A325618 -id:A325618 - cp's OEIS frontend

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

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

Gus Wiseman, May 12 2019

			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)

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.

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

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

A325619 Heinz numbers of integer partitions whose reciprocal factorial sum is 1.

Original entry on oeis.org

2, 9, 375, 15625

Offset: 1

Gus Wiseman, May 13 2019

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!.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      9: {2,2}
    375: {2,3,3,3}
  15625: {3,3,3,3,3,3}

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.

Reciprocal factorial sum: A002966, A051908, A316855, A325618, A325624.

Select[Range[100000],Total[Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]!]]==1&]

Contains prime(n)^(n!) for all n > 0, including 191581231380566414401 for n = 4.

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

Gus Wiseman, May 13 2019

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

			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)

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.

Reciprocal factorial sum: A002966, A051908, A058360, A316854, A316856, A325618, A325621, A325622.

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

A325622 Number of integer partitions of n whose reciprocal factorial sum is the reciprocal of an integer.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 3, 3, 2, 2, 3, 3, 3, 5, 4, 4, 3, 3, 4, 6, 3, 4, 5, 5, 5, 6, 3, 7, 6, 5, 6, 6, 6, 5, 6, 8, 5, 7, 5, 4, 8, 7, 7, 7, 7, 9, 9, 9, 10, 12, 6, 12, 8, 10, 7, 14, 10, 8, 11, 11, 12, 11, 10, 10, 12, 14, 11, 10, 9, 10, 12, 10, 15, 14, 11, 10

Offset: 1

Gus Wiseman, May 13 2019

nonn

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

			The initial terms count the following partitions:
   1: (1)
   2: (2)
   3: (3)
   4: (4)
   4: (2,2)
   5: (5)
   6: (6)
   6: (3,3)
   7: (7)
   8: (8)
   8: (4,4)
   9: (9)
   9: (5,4)
   9: (3,3,3)
  10: (10)
  10: (5,5)
  11: (11)
  11: (4,4,3)
  11: (3,3,3,2)
  12: (12)
  12: (6,6)
  12: (4,4,4)

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.

Reciprocal factorial sum: A002966, A316854, A316857, A325618, A325620, A325623.

f:= proc(n) nops(select(proc(t) local i; (1/add(1/i!,i=t))::integer end proc, combinat:-partition(n))) end proc:
map(f, [$1..70]); # Robert Israel, May 09 2024

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

a(n) = my(c=0); forpart(v=n, if(numerator(sum(i=1, #v, 1/v[i]!))==1, c++)); c; \\ Jinyuan Wang, Feb 25 2025

a(61)-a(70) from Robert Israel, May 09 2024

a(71)-a(80) from Jinyuan Wang, Feb 25 2025

A325624 a(n) = prime(n)^(n!).

Original entry on oeis.org

2, 9, 15625, 191581231380566414401, 92709068817830061978520606494193845859707401497097037749844778027824097442147966967457359038488841338006006032592594389655201

Offset: 1

Gus Wiseman, May 13 2019

nonn

A subsequence of A325619 (numbers whose prime indices have reciprocal factorial sum equal to 1).

Cf. A056239, A112798.
Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.
Reciprocal factorial sum: A002966, A051908, A316855, A325618, A325619.

Mathematica
```
Table[Prime[n]^n!,{n,5}]
```

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

Gus Wiseman, May 18 2019

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

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.

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k/PrimePi[p]!]],{n,100}]//Numerator

A325704(n) = { my(f=factor(n)); numerator(sum(i=1,#f~,f[i, 2]/(primepi(f[i, 1])!))); }; \\ Antti Karttunen, Nov 17 2019

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

A325709 Replace k with k! in the prime indices of n.

Original entry on oeis.org

1, 2, 3, 4, 13, 6, 89, 8, 9, 26, 659, 12, 5443, 178, 39, 16, 49033, 18, 484037, 52, 267, 1318, 5222429, 24, 169, 10886, 27, 356, 61194647, 78, 774825383, 32, 1977, 98066, 1157, 36, 10552185239, 968074, 16329, 104, 153903050137, 534, 2394322471421, 2636, 117

Offset: 1

Gus Wiseman, May 19 2019

The union is A308299.

			The sequence of terms together with their prime indices begins:
       1: {}
       2: {1}
       3: {2}
       4: {1,1}
      13: {6}
       6: {1,2}
      89: {24}
       8: {1,1,1}
       9: {2,2}
      26: {1,6}
     659: {120}
      12: {1,1,2}
    5443: {720}
     178: {1,24}
      39: {2,6}
      16: {1,1,1,1}
   49033: {5040}
      18: {1,2,2}
  484037: {40320}
      52: {1,1,6}.

Amiram Eldar, Table of n, a(n) for n = 1..78 (calculated using the b-file at A062439)
Index entries for sequences computed from indices in prime factorization.

Cf. A000142, A056239, A062439, A064986, A076934, A112798, A115944, A284605, A308299, A322583, A325509, A325616, A325618, A325704.

Table[Times@@Prime/@(If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]!),{n,20}]

A325709(n) = { my(f=factor(n)); prod(i=1,#f~,prime(primepi(f[i, 1])!)^f[i, 2]); }; \\ Antti Karttunen, Nov 17 2019

from math import prod, factorial
from sympy import prime, primepi, factorint
def A325709(n): return prod(prime(factorial(primepi(p)))**e for p, e in factorint(n).items()) # Chai Wah Wu, Dec 26 2022

Completely multiplicative with a(prime(n)) = prime(n!).

Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(k!)) = 3.292606708493... . - Amiram Eldar, Dec 09 2022

Keyword:mult added by Antti Karttunen, Nov 17 2019

A308299 Numbers whose prime indices are factorial numbers.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 24, 26, 27, 32, 36, 39, 48, 52, 54, 64, 72, 78, 81, 89, 96, 104, 108, 117, 128, 144, 156, 162, 169, 178, 192, 208, 216, 234, 243, 256, 267, 288, 312, 324, 338, 351, 356, 384, 416, 432, 468, 486, 507, 512, 534, 576, 624, 648

Offset: 1

Gus Wiseman, May 19 2019

nonn

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. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions using factorial numbers. The enumeration of these partitions by sum is given by A064986.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   12: {1,1,2}
   13: {6}
   16: {1,1,1,1}
   18: {1,2,2}
   24: {1,1,1,2}
   26: {1,6}
   27: {2,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   39: {2,6}
   48: {1,1,1,1,2}
   52: {1,1,6}
   54: {1,2,2,2}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000142, A056239, A062439, A064986, A112798, A115944, A284605, A322583, A325616, A325709, A325618.

nn=5;
facts=Array[Factorial,nn];
Select[Range[Prime[Max@@facts]],SubsetQ[facts,PrimePi/@First/@FactorInteger[#]]&]

Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(k!)) = 3.292606708493... . - Amiram Eldar, Dec 03 2022

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

Gus Wiseman, May 18 2019

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

Robert Israel, Table of n, a(n) for n = 1..3168
Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

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.

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

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k/PrimePi[p]!]],{n,100}]//Denominator

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

Gus Wiseman, May 17 2019

nonn

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!.

			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.

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.

Reciprocal factorial sum: A002966, A316854, A316857, A325618, A325620, A325622, A325623.

Select[Range[1000],!PrimeQ[#]&&IntegerQ[1/Total[Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]!]]]&]

cp's OEIS Frontend

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

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A325619 Heinz numbers of integer partitions whose reciprocal factorial sum is 1.

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

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A325622 Number of integer partitions of n whose reciprocal factorial sum is the reciprocal of an integer.

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A325624 a(n) = prime(n)^(n!).

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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!.

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A325709 Replace k with k! in the prime indices of n.

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A308299 Numbers whose prime indices are factorial numbers.

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