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.

Showing 1-6 of 6 results.

A320340 Heinz numbers of double-free integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83
Offset: 1

Views

Author

Gus Wiseman, Jan 07 2019

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
An integer partition is double-free if no part is twice any other part.
Also numbers n such that if prime(m) divides n then prime(2m) does not divide n, i.e., numbers not divisible by any element of A319613.

Examples

			The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (), (1), (2), (11), (3), (4), (111), (22), (31), (5), (6), (41), (32), (1111), (7), (8), (311), (51), (9), (33), (61), (222), (411).

Links

Crossrefs

Cf. A018819, A056239, A087897, A101417, A112798, A120641, A276431, A305148, A319613, A323092, A323093.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[primeMS[#],2*primeMS[#]]=={}&]

A339116 Triangle of all squarefree semiprimes grouped by greater prime factor, read by rows.

Original entry on oeis.org

6, 10, 15, 14, 21, 35, 22, 33, 55, 77, 26, 39, 65, 91, 143, 34, 51, 85, 119, 187, 221, 38, 57, 95, 133, 209, 247, 323, 46, 69, 115, 161, 253, 299, 391, 437, 58, 87, 145, 203, 319, 377, 493, 551, 667, 62, 93, 155, 217, 341, 403, 527, 589, 713, 899
Offset: 2

Views

Author

Gus Wiseman, Dec 01 2020

Keywords

Comments

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

Examples

			Triangle begins:
   6
  10  15
  14  21  35
  22  33  55  77
  26  39  65  91 143
  34  51  85 119 187 221
  38  57  95 133 209 247 323
  46  69 115 161 253 299 391 437
  58  87 145 203 319 377 493 551 667
  62  93 155 217 341 403 527 589 713 899

Links

Crossrefs

A339194 gives row sums.
A100484 is column k = 1.
A001748 is column k = 2.
A001750 is column k = 3.
A006094 is column k = n - 1.
A090076 is column k = n - 2.
A319613 is the central column k = 2*n.
A087112 is the not necessarily squarefree version.
A338905 is a different triangle of squarefree semiprimes.
A339195 is the generalization to all squarefree numbers, row sums A339360.
A001358 lists semiprimes.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd terms A046388.
A024697 is the sum of semiprimes of weight n.
A025129 is the sum of squarefree semiprimes of weight n.
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, A112798, A168472, A320656, A338901, A339003, A339114/A339115.
Subsequence of A019565.

Programs

  • Mathematica
    Table[Prime[i]*Prime[j],{i,2,10},{j,i-1}]
  • PARI
    row(n) = {prime(n)*primes(n-1)}
{ for(n=2, 10, print(row(n))) } \\ Andrew Howroyd, Jan 19 2023

Formula

T(n,k) = prime(n) * prime(k) for k < n.

Extensions

Offset corrected by Andrew Howroyd, Jan 19 2023

A350838 Heinz numbers of partitions with no adjacent parts of quotient 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83
Offset: 1

Views

Author

Gus Wiseman, Jan 18 2022

Keywords

Comments

Differs from A320340 in having 105: (4,3,2), 315: (4,3,2,2), 455: (6,4,3), etc.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with no adjacent prime indices of quotient 1/2.

Examples

			The terms and their prime indices begin:
      1: {}            19: {8}             38: {1,8}
      2: {1}           20: {1,1,3}         39: {2,6}
      3: {2}           22: {1,5}           40: {1,1,1,3}
      4: {1,1}         23: {9}             41: {13}
      5: {3}           25: {3,3}           43: {14}
      7: {4}           26: {1,6}           44: {1,1,5}
      8: {1,1,1}       27: {2,2,2}         45: {2,2,3}
      9: {2,2}         28: {1,1,4}         46: {1,9}
     10: {1,3}         29: {10}            47: {15}
     11: {5}           31: {11}            49: {4,4}
     13: {6}           32: {1,1,1,1,1}     50: {1,3,3}
     14: {1,4}         33: {2,5}           51: {2,7}
     15: {2,3}         34: {1,7}           52: {1,1,6}
     16: {1,1,1,1}     35: {3,4}           53: {16}
     17: {7}           37: {12}            55: {3,5}

Crossrefs

The version with quotients >= 2 is counted by A000929, sets A018819.
<= 2 is A342191, counted by A342094.
< 2 is counted by A342096, sets A045690.
> 2 is counted by A342098, sets A040039.
The sets version (subsets of prescribed maximum) is counted by A045691.
These partitions are counted by A350837.
The strict case is counted by A350840.
For differences instead of quotients we have A350842, strict A350844.
The complement is A350845, counted by A350846.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
A003114 = strict partitions with no successions, ranked by A325160.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free integer partitions, ranked by A320340.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A000302, A001105, A003000, A018819, A094537, A120641, A154402, A319613, A323093, A337135, A342097, A342095.

Programs

  • Mathematica
    primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],And@@Table[FreeQ[Divide@@@Partition[primeptn[#],2,1],2],{i,2,PrimeOmega[#]}]&]

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

A350845 Heinz numbers of integer partitions with at least two adjacent parts of quotient 2.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 130, 132, 133, 138, 144, 147, 150, 156, 162, 168, 174, 180, 186, 189, 192, 195, 198, 204, 210, 216, 222, 228, 231, 234, 240, 246, 252, 258, 260, 264, 266, 270
Offset: 1

Views

Author

Gus Wiseman, Jan 20 2022

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with at least two adjacent prime indices of quotient 1/2.

Examples

			The terms and corresponding partitions begin:
   6: (2,1)
  12: (2,1,1)
  18: (2,2,1)
  21: (4,2)
  24: (2,1,1,1)
  30: (3,2,1)
  36: (2,2,1,1)
  42: (4,2,1)
  48: (2,1,1,1,1)
  54: (2,2,2,1)
  60: (3,2,1,1)
  63: (4,2,2)
  65: (6,3)
  66: (5,2,1)
  72: (2,2,1,1,1)
  78: (6,2,1)
  84: (4,2,1,1)
  90: (3,2,2,1)
  96: (2,1,1,1,1,1)

Crossrefs

The complement is A350838, counted by A350837.
The strict complement is counted by A350840.
These partitions are counted by A350846.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free integer partitions, ranked by A320340.
A325160 ranks strict partitions with no successions, counted by A003114.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A000929, A001105, A018819, A045690, A045691, A094537, A154402, A319613, A323093, A337135, A342094, A342095, A342098, A342191.

Programs

  • Mathematica
    primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],MemberQ[Divide@@@Partition[primeptn[#],2,1],2]&]

A143215 a(n) = prime(n) * Sum_{i=1..n} prime(i).

Original entry on oeis.org

4, 15, 50, 119, 308, 533, 986, 1463, 2300, 3741, 4960, 7289, 9758, 12083, 15416, 20193, 25960, 30561, 38056, 45369, 51976, 62489, 72542, 85707, 102820, 117261, 130192, 146697, 161320, 180009, 218440, 242481, 272356, 295653, 339124, 366477
Offset: 1

Views

Author

Gary W. Adamson, Jul 30 2008

Keywords

Comments

Row sums of triangle A087112.
Sum of semiprimes (A001358) with greater prime factor prime(n). - Gus Wiseman, Dec 06 2020

Examples

			The series begins (4, 15, 50, 119, 308,...) since the primes = (2, 3, 5, 7, 11,...) and partial sum of primes = (2, 5, 10, 17, 28,...).
a(5) = 308 = 11 * 28.
a(4) = 119 = sum of row 4 terms of triangle A087112: (14 + 21 + 35 + 49).

Links

Crossrefs

Row sums of A087112.
The squarefree version is A339194, row sums of A339116.
Semiprimes grouped by weight are A338904, with row sums A024697.
Squarefree semiprimes grouped by weight are A338905, with row sums A025129.
Squarefree numbers grouped by greatest prime factor are A339195, with row sums A339360.
A001358 lists semiprimes.
A006881 lists squarefree semiprimes.
A332765 is the greatest semiprime of weight n.
A338898/A338912/A338913 give the prime indices of semiprimes.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
Cf. A000040, A001222, A001748, A007504, A014342, A098350, A100484, A168472, A319613, A339003, A339114/A339115.

Programs

Formula

a(n) = A000040(n) * A007504(n).

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Sep 21 2009
Showing 1-6 of 6 results.

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