A270650 -id:A270650 - cp's OEIS frontend

A338911 Numbers of the form prime(x) * prime(y) where x and y are both even.

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

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}]

A339195 Triangle of squarefree numbers grouped by greatest prime factor, read by rows.

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 66, 77, 110, 154, 165, 231, 330, 385, 462, 770, 1155, 2310, 13, 26, 39, 65, 78, 91, 130, 143, 182, 195, 273, 286, 390, 429, 455, 546, 715, 858, 910, 1001, 1365, 1430, 2002, 2145, 2730, 3003, 4290, 5005, 6006, 10010, 15015, 30030

Offset: 0

Gus Wiseman, Dec 02 2020

Also Heinz numbers of subsets of {1..n} that contain n if n>0, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
A019565 in its triangle form, with each row's terms in increasing order. - Peter Munn, Feb 26 2021
From David James Sycamore, Jan 09 2025: (Start)
Alternative definition, with offset = 1: a(1) = 1. For n>1 if a(n-1) = A002110(k), a(n) = prime(k+1). Otherwise a(n) is the smallest novel squarefree number whose prime factors have already occurred as previous terms.
Permutation of A005117, Squarefree version A379746. (End)

			Triangle begins:
   1
   2
   3   6
   5  10  15  30
   7  14  21  35  42  70  105  210

Alois P. Heinz, Rows n = 0..14, flattened
Michael De Vlieger, Plot p | a(n) at (x,y) = (n,pi(p)), n = 0..2047, 12X vertical exaggeration.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2047, with a color function related to the order of a(n) in A019565.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2047, with a color function showing 1 in gray, primes in red, primorials in bright green, even squarefree semiprimes in yellow, odd squarefree semiprimes in light green, thereafter, progressively deeper green related to omega(a(n)) = m until m >= 6.

A011782 gives row lengths.
A339360 gives row sums.
A008578 (shifted) is column k = 1.
A100484 is column k = 2.
A001748 is column k = 3.
A002110 is column k = 2^(n-1).
A070826 is column k = 2^(n-1) - 1.
A209862 takes prime indices to binary indices in these terms.
A246867 groups squarefree numbers by Heinz weight, with row sums A147655.
A261144 divides the n-th row by prime(n), with row sums A054640.
A339116 is the restriction to semiprimes, with row sums A339194.
A005117 lists squarefree numbers, ordered lexicographically by prime factors: A019565.
A006881 lists squarefree semiprimes.
A072047 counts prime factors of squarefree numbers.
A319246 is the sum of prime indices of the n-th squarefree number.
A329631 lists prime indices of squarefree numbers, reversed: A319247.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
Cf. A001221, A014342, A014466, A019565, A098350, A112798, A320656, A326882, A338901, A379770.
Cf. A379746.

T:= proc(n) option remember; `if`(n=0, 1, (p-> map(
      x-> x*p, {seq(T(i), i=0..n-1)})[])(ithprime(n)))
    end:
seq(T(n), n=0..6);  # Alois P. Heinz, Jan 08 2025

Table[Prime[n]*Sort[Times@@Prime/@#&/@Subsets[Range[n-1]]],{n,5}]

For n > 1, T(n,k) = prime(n) * A261144(n-1,k).

a(n) = A019565(A379770(n)). - Michael De Vlieger, Jan 08 2025

Row n=0 (term 1) prepended by Alois P. Heinz, Jan 08 2025

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

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:
     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[#]]]&]

A339005 Numbers of the form prime(x) * prime(y) where x properly divides y. Squarefree semiprimes with divisible prime indices.

Original entry on oeis.org

6, 10, 14, 21, 22, 26, 34, 38, 39, 46, 57, 58, 62, 65, 74, 82, 86, 87, 94, 106, 111, 115, 118, 122, 129, 133, 134, 142, 146, 158, 159, 166, 178, 183, 185, 194, 202, 206, 213, 214, 218, 226, 235, 237, 254, 259, 262, 267, 274, 278, 298, 302, 303, 305, 314, 319

Offset: 1

Gus Wiseman, Dec 05 2020

nonn

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.

			The sequence of terms together with their prime indices begins:
    6: {1,2}    82: {1,13}  159: {2,16}  259: {4,12}
   10: {1,3}    86: {1,14}  166: {1,23}  262: {1,32}
   14: {1,4}    87: {2,10}  178: {1,24}  267: {2,24}
   21: {2,4}    94: {1,15}  183: {2,18}  274: {1,33}
   22: {1,5}   106: {1,16}  185: {3,12}  278: {1,34}
   26: {1,6}   111: {2,12}  194: {1,25}  298: {1,35}
   34: {1,7}   115: {3,9}   202: {1,26}  302: {1,36}
   38: {1,8}   118: {1,17}  206: {1,27}  303: {2,26}
   39: {2,6}   122: {1,18}  213: {2,20}  305: {3,18}
   46: {1,9}   129: {2,14}  214: {1,28}  314: {1,37}
   57: {2,8}   133: {4,8}   218: {1,29}  319: {5,10}
   58: {1,10}  134: {1,19}  226: {1,30}  321: {2,28}
   62: {1,11}  142: {1,20}  235: {3,15}  326: {1,38}
   65: {3,6}   146: {1,21}  237: {2,22}  334: {1,39}
   74: {1,12}  158: {1,22}  254: {1,31}  339: {2,30}

A300912 is the version for relative primality.
A318990 is the not necessarily squarefree version.
A339002 is the version for non-relative primality.
A339003 is the version for odd indices.
A339004 is the version for even indices
A001358 lists semiprimes.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
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.
Cf. A001221, A112798, A166237, A320911, A338901, A338904, A338909.

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

Equals A318990 \ A000290.

A338903 Number of integer partitions of the n-th squarefree semiprime into squarefree semiprimes.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 5, 4, 6, 5, 12, 14, 19, 22, 27, 36, 38, 51, 77, 86, 128, 141, 163, 163, 207, 233, 259, 260, 514, 657, 813, 983, 1010, 1215, 1255, 1720, 2112, 2256, 3171, 3370, 3499, 3864, 4103, 6292, 7313, 7620, 8374, 10650, 17579, 18462, 23034, 25180

Offset: 1

Gus Wiseman, Nov 24 2020

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

			The a(n) partitions for n = 1, 5, 7, 9, 10, 11, 13:
  6  21    26       34          35        38           46
     15,6  14,6,6   22,6,6      21,14     26,6,6       34,6,6
           10,10,6  14,14,6     15,14,6   22,10,6      26,14,6
                    14,10,10    15,10,10  14,14,10     21,15,10
                    10,6,6,6,6            14,6,6,6,6   22,14,10
                                          10,10,6,6,6  26,10,10
                                                       15,15,10,6
                                                       22,6,6,6,6
                                                       14,14,6,6,6
                                                       14,10,10,6,6
                                                       10,10,10,10,6
                                                       10,6,6,6,6,6,6

A002100 counts partitions into squarefree semiprimes.
A056768 uses primes instead of squarefree semiprimes.
A101048 counts partitions into semiprimes.
A338902 is the not necessarily squarefree version.
A339113 includes the Heinz numbers of these partitions.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A006881 lists squarefree semiprimes, with odd and even terms A046388 and A100484.
A320656 counts factorizations into squarefree semiprimes.
A338898/A338912/A338913 give prime indices of semiprimes, with sum/difference/product A176504/A176506/A087794.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes.
Cf. A000041, A000607, A065516, A115392, A128301, A320655, A320732, A320892, A320912, A338915, A338916.

nn=100;
sqs=Select[Range[nn],SquareFreeQ[#]&&PrimeOmega[#]==2&];
Table[Length[IntegerPartitions[n,All,sqs]],{n,sqs}]

a(n) = A002100(A006881(n)).

A339360 Sum of all squarefree numbers with greatest prime factor prime(n).

Original entry on oeis.org

1, 2, 9, 60, 504, 6336, 89856, 1645056, 33094656, 801239040, 24246190080, 777550233600, 29697402470400, 1250501433753600, 55083063155097600, 2649111037319577600, 143390180403000115200, 8619643674791667302400, 534710099148093259776000, 36412881178052121329664000

Offset: 0

Gus Wiseman, Dec 04 2020

nonn

			The initial terms are:
   1 = 1,
   2 = 2,
   9 = 3 + 6,
  60 = 5 + 10 + 15 + 30.

Robert Israel, Table of n, a(n) for n = 0..349

A010036 takes prime indices here to binary indices, row sums of A209862.
A048672 takes prime indices to binary indices in squarefree numbers.
A054640 divides the n-th term by prime(n), row sums of A261144.
A072047 counts prime factors of squarefree numbers.
A339194 is the restriction to semiprimes, row sums of A339116.
A339195 has this as row sums.
A002110 lists primorials.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A056239 is the sum of prime indices of n (Heinz weight).
A246867 groups squarefree numbers by weight, with row sums A147655.
A319246 is the sum of prime indices of the n-th squarefree number.
A319247 lists reversed prime indices of squarefree numbers.
A329631 lists prime indices of squarefree numbers.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
Cf. A001221, A014466, A098350, A168472, A282935, A302590, A326878, A338901.

f:= proc(n) local i;
  `if`(n=0, 1, ithprime(n)) *mul(1+ithprime(i),i=1..n-1)
end proc:
map(f, [$0..20]); # Robert Israel, Dec 08 2020

Table[Sum[Times@@Prime/@stn,{stn,Select[Subsets[Range[n]],MemberQ[#,n]&]}],{n,10}]

For n >= 1, a(n) = A054640(n-1) * prime(n).

a(0)=1 prepended by Alois P. Heinz, Jan 08 2025

cp's OEIS Frontend

A338911 Numbers of the form prime(x) * prime(y) where x and y are both even.

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

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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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A339195 Triangle of squarefree numbers grouped by greatest prime factor, read by rows.

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

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A339005 Numbers of the form prime(x) * prime(y) where x properly divides y. Squarefree semiprimes with divisible prime indices.

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