A338906 -id:A338906 - cp's OEIS frontend

A339003 Numbers of the form prime(x) * prime(y) where x and y are distinct and both odd.

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

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

Original entry on oeis.org

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

Offset: 3

Gus Wiseman, Nov 28 2020

A squarefree semiprime is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

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

A004526 (shifted right) gives row lengths.
A025129 (shifted right) gives row sums.
A056239 gives sum of prime indices (Heinz weight).
A339116 is a different triangle whose diagonals are these rows.
A338904 is the not necessarily squarefree version, with row sums A024697.
A338907/A338908 are the union of odd/even rows.
A339114/A332765 are the row minima/maxima.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A087112 groups semiprimes by greater factor.
A168472 gives partial sums of squarefree semiprimes.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
Cf. A000040, A001221, A014342, A098350, A112798, A320656, A338901, A338906, A339003, A339004, A339005, A339115.

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

A339004 Numbers of the form prime(x) * prime(y) where x and y are distinct and both even.

Original entry on oeis.org

21, 39, 57, 87, 91, 111, 129, 133, 159, 183, 203, 213, 237, 247, 259, 267, 301, 303, 321, 339, 371, 377, 393, 417, 427, 453, 481, 489, 497, 519, 543, 551, 553, 559, 579, 597, 623, 669, 687, 689, 703, 707, 717, 749, 753, 789, 791, 793, 813, 817, 843, 879, 917

Offset: 1

Gus Wiseman, Nov 22 2020

nonn

The squarefree semiprimes in A332821. - Peter Munn, Dec 25 2020

			The sequence of terms together with their prime indices begins:
     21: {2,4}     267: {2,24}    543: {2,42}
     39: {2,6}     301: {4,14}    551: {8,10}
     57: {2,8}     303: {2,26}    553: {4,22}
     87: {2,10}    321: {2,28}    559: {6,14}
     91: {4,6}     339: {2,30}    579: {2,44}
    111: {2,12}    371: {4,16}    597: {2,46}
    129: {2,14}    377: {6,10}    623: {4,24}
    133: {4,8}     393: {2,32}    669: {2,48}
    159: {2,16}    417: {2,34}    687: {2,50}
    183: {2,18}    427: {4,18}    689: {6,16}
    203: {4,10}    453: {2,36}    703: {8,12}
    213: {2,20}    481: {6,12}    707: {4,26}
    237: {2,22}    489: {2,38}    717: {2,52}
    247: {6,8}     497: {4,20}    749: {4,28}
    259: {4,12}    519: {2,40}    753: {2,54}

A338911 is the not necessarily squarefree version.
A339003 is the odd instead of even version, with not necessarily squarefree version A338910.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A289182/A115392 list the positions of odd/even terms in A001358.
A300912 lists products of pairs of primes with relatively prime indices.
A318990 lists products of pairs of primes with divisible indices.
A320656 counts factorizations into squarefree semiprimes.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338904 groups semiprimes by weight.
A338906/A338907 list semiprimes of even/odd weight.
Cf. A000040, A001221, A001222, A056239, A112798, A166237, A195017, A320911, A338901, A338903, A339002.
Subsequence of A332821.

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

from math import isqrt
from sympy import primepi, primerange
def A339004(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p)-a>>1 for a,p in enumerate(primerange(isqrt(x)+1),1) if a&1^1)
    return bisection(f,n,n) # Chai Wah Wu, Apr 03 2025

Numbers m such that A001221(m) = A001222(m) = 2 and A195017(m) = -2. - Peter Munn, Dec 31 2020

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}]

A338908 Squarefree semiprimes whose prime indices sum to an even number.

Original entry on oeis.org

10, 21, 22, 34, 39, 46, 55, 57, 62, 82, 85, 87, 91, 94, 111, 115, 118, 129, 133, 134, 146, 155, 159, 166, 183, 187, 194, 203, 205, 206, 213, 218, 235, 237, 247, 253, 254, 259, 267, 274, 295, 298, 301, 303, 314, 321, 334, 335, 339, 341, 358, 365, 371, 377, 382

Offset: 1

Gus Wiseman, Nov 28 2020

nonn

			The sequence of terms together with their prime indices begins:
     10: {1,3}     115: {3,9}     213: {2,20}
     21: {2,4}     118: {1,17}    218: {1,29}
     22: {1,5}     129: {2,14}    235: {3,15}
     34: {1,7}     133: {4,8}     237: {2,22}
     39: {2,6}     134: {1,19}    247: {6,8}
     46: {1,9}     146: {1,21}    253: {5,9}
     55: {3,5}     155: {3,11}    254: {1,31}
     57: {2,8}     159: {2,16}    259: {4,12}
     62: {1,11}    166: {1,23}    267: {2,24}
     82: {1,13}    183: {2,18}    274: {1,33}
     85: {3,7}     187: {5,7}     295: {3,17}
     87: {2,10}    194: {1,25}    298: {1,35}
     91: {4,6}     203: {4,10}    301: {4,14}
     94: {1,15}    205: {3,13}    303: {2,26}
    111: {2,12}    206: {1,27}    314: {1,37}

A031215 looks at primes instead of semiprimes.
A300061 and A319241 (squarefree) look all numbers (not just semiprimes).
A338905 has this as union of even-indexed rows.
A338906 is the nonsquarefree version.
A338907 is the odd version.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A024697 is the sum of semiprimes of weight n.
A025129 is the sum of squarefree semiprimes of weight n.
A056239 gives the sum of prime indices of n.
A289182/A115392 list the positions of odd/even terms in A001358.
A320656 counts factorizations into squarefree semiprimes.
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.
A338904 groups semiprimes by weight.
A338911 lists products of pairs of primes both of even index.
A339114/A339115 give the least/greatest semiprime of weight n.
A339116 groups squarefree semiprimes by greater prime factor.
Cf. A000040, A001221, A001222, A087112, A098350, A112798, A168472, A338901, A338904, A339004, A339005.

Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&& EvenQ[Total[PrimePi/@First/@FactorInteger[#]]]&]

cp's OEIS Frontend

A339003 Numbers of the form prime(x) * prime(y) where x and y are distinct and both odd.

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

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A339004 Numbers of the form prime(x) * prime(y) where x and y are distinct and both even.

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A339115 Greatest semiprime whose prime indices sum to n.

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A338908 Squarefree semiprimes whose prime indices sum to an even number.

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