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 1-7 of 7 results.

A325618 Numbers m such that there exists an integer partition of m whose reciprocal factorial sum is 1.

Original entry on oeis.org

1, 4, 11, 18, 24, 31, 37, 44, 45, 50, 52, 57, 58, 65, 66, 70, 71, 73, 76, 78, 79, 83, 86, 87, 89, 91, 92, 94, 96, 97, 99, 100, 102, 104, 107, 108, 109, 110, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 128, 130, 131
Offset: 1

Views

Author

Gus Wiseman, May 13 2019

Keywords

Comments

The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.
Conjecture: 137 is the greatest integer not in this sequence. - Charlie Neder, May 14 2019

Examples

			The sequence of terms together with an integer partition of each whose reciprocal factorial sum is 1 begins:
   1: (1)
   4: (2,2)
  11: (3,3,3,2)
  18: (3,3,3,3,3,3)
  24: (4,4,4,4,3,3,2)
  31: (4,4,4,4,3,3,3,3,3)
  37: (4,4,4,4,4,4,4,4,3,2)
  44: (4,4,4,4,4,4,4,4,3,3,3,3)
  45: (5,5,5,5,5,4,4,4,3,3,2)
  50: (4,4,4,4,4,4,4,4,4,4,4,4,2)

Links

Crossrefs

Factorial numbers: A000142, A002982, A007489, A011371, A022559, A064986, A115627, A284605, A325616.
Reciprocal factorial sum: A002966, A051908, A058360, A316854, A316855, A325619, A325620, A325621, A325622, A325623, A325624.

Extensions

a(11)-a(55) from Charlie Neder, May 14 2019

A325616 Triangle read by rows where T(n,k) is the number of length-k integer partitions of n into factorial numbers.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 2, 2, 1
Offset: 0

Views

Author

Gus Wiseman, May 12 2019

Keywords

Examples

			Triangle begins:
  1
  0 1
  0 1 1
  0 0 1 1
  0 0 1 1 1
  0 0 0 1 1 1
  0 1 0 1 1 1 1
  0 0 1 0 1 1 1 1
  0 0 1 1 1 1 1 1 1
  0 0 0 1 1 1 1 1 1 1
  0 0 0 1 1 2 1 1 1 1 1
  0 0 0 0 1 1 2 1 1 1 1 1
  0 0 1 0 1 1 2 2 1 1 1 1 1
  0 0 0 1 0 1 1 2 2 1 1 1 1 1
  0 0 0 1 1 1 1 2 2 2 1 1 1 1 1
  0 0 0 0 1 1 1 1 2 2 2 1 1 1 1 1
  0 0 0 0 1 1 2 1 2 2 2 2 1 1 1 1 1
  0 0 0 0 0 1 1 2 1 2 2 2 2 1 1 1 1 1
  0 0 0 1 0 1 1 2 2 2 2 2 2 2 1 1 1 1 1
  0 0 0 0 1 0 1 1 2 2 2 2 2 2 2 1 1 1 1 1
  0 0 0 0 1 1 1 1 2 2 3 2 2 2 2 2 1 1 1 1 1
Row n = 12 counts the following partitions:
  (66)
  (6222)
  (62211)
  (222222) (621111)
  (2222211) (6111111)
  (22221111)
  (222111111)
  (2211111111)
  (21111111111)
  (111111111111)

Crossrefs

Row sums are A064986.
Cf. A008284.
Factorial numbers: A000142, A007489, A076934, A108731, A115944, A227157, A284605, A322583, A325509, A325617.
Reciprocal factorial sum: A325618, A325619, A325620, A325622.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1-y*x^(i!)),{i,1,n}],{x,0,n},{y,0,k}],{n,0,15},{k,0,n}]

Formula

T(n,k) is the coefficient of x^n * y^k in the expansion of Product_{i > 0} 1/(1 - y * x^(i!)).

A325620 Number of integer partitions of n whose reciprocal factorial sum is an integer.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 14, 14, 15, 16, 18, 19, 20, 22, 24, 25, 26, 28, 31, 33, 34, 36, 39, 41, 43, 45, 49, 52, 54, 57, 61, 65, 68, 71, 76, 80, 84, 88, 93, 98, 103, 107, 113
Offset: 1

Views

Author

Gus Wiseman, May 13 2019

Keywords

Comments

The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.

Examples

			The initial terms count the following partitions:
  1: (1)
  2: (1,1)
  3: (1,1,1)
  4: (2,2)
  4: (1,1,1,1)
  5: (2,2,1)
  5: (1,1,1,1,1)
  6: (2,2,1,1)
  6: (1,1,1,1,1,1)
  7: (2,2,1,1,1)
  7: (1,1,1,1,1,1,1)
  8: (2,2,2,2)
  8: (2,2,1,1,1,1)
  8: (1,1,1,1,1,1,1,1)
  9: (2,2,2,2,1)
  9: (2,2,1,1,1,1,1)
  9: (1,1,1,1,1,1,1,1,1)

Crossrefs

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.
Reciprocal factorial sum: A002966, A051908, A058360, A316854, A316856, A325618, A325621, A325622.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],IntegerQ[Total[1/(#!)]]&]],{n,30}]

A325623 Heinz numbers of integer partitions whose reciprocal factorial sum is the reciprocal of an integer.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 221, 223, 227, 229
Offset: 1

Views

Author

Gus Wiseman, May 13 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   29: {10}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   49: {4,4}
   53: {16}

Crossrefs

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.
Reciprocal factorial sum: A002966, A051908, A058360, A316854, A316857, A325619, A325621, A325622.

Programs

  • Mathematica
    Select[Range[100],IntegerQ[1/Total[Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]!]]]&]

A325704 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the numerator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 1, 7, 1, 5, 1, 25, 2, 4, 1, 2, 1, 13, 13, 121, 1, 7, 1, 721, 3, 49, 1, 5, 1, 5, 61, 5041, 5, 3, 1, 40321, 361, 19, 1, 37, 1, 241, 7, 362881, 1, 9, 1, 4, 2521, 1441, 1, 5, 7, 73, 20161, 3628801, 1, 8, 1, 39916801, 25, 6, 121, 181, 1
Offset: 1

Views

Author

Gus Wiseman, May 18 2019

Keywords

Comments

Alternatively, if n = prime(i_1) * ... * prime(i_k), then a(n) is the numerator of 1/i_1! + ... + 1/i_k!.

Links

Crossrefs

Factorial numbers: A000142, A002982, A011371, A022559, A071626, A076934, A108731, A325272, A325508, A325709.
Reciprocal sum: A002966, A316855, A316856, A316857, A318573, A318574, A325618, A325619, A325620, A325621, A325622, A325623, A325624, A325703.

Programs

  • Mathematica
    Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k/PrimePi[p]!]],{n,100}]//Numerator
  • PARI
    A325704(n) = { my(f=factor(n)); numerator(sum(i=1,#f~,f[i, 2]/(primepi(f[i, 1])!))); }; \\ Antti Karttunen, Nov 17 2019

Formula

a(n) = A318573(A325709(n)).

A325703 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the denominator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 24, 1, 1, 6, 120, 2, 720, 24, 3, 1, 5040, 1, 40320, 6, 24, 120, 362880, 2, 3, 720, 2, 24, 3628800, 3, 39916800, 1, 120, 5040, 24, 1, 479001600, 40320, 720, 6, 6227020800, 24, 87178291200, 120, 6, 362880, 1307674368000, 2, 12, 3, 5040, 720
Offset: 1

Views

Author

Gus Wiseman, May 18 2019

Keywords

Comments

Alternatively, if n = prime(i_1) * ... * prime(i_k), then a(n) is the denominator of 1/i_1! + ... + 1/i_k!.

Links

Crossrefs

Factorial numbers: A000142, A002982, A011371, A022559, A071626, A076934, A108731, A325272, A325508, A325709.
Reciprocal sum: A002966, A316855, A316856, A316857, A318573, A318574, A325618, A325619, A325620, A325621, A325622, A325623, A325624, A325704.

Programs

  • Maple
    f:= proc(n) local F,t;
    F:= ifactors(n)[2];
    denom(add(t[2]/numtheory:-pi(t[1])!,t=F))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 13 2024
  • Mathematica
    Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k/PrimePi[p]!]],{n,100}]//Denominator

Formula

a(n) = A318574(A325709(n)).

A325701 Nonprime Heinz numbers of integer partitions whose reciprocal factorial sum is the reciprocal of an integer.

Original entry on oeis.org

1, 9, 25, 49, 77, 121, 125, 169, 221, 245, 289, 323, 343, 361, 375, 437, 529, 841, 899, 961, 1331, 1369, 1517, 1681, 1763, 1849, 1859, 2021, 2197, 2209, 2401, 2773, 2809, 2873, 3127, 3481, 3721, 3757, 4087, 4489, 4757, 4913, 5041, 5183, 5329, 5929, 6137, 6241
Offset: 1

Views

Author

Gus Wiseman, May 17 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.

Examples

			The sequence of terms together with their prime indices begins:
     1: {}
     9: {2,2}
    25: {3,3}
    49: {4,4}
    77: {4,5}
   121: {5,5}
   125: {3,3,3}
   169: {6,6}
   221: {6,7}
   245: {3,4,4}
   289: {7,7}
   323: {7,8}
   343: {4,4,4}
   361: {8,8}
   375: {2,3,3,3}
   437: {8,9}
   529: {9,9}
   841: {10,10}
   899: {10,11}
   961: {11,11}
For example, the sequence contains 245 because the prime indices of 245 are {3,4,4}, with reciprocal sum 1/6 + 1/24 + 1/24 = 1/4.

Crossrefs

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.
Reciprocal factorial sum: A002966, A316854, A316857, A325618, A325620, A325622, A325623.

Programs

  • Mathematica
    Select[Range[1000],!PrimeQ[#]&&IntegerQ[1/Total[Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]!]]]&]
Showing 1-7 of 7 results.

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

Content is available under The OEIS End-User License Agreement.