A084126 -id:A084126 - cp's OEIS frontend

A338907 Semiprimes whose prime indices sum to an odd number.

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.

A112141 Product of the first n semiprimes.

Original entry on oeis.org

4, 24, 216, 2160, 30240, 453600, 9525600, 209563200, 5239080000, 136216080000, 4495130640000, 152834441760000, 5349205461600000, 203269807540800000, 7927522494091200000, 364666034728195200000, 17868635701681564800000, 911300420785759804800000

Offset: 1

Jonathan Vos Post, Nov 28 2005

Semiprime analog of primorial (A002110). Equivalent for product of what A062198 is for sum.

			a(10) = 4*6*9*10*14*15*21*22*25*26 = 136216080000, the product of the first 10 semiprimes.
From _Gus Wiseman_, Dec 06 2020: (Start)
The sequence of terms together with their prime signatures begins:
                        4: (2)
                       24: (3,1)
                      216: (3,3)
                     2160: (4,3,1)
                    30240: (5,3,1,1)
                   453600: (5,4,2,1)
                  9525600: (5,5,2,2)
                209563200: (6,5,2,2,1)
               5239080000: (6,5,4,2,1)
             136216080000: (7,5,4,2,1,1)
            4495130640000: (7,6,4,2,2,1)
          152834441760000: (8,6,4,2,2,1,1)
         5349205461600000: (8,6,5,3,2,1,1)
       203269807540800000: (9,6,5,3,2,1,1,1)
      7927522494091200000: (9,7,5,3,2,2,1,1)
    364666034728195200000: (10,7,5,3,2,2,1,1,1)
  17868635701681564800000: (10,7,5,5,2,2,1,1,1)
(End)

T. D. Noe, Table of n, a(n) for n = 1..100

Partial sums of semiprimes are A062198.
First differences of semiprimes are A065516.
A000040 lists primes, with partial products A002110 (primorials).
A000142 lists factorials, with partial products A000178 (superfactorials).
A001358 lists semiprimes, with partial products A112141 (this sequence).
A005117 lists squarefree numbers, with partial products A111059.
A006881 lists squarefree semiprimes, with partial products A339191.
A101048 counts partitions into semiprimes (restricted: A338902).
A320655 counts factorizations into semiprimes.
A338898/A338912/A338913 give the prime indices of semiprimes, with product/sum/difference A087794/A176504/A176506.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with product/sum/difference A339361/A339362/A338900.
Cf. A001222, A084126, A084127, A115392, A168472, A320732, A320892.

A112141 := proc(n)
    mul(A001358(i),i=1..n) ;
end proc:
seq(A112141(n),n=1..10) ; # R. J. Mathar, Jun 30 2020

NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; f[n_] := Times @@ NestList[ NextSemiPrime@# &, 2^2, n - 1]; Array[f, 18] (* Robert G. Wilson v, Jun 13 2013 *)
FoldList[Times,Select[Range[30],PrimeOmega[#]==2&]] (* Gus Wiseman, Dec 06 2020 *)

a(n)=my(v=vector(n),i,k=3);while(iCharles R Greathouse IV, Apr 04 2013

from sympy import factorint
def aupton(terms):
    alst, k, p = [], 1, 1
    while len(alst) < terms:
        if sum(factorint(k).values()) == 2:
            p *= k
            alst.append(p)
        k += 1
    return alst
print(aupton(18)) # Michael S. Branicky, Aug 31 2021

a(n) = Product_{i=1..n} A001358(i).

A001222(a(n)) = 2*n.

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

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.

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]]]&]

A068318 Sum of prime factors of n-th semiprime.

Original entry on oeis.org

4, 5, 6, 7, 9, 8, 10, 13, 10, 15, 14, 19, 12, 21, 16, 25, 14, 20, 16, 22, 31, 33, 18, 26, 39, 18, 43, 22, 45, 32, 20, 34, 49, 24, 55, 40, 28, 61, 24, 22, 63, 44, 46, 26, 69, 50, 73, 24, 34, 75, 36, 81, 56, 30, 85, 26, 62, 91, 64, 42, 28, 99, 70, 103, 36, 46, 105, 30, 74, 109

Offset: 1

Reinhard Zumkeller, Feb 27 2002

nonn

Odd k is a term if and only if k - 2 is prime. Goldbach's conjecture implies that every even number k >= 4 is a term. - Jianing Song, May 26 2021

			a(2) = 5 because A001358(2) = 6 = 2*3 and 2+3 = 5.

T. D. Noe, Table of n, a(n) for n=1..1000
Robert G. Wilson v, Graph of n and a(n).

Semiprimes are in A001358.

Cf. A084126, A084127, A120831, A120832, A120833, A120834, A109313.

with(numtheory): a:=proc(n) if bigomega(n)=2 and nops(factorset(n))=2 then factorset(n)[1]+factorset(n)[2] elif bigomega(n)=2 then 2*sqrt(n) else fi end: seq(a(n),n=1..214); # Emeric Deutsch

f[n_] := Total[#1*#2 & @@@ FactorInteger@ n]; f@# & /@ Select[Range@300, PrimeOmega@# == 2 &] (* Robert G. Wilson v, Jan 23 2013 *)

s(n) = my(f = factor(n)); if(bigomega(f) == 2, f[,1]~*f[,2], 0);
list(lim) = select(x -> x > 0, apply(s, vector(lim, i, i))); \\ Amiram Eldar, May 15 2025

from math import isqrt
from sympy import primepi, primerange, factorint
def A068318(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 int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//p) for p in primerange(s+1)))
    return sum(p*e for p,e in factorint(bisection(f,n,n)).items()) # Chai Wah Wu, Apr 03 2025

a(n) = A001414(A001358(n)).
a(n) = A003415(A001358(n)), the arithmetic derivative.
If A001358(n) = s*p, then in this sequence a(n) = s+p.
a(n) = A084126(n)+A084127(n). - Reinhard Zumkeller, Jul 24 2006 [Typo in formula fixed by Zak Seidov, Aug 23 2014]

A096916 Lesser prime factor of n-th product of two distinct primes.

Original entry on oeis.org

2, 2, 2, 3, 3, 2, 2, 3, 2, 5, 2, 3, 2, 3, 5, 3, 2, 2, 5, 3, 2, 7, 2, 5, 2, 3, 7, 3, 2, 5, 2, 3, 5, 2, 7, 2, 3, 3, 7, 2, 3, 2, 11, 5, 2, 5, 2, 3, 7, 2, 3, 2, 3, 5, 11, 2, 3, 2, 7, 5, 2, 11, 3, 2, 5, 7, 2, 3, 13, 2, 5, 3, 13, 3, 11, 2, 7, 2, 5, 3, 2, 2, 7, 3, 5, 2, 13, 7, 2, 3, 5, 3, 2, 11, 3, 17, 2, 3

Offset: 1

Reinhard Zumkeller, Jul 15 2004

a(n)*A070647(n) = A006881(n); a(n) < A070647(n);

a(n) = A020639(A006881(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A084126, A195758.

a096916 = a020639 . a006881 -- Reinhard Zumkeller, Sep 23 2011

f[n_]:=Last/@FactorInteger[n]=={1,1};f1[n_]:=Min[First/@FactorInteger[n]];f2[n_]:=Max[First/@FactorInteger[n]];lst={};Do[If[f[n],AppendTo[lst,f1[n]]],{n,0,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 10 2010 *)

go(x)=my(v=List()); forprime(p=2, sqrtint(x\1), forprime(q=p+1, x\p, listput(v, [p*q,p]))); apply(v->v[2], vecsort(Vec(v),1)) \\ Charles R Greathouse IV, Sep 14 2015

A339115 Greatest semiprime whose prime indices sum to n.

Original entry on oeis.org

4, 6, 10, 15, 25, 35, 55, 77, 121, 143, 187, 221, 289, 323, 391, 493, 551, 667, 841, 899, 1073, 1189, 1369, 1517, 1681, 1763, 1961, 2183, 2419, 2537, 2809, 3127, 3481, 3599, 3953, 4189, 4489, 4757, 5041, 5293, 5723, 5963, 6499, 6887, 7171, 7663, 8051, 8633

Offset: 2

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}      493: {7,10}      2809: {16,16}
        6: {1,2}      551: {8,10}      3127: {16,17}
       10: {1,3}      667: {9,10}      3481: {17,17}
       15: {2,3}      841: {10,10}     3599: {17,18}
       25: {3,3}      899: {10,11}     3953: {17,19}
       35: {3,4}     1073: {10,12}     4189: {17,20}
       55: {3,5}     1189: {10,13}     4489: {19,19}
       77: {4,5}     1369: {12,12}     4757: {19,20}
      121: {5,5}     1517: {12,13}     5041: {20,20}
      143: {5,6}     1681: {13,13}     5293: {19,22}
      187: {5,7}     1763: {13,14}     5723: {17,25}
      221: {6,7}     1961: {12,16}     5963: {19,24}
      289: {7,7}     2183: {12,17}     6499: {19,25}
      323: {7,8}     2419: {13,17}     6887: {20,25}
      391: {7,9}     2537: {14,17}     7171: {20,26}

Robert Israel, Table of n, a(n) for n = 2..10000

A024697 is the sum of the same semiprimes.
A332765/A332877 is the squarefree case.
A338904 has this sequence as row maxima.
A339114 is the least 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.
A084126 and A084127 give the prime factors of semiprimes.
A087112 groups semiprimes by greater factor.
A320655 counts factorizations into 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.
A338907/A338906 list semiprimes of odd/even weight.
A338907/A338908 list squarefree semiprimes of odd/even weight.
Cf. A000040, A001221, A001222, A014342, A025129, A056239, A062198, A098350, A112798, A338905, A339116.

P:= [seq(ithprime(i),i=1..200)]:
[seq(max(seq(P[i]*P[j-i],i=1..j-1)),j=2..200)]; # Robert Israel, Dec 06 2020

Table[Max@@Table[Prime[k]*Prime[n-k],{k,n-1}],{n,2,30}]

A108541 Lesser prime factor of n-th golden semiprime.

Original entry on oeis.org

2, 3, 7, 11, 19, 23, 29, 37, 41, 67, 79, 97, 101, 103, 107, 131, 149, 163, 167, 181, 193, 227, 241, 271, 277, 283, 311, 353, 367, 379, 383, 397, 409, 419, 433, 449, 457, 487, 509, 563, 607, 613, 631, 661, 719, 733, 761, 797, 853, 883, 919, 941, 971, 997, 1031

Offset: 1

Reinhard Zumkeller, Jun 09 2005

nonn

abs(phi*a(n) - A108542(n)) < 1, where phi = golden ratio = (1+sqrt(5))/2.

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

Cf. A001622, A084126, A108539, A108540, A108542, A108543, A108544.

f[p_] := Module[{x = GoldenRatio * p}, p1 = NextPrime[x, -1]; p2 = NextPrime[p1]; q = If[x - p1 < p2 - x, p1, p2]; If[Abs[q - x] < 1, q, 0]]; seq = {}; p=1; Do[p = NextPrime[p]; If[f[p] > 0, AppendTo[seq, p]], {200}]; seq (* Amiram Eldar, Nov 28 2019 *)

a(n) = A108540(n)/A108542(n).

A089995 Products of pairs of distinct, non-consecutive primes.

Original entry on oeis.org

10, 14, 21, 22, 26, 33, 34, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 213

Offset: 1

Reinhard Zumkeller, Nov 19 2003

nonn

m belongs to the sequence iff m=A001358(k) for some k and A089994(k)>0.

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

Cf. A006881, A001358, A084126, A084127, A089994, A033476.

With[{nn = 213}, TakeWhile[#, # <= nn &] &@ Union@ Flatten@ Table[Function[p, p Prime@ Range[n + 2, Prime@ PrimePi[nn/p]]]@ Prime@ n, {n, Sqrt@ nn}]] (* Michael De Vlieger, Feb 02 2017 *)

cp's OEIS Frontend

A338907 Semiprimes whose prime indices sum to an odd number.

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A112141 Product of the first n semiprimes.

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A339114 Least semiprime whose prime indices sum to n.

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A338906 Semiprimes whose prime indices sum to an even number.

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A339112 Products of primes of semiprime index (A106349).

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A068318 Sum of prime factors of n-th semiprime.

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A096916 Lesser prime factor of n-th product of two distinct primes.

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