cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Gus Wiseman, Dec 05 2020

Keywords

Comments

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.

Examples

			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}

Crossrefs

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.

Programs

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

Formula

Equals A318990 \ A000290.

A338909 Numbers of the form prime(x) * prime(y) where x and y have a common divisor > 1.

Original entry on oeis.org

9, 21, 25, 39, 49, 57, 65, 87, 91, 111, 115, 121, 129, 133, 159, 169, 183, 185, 203, 213, 235, 237, 247, 259, 267, 289, 299, 301, 303, 305, 319, 321, 339, 361, 365, 371, 377, 393, 417, 427, 445, 453, 481, 489, 497, 515, 517, 519, 529, 543, 551, 553, 559, 565
Offset: 1

Views

Author

Gus Wiseman, Nov 20 2020

Keywords

Examples

			The sequence of terms together with their prime indices begins:
      9: {2,2}     169: {6,6}     319: {5,10}
     21: {2,4}     183: {2,18}    321: {2,28}
     25: {3,3}     185: {3,12}    339: {2,30}
     39: {2,6}     203: {4,10}    361: {8,8}
     49: {4,4}     213: {2,20}    365: {3,21}
     57: {2,8}     235: {3,15}    371: {4,16}
     65: {3,6}     237: {2,22}    377: {6,10}
     87: {2,10}    247: {6,8}     393: {2,32}
     91: {4,6}     259: {4,12}    417: {2,34}
    111: {2,12}    267: {2,24}    427: {4,18}
    115: {3,9}     289: {7,7}     445: {3,24}
    121: {5,5}     299: {6,9}     453: {2,36}
    129: {2,14}    301: {4,14}    481: {6,12}
    133: {4,8}     303: {2,26}    489: {2,38}
    159: {2,16}    305: {3,18}    497: {4,20}

Crossrefs

A082023 counts partitions with these as Heinz numbers, complement A023022.
A300912 is the complement in A001358.
A339002 is the squarefree case.
A001221 counts distinct prime indices.
A001222 counts prime indices.
A001358 lists semiprimes, with odds A046315 and evens A100484.
A004526 counts 2-part partitions, with strict case A140106 (shifted left).
A006881 lists squarefree semiprimes, with odds A046388 and evens A100484.
A176504/A176506/A087794 give sum/difference/product of semiprime indices.
A318990 lists semiprimes with divisible indices.
A320655 counts factorizations into semiprimes.
A338898, A338912, and A338913 give semiprime indices.
A338899, A270650, and A270652 give squarefree semiprime indices.
A338910 lists semiprimes with odd indices.
A338911 lists semiprimes with even indices.
Cf. A005117, A037143, A055684, A056239, A065516, A112798, A115392, A128301, A289182, A338900, A338904.

Programs

  • Mathematica
    Select[Range[100],PrimeOmega[#]==2&&GCD@@PrimePi/@First/@FactorInteger[#]>1&]

Formula

Equals A001358 \ A300912.
Equals A339002 \/ (A001248 \ {4}).

A339194 Sum of all squarefree semiprimes with greater prime factor prime(n).

Original entry on oeis.org

0, 6, 25, 70, 187, 364, 697, 1102, 1771, 2900, 3999, 5920, 8077, 10234, 13207, 17384, 22479, 26840, 33567, 40328, 46647, 56248, 65653, 77786, 93411, 107060, 119583, 135248, 149439, 167240, 202311, 225320, 253587, 276332, 316923, 343676, 381039, 421192, 458749
Offset: 1

Views

Author

Gus Wiseman, Dec 02 2020

Keywords

Examples

			The triangle A339116 with row sums equal to this sequence begins (n > 1):
    6 = 6
   25 = 10 + 15
   70 = 14 + 21 + 35
  187 = 22 + 33 + 55 + 77

Crossrefs

A025129 gives sums of squarefree semiprimes by weight, row sums of A338905.
A143215 is the not necessarily squarefree version, row sums of A087112.
A339116 is a triangle of squarefree semiprimes with these row sums.
A339360 looks at all squarefree numbers, row sums of A339195.
A001358 lists semiprimes.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd terms A046388.
A024697 is the sum of semiprimes of weight n.
A168472 gives partial sums of squarefree semiprimes.
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, A100484, A319613, A320656, A338901, A339003, A339114/A339115.

Programs

  • Mathematica
    Table[Sum[Prime[i]*Prime[j],{j,i-1}],{i,10}]
  • PARI
    a(n) = prime(n)*vecsum(primes(n-1)); \\ Michel Marcus, Jun 15 2024

Formula

a(n) = prime(n) * Sum_{k=1..n-1} prime(k) = prime(n) * A007504(n-1).
a(n) = A024447(n) - A024447(n-1).
a(n) = A034960(n) - A143215(n). - Marco Zárate, Jun 14 2024

A339362 Sum of prime indices of the n-th squarefree semiprime.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Dec 06 2020

Keywords

Comments

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.

Examples

			The sequence of all squarefree semiprimes together with the sums of their prime indices begins:
   6: 1 + 2 = 3
  10: 1 + 3 = 4
  14: 1 + 4 = 5
  15: 2 + 3 = 5
  21: 2 + 4 = 6
  22: 1 + 5 = 6
  26: 1 + 6 = 7
  33: 2 + 5 = 7
  34: 1 + 7 = 8
  35: 3 + 4 = 7

Crossrefs

A001358 lists semiprimes.
A003963 gives the product of prime indices of n.
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.
A332765/A339114 give the greatest/least squarefree semiprime of weight n.
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.
A338904 groups semiprimes by weight.
A338905 groups squarefree semiprimes by weight.
A338907/A338908 list squarefree semiprimes of odd/even weight.
A339116 groups squarefree semiprimes by greater prime factor.
Cf. A001221, A046388, A065516, A112798, A115392, A166237, A320656, A338901, A339003, A339004.

Programs

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

Formula

a(n) = A056239(A006881(n)).
a(n) = A270650(n) + A270652(n).

A339002 Numbers of the form prime(x) * prime(y) where x and y are distinct and have a common divisor > 1.

Original entry on oeis.org

21, 39, 57, 65, 87, 91, 111, 115, 129, 133, 159, 183, 185, 203, 213, 235, 237, 247, 259, 267, 299, 301, 303, 305, 319, 321, 339, 365, 371, 377, 393, 417, 427, 445, 453, 481, 489, 497, 515, 517, 519, 543, 551, 553, 559, 565, 579, 597, 611, 623, 669, 685, 687
Offset: 1

Views

Author

Gus Wiseman, Nov 22 2020

Keywords

Examples

			The sequence of terms together with their prime indices begins:
     21: {2,4}     235: {3,15}    393: {2,32}
     39: {2,6}     237: {2,22}    417: {2,34}
     57: {2,8}     247: {6,8}     427: {4,18}
     65: {3,6}     259: {4,12}    445: {3,24}
     87: {2,10}    267: {2,24}    453: {2,36}
     91: {4,6}     299: {6,9}     481: {6,12}
    111: {2,12}    301: {4,14}    489: {2,38}
    115: {3,9}     303: {2,26}    497: {4,20}
    129: {2,14}    305: {3,18}    515: {3,27}
    133: {4,8}     319: {5,10}    517: {5,15}
    159: {2,16}    321: {2,28}    519: {2,40}
    183: {2,18}    339: {2,30}    543: {2,42}
    185: {3,12}    365: {3,21}    551: {8,10}
    203: {4,10}    371: {4,16}    553: {4,22}
    213: {2,20}    377: {6,10}    559: {6,14}

Crossrefs

A300912 is the complement in A001358.
A338909 is the not necessarily squarefree version.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A339005 lists products of pairs of distinct primes of divisible 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.
A338910/A338911 list products of pairs of primes both of odd/even index.
A339003/A339004 list squarefree semiprimes of odd/even index.
Cf. A001221, A001222, A055684, A056239, A112798, A115392, A166237, A167171, A318990, A320892, A320911, A338901.

Programs

  • Mathematica
    Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&&GCD@@PrimePi/@First/@FactorInteger[#]>1&]

A339361 Product of prime indices of the n-th squarefree semiprime.

Original entry on oeis.org

2, 3, 4, 6, 8, 5, 6, 10, 7, 12, 8, 12, 9, 14, 15, 16, 10, 11, 18, 18, 12, 20, 13, 21, 14, 20, 24, 22, 15, 24, 16, 24, 27, 17, 28, 18, 26, 28, 32, 19, 30, 20, 30, 30, 21, 33, 22, 32, 36, 23, 34, 24, 36, 36, 35, 25, 38, 26, 40, 39, 27, 40, 40, 28, 42, 44, 29, 42
Offset: 1

Views

Author

Gus Wiseman, Dec 06 2020

Keywords

Comments

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.

Examples

			The sequence of all squarefree semiprimes together with the products of their prime indices begins:
   6: 1 * 2 = 2
  10: 1 * 3 = 3
  14: 1 * 4 = 4
  15: 2 * 3 = 6
  21: 2 * 4 = 8
  22: 1 * 5 = 5
  26: 1 * 6 = 6
  33: 2 * 5 = 10
  34: 1 * 7 = 7
  35: 3 * 4 = 12

Crossrefs

A001358 lists semiprimes.
A003963 gives the product of prime indices of n.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A025129 is the sum of squarefree semiprimes of weight n.
A332765/A339114 give the greatest/least squarefree semiprime of weight n.
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.
A338905 groups squarefree semiprimes by weight.
A338907/A338908 list squarefree semiprimes of odd/even weight.
A339116 groups squarefree semiprimes by greater prime factor.
Cf. A001221, A046388, A056239 (weight), A112798, A166237, A320656, A320911, A338901, A339002, A339003, A339004.

Programs

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

Formula

a(n) = A003963(A006881(n)).
a(n) = A270650(n) * A270652(n).

A339191 Partial products of squarefree semiprimes (A006881).

Original entry on oeis.org

6, 60, 840, 12600, 264600, 5821200, 151351200, 4994589600, 169816046400, 5943561624000, 225855341712000, 8808358326768000, 405184483031328000, 20664408634597728000, 1136542474902875040000, 64782921069463877280000, 3757409422028904882240000
Offset: 1

Views

Author

Gus Wiseman, Nov 30 2020

Keywords

Comments

A squarefree semiprime is a product of any two distinct prime numbers.
Do all terms belong to A242031 (weakly decreasing prime signature)?

Examples

			The sequence of terms together with their prime indices begins:
          6: {1,2}
         60: {1,1,2,3}
        840: {1,1,1,2,3,4}
      12600: {1,1,1,2,2,3,3,4}
     264600: {1,1,1,2,2,2,3,3,4,4}
    5821200: {1,1,1,1,2,2,2,3,3,4,4,5}
  151351200: {1,1,1,1,1,2,2,2,3,3,4,4,5,6}
The sequence of terms together with their prime signatures begins:
                   6: (1,1)
                  60: (2,1,1)
                 840: (3,1,1,1)
               12600: (3,2,2,1)
              264600: (3,3,2,2)
             5821200: (4,3,2,2,1)
           151351200: (5,3,2,2,1,1)
          4994589600: (5,4,2,2,2,1)
        169816046400: (6,4,2,2,2,1,1)
       5943561624000: (6,4,3,3,2,1,1)
     225855341712000: (7,4,3,3,2,1,1,1)
    8808358326768000: (7,5,3,3,2,2,1,1)
  405184483031328000: (8,5,3,3,2,2,1,1,1)

Crossrefs

A000040 lists the primes, with partial products A002110 (primorials).
A001358 lists semiprimes, with partial products A112141.
A002100 counts partitions into squarefree semiprimes (restricted: A338903)
A000142 lists factorial numbers, with partial products A000178.
A005117 lists squarefree numbers, with partial products A111059.
A006881 lists squarefree semiprimes, with partial sums A168472.
A166237 gives first differences of squarefree semiprimes.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A338898/A338912/A338913 give prime indices of semiprimes.
A338899/A270650/A270652 give prime indices of squarefree semiprimes.
A338901 gives first appearances in the list of squarefree semiprimes.
A339113 gives products of primes of squarefree semiprime index.
Cf. A001221, A112798, A167171, A320732, A320891, A320892, A320894, A320911, A338900, A338902, A339003, A339004.

Programs

  • Mathematica
    FoldList[Times,Select[Range[20],SquareFreeQ[#]&&PrimeOmega[#]==2&]]

A347047 Smallest squarefree semiprime whose prime indices sum to n.

Original entry on oeis.org

6, 10, 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: 3

Views

Author

Gus Wiseman, Aug 22 2021

Keywords

Comments

Compared to A001747, we have 21 instead of 22 and lack 2 and 4.
Compared to A100484 (shifted) we have 21 instead of 22 and lack 4.
Compared to A161344, we have 21 instead of 22 and lack 4 and 8.
Compared to A339114, we have 11 instead of 9 and lack 4.
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.
A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

Examples

			The initial terms and their prime indices:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   21: {2,4}
   26: {1,6}
   34: {1,7}
   38: {1,8}
   46: {1,9}

Crossrefs

The opposite version (greatest instead of smallest) is A332765.
These are the minima of rows of A338905.
The nonsquarefree version is A339114 (opposite: A339115).
A001358 lists semiprimes (squarefree: A006881).
A024697 adds up semiprimes by weight (squarefree: A025129).
A056239 adds up prime indices, row sums of A112798.
A246868 gives the greatest squarefree number whose prime indices sum to n.
A320655 counts factorizations into semiprimes (squarefree: A320656).
A338898, A338912, A338913 give the prime indices of semiprimes.
A338899, A270650, A270652 give the prime indices of squarefree semiprimes.
A339116 groups squarefree semiprimes by greater factor, sums A339194.
A339362 adds up prime indices of squarefree semiprimes.
Cf. A001221, A087112, A089994, A098350, A176504, A338900, A338901, A338904, A338907/A338908, A339005, A339191.

Programs

  • Mathematica
    Table[Min@@Select[Table[Times@@Prime/@y,{y,IntegerPartitions[n,{2}]}],SquareFreeQ],{n,3,50}]
  • Python
    from sympy import prime, sieve
def a(n):
    p = [0] + list(sieve.primerange(1, prime(n)+1))
    return min(p[i]*p[n-i] for i in range(1, (n+1)//2))
print([a(n) for n in range(3, 58)]) # Michael S. Branicky, Sep 05 2021
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