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
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)
Reciprocal factorial sum:
A002966,
A051908,
A058360,
A316854,
A316855,
A325619,
A325620,
A325621,
A325622,
A325623,
A325624.
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
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)
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
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}
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
Reciprocal sum:
A002966,
A316855,
A316856,
A316857,
A318573,
A318574,
A325618,
A325619,
A325620,
A325621,
A325622,
A325623,
A325624,
A325703.
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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
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
Reciprocal sum:
A002966,
A316855,
A316856,
A316857,
A318573,
A318574,
A325618,
A325619,
A325620,
A325621,
A325622,
A325623,
A325624,
A325704.
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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
Showing 1-5 of 5 results.
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