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 1-4 of 4 results.

A336426 Numbers that cannot be written as a product of superprimorials {2, 12, 360, 75600, ...}.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76
Offset: 1

Views

Author

Gus Wiseman, Jul 26 2020

Keywords

Comments

The n-th superprimorial is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

Examples

			We have 288 = 2*12*12 so 288 is not in the sequence.

Crossrefs

A181818 is the complement.
A336497 is the version for superfactorials.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A022915 counts permutations of prime indices of superprimorials.
A317829 counts factorizations of superprimorials.
A336417 counts perfect-power divisors of superprimorials.
Cf. A000325, A005117, A076954, A124010, A294068, A336419, A336420, A336421, A336496, A336500, A336568.

Programs

  • Mathematica
    chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[100],facsusing[Array[chern,30],#]=={}&]

A336496 Products of superfactorials (A000178).

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 384, 512, 576, 768, 1024, 1152, 1536, 1728, 2048, 2304, 3072, 3456, 4096, 4608, 6144, 6912, 8192, 9216, 12288, 13824, 16384, 18432, 20736, 24576, 27648, 32768, 34560, 36864, 41472, 49152, 55296
Offset: 1

Views

Author

Gus Wiseman, Aug 03 2020

Keywords

Comments

First differs from A317804 in having 34560, which is the first term with more than two distinct prime factors.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    4: {1,1}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   48: {1,1,1,1,2}
   64: {1,1,1,1,1,1}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
  144: {1,1,1,1,2,2}
  192: {1,1,1,1,1,1,2}
  256: {1,1,1,1,1,1,1,1}
  288: {1,1,1,1,1,2,2}
  384: {1,1,1,1,1,1,1,2}
  512: {1,1,1,1,1,1,1,1,1}

Crossrefs

A001013 is the version for factorials, with complement A093373.
A181818 is the version for superprimorials, with complement A336426.
A336497 is the complement.
A000178 lists superfactorials.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A049711 is the minimum prime multiplicity in A000178.
A174605 is the maximum prime multiplicity in A000178.
A303279 counts prime factors of superfactorials.
A317829 counts factorizations of superprimorials.
A322583 counts factorizations into factorials.
A325509 counts factorizations of factorials into factorials.
Cf. A000142, A000720, A007489, A011371, A022559, A022915, A027423, A034878, A034876, A076954, A115627, A294068.

Programs

  • Mathematica
    supfac[n_]:=Product[k!,{k,n}];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[1000],facsusing[Rest[Array[supfac,30]],#]!={}&]

A336620 Numbers that are not a product of elements of A304711.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 42, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 78, 79, 81, 83, 87, 89, 91, 97, 101, 103, 105, 107, 109, 111, 113, 114, 115, 117, 121, 125, 126, 127, 129, 130, 131, 133, 137, 139, 147, 149
Offset: 1

Views

Author

Gus Wiseman, Aug 02 2020

Keywords

Comments

A304711 lists numbers whose distinct prime indices are pairwise coprime.
The first term divisible by 4 is a(421) = 1092.

Examples

			The sequence of terms together with their prime indices begins:
      3: {2}         39: {2,6}       78: {1,2,6}
      5: {3}         41: {13}        79: {22}
      7: {4}         42: {1,2,4}     81: {2,2,2,2}
      9: {2,2}       43: {14}        83: {23}
     11: {5}         47: {15}        87: {2,10}
     13: {6}         49: {4,4}       89: {24}
     17: {7}         53: {16}        91: {4,6}
     19: {8}         57: {2,8}       97: {25}
     21: {2,4}       59: {17}       101: {26}
     23: {9}         61: {18}       103: {27}
     25: {3,3}       63: {2,2,4}    105: {2,3,4}
     27: {2,2,2}     65: {3,6}      107: {28}
     29: {10}        67: {19}       109: {29}
     31: {11}        71: {20}       111: {2,12}
     37: {12}        73: {21}       113: {30}

Links

Crossrefs

A336426 is the version for superprimorials, with complement A181818.
A336497 is the version for superfactorials, with complement A336496.
A336735 is the complement.
A000837 counts relatively prime partitions, with strict case A007360.
A001055 counts factorizations.
A302696 lists numbers with coprime prime indices.
A304711 lists numbers with coprime distinct prime indices.
Cf. A007916, A112798, A302569, A327516, A333228, A335238, A335239, A335240, A335241, A336424, A336568, A336736.

Programs

  • Mathematica
    nn=100;
dat=Select[Range[nn],CoprimeQ@@PrimePi/@First/@FactorInteger[#]&];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[nn],facsusing[dat,#]=={}&]

A336735 Products of elements of A304711.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 34, 35, 36, 38, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 75, 76, 77, 80, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 106
Offset: 1

Views

Author

Gus Wiseman, Aug 02 2020

Keywords

Comments

A304711 lists numbers whose distinct prime indices are pairwise coprime.
First differs from A304711 in having 84.

Examples

			The sequence of terms together with their prime indices begins:
      1: {}            28: {1,1,4}         52: {1,1,6}
      2: {1}           30: {1,2,3}         54: {1,2,2,2}
      4: {1,1}         32: {1,1,1,1,1}     55: {3,5}
      6: {1,2}         33: {2,5}           56: {1,1,1,4}
      8: {1,1,1}       34: {1,7}           58: {1,10}
     10: {1,3}         35: {3,4}           60: {1,1,2,3}
     12: {1,1,2}       36: {1,1,2,2}       62: {1,11}
     14: {1,4}         38: {1,8}           64: {1,1,1,1,1,1}
     15: {2,3}         40: {1,1,1,3}       66: {1,2,5}
     16: {1,1,1,1}     44: {1,1,5}         68: {1,1,7}
     18: {1,2,2}       45: {2,2,3}         69: {2,9}
     20: {1,1,3}       46: {1,9}           70: {1,3,4}
     22: {1,5}         48: {1,1,1,1,2}     72: {1,1,1,2,2}
     24: {1,1,1,2}     50: {1,3,3}         74: {1,12}
     26: {1,6}         51: {2,7}           75: {2,3,3}

Links

Crossrefs

A181818 is the version for superprimorials, with complement A336426.
A336496 is the version for superfactorials, with complement A336497.
A336620 is the complement.
A000837 counts relatively prime partitions, with strict case A007360.
A001055 counts factorizations.
A302696 lists numbers with coprime prime indices.
A304711 lists numbers with coprime distinct prime indices.
Cf. A001221, A007360, A007916, A056239, A112798, A302569, A327516, A333228, A335238, A335239, A335240, A336424, A336497, A336736.

Programs

  • Mathematica
    nn=100;
dat=Select[Range[nn],CoprimeQ@@PrimePi/@First/@FactorInteger[#]&];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[nn],facsusing[dat,#]!={}&]
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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