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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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&]]
Previous Showing 11-13 of 13 results.

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