}    3127: {16,17}
     10: {1,3}     667: {9,10}    3233: {16,18}
     15: {2,3}     713: {9,11}    3599: {17,18}
     22: {1,5}     899: {10,11}   3953: {17,19}
     35: {3,4}    1073: {10,12}   4189: {17,20}
     55: {3,5}    1189: {10,13}   4331: {18,20}
     77: {4,5}    1271: {11,13}   4757: {19,20}
     91: {4,6}    1517: {12,13}   4897: {17,23}
    143: {5,6}    1591: {12,14}   5293: {19,22}
    187: {5,7}    1763: {13,14}   5723: {17,25}
    221: {6,7}    1961: {12,16}   5963: {19,24}
    253: {5,9}    2183: {12,17}   6499: {19,25}
    323: {7,8}    2419: {13,17}   6887: {20,25}
    391: {7,9}    2537: {14,17}   7171: {20,26}
    493: {7,10}   2773: {15,17}   7663: {22,25}
(End)

Cf. A332877, A333747.
A338904 and A338905 have this sequence as row maxima.
A339115 is the not necessarily squarefree version.
A001358 lists semiprimes.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A025129 gives the sum of squarefree semiprimes of weight n.
A056239 (weight) gives the sum of prime indices of n.
A320656 counts factorizations into squarefree 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.
A338907/A338908 list squarefree semiprimes of odd/even weight.
A339114 is the least (squarefree) semiprime of weight n.
A339116 groups squarefree semiprimes by greater prime factor.
Cf. A001221, A014342, A024697, A046388, A062198, A098350, A112798, A168472, A320655, A338901.

primes[n_]:=Reverse[Prime/@Range[n]]; partition[n_]:=Partition[primes[n],UpTo[Ceiling[n/2]]];
riffle[n_]:=Riffle[partition[n][[1]],Reverse[partition[n][[2]]]];
a[n_]:=Max[Table[riffle[n][[i]]*riffle[n][[i+1]],{i,1,n-1}]];a/@Range[2,53]
(* Ivan N. Ianakiev, Apr 28 2020 *)

It appears that a(n) = A332877(n - 1) for n > 5.

a(12)-a(13) from Jinyuan Wang, Apr 24 2020

More terms from Ivan N. Ianakiev, Apr 28 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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A338898 Concatenated sequence of prime indices of semiprimes (A001358).

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A176506 Difference between the prime indices of the two factors of the n-th semiprime.

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A176504 a(n) = m + k where prime(m)*prime(k) = semiprime(n).

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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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A332820 Integers in the multiplicative subgroup of positive rationals generated by the products of two consecutive primes and the cubes of primes. Numbers k for which A048675(k) is a multiple of three.

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