Displaying 1-10 of 61 results found.
Number of ways to choose a sequence of all different divisors, one of each prime index of n (with multiplicity).
+10
79
1, 1, 2, 0, 2, 1, 3, 0, 2, 1, 2, 0, 4, 2, 3, 0, 2, 0, 4, 0, 4, 1, 3, 0, 2, 3, 0, 0, 4, 1, 2, 0, 3, 1, 5, 0, 6, 3, 6, 0, 2, 1, 4, 0, 2, 2, 4, 0, 6, 0, 3, 0, 5, 0, 3, 0, 6, 3, 2, 0, 6, 1, 2, 0, 6, 1, 2, 0, 5, 2, 6, 0, 4, 5, 2, 0, 5, 2, 4, 0, 0, 1, 2, 0, 3, 3, 6
COMMENTS
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.
EXAMPLE
The a(49) = 6 ways are: (1,2), (1,4), (2,1), (2,4), (4,1), (4,2).
The a(182) = 5 ways are: (1,2,3), (1,2,6), (1,4,2), (1,4,3), (1,4,6).
The a(546) = 2 ways are: (1,2,4,3), (1,2,4,6).
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Select[Tuples[Divisors/@primeMS[n]], UnsameQ@@#&]], {n, 100}]
CROSSREFS
For relatively prime instead of strict we have A355737, firsts A355738. A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A120383 lists numbers divisible by all of their prime indices.
Cf. A000720, A076610, A302796, A355535, A355537, A355733, A355735, A355741, A355742, A355744, A355748.
Number of ways to choose a sequence of prime factors, one of each prime index of n.
+10
79
1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2
COMMENTS
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.
FORMULA
Totally multiplicative with a(prime(k)) = A001221(k).
EXAMPLE
The prime indices of 1131 are {2,6,10}, and the a(1131) = 4 choices are: {2,2,2}, {2,2,5}, {2,3,2}, {2,3,5}.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Times@@PrimeNu/@primeMS[n], {n, 100}]
CROSSREFS
Choosing prime-power divisors gives A355742. Counting multisets instead of sequences gives A355744. A003963 multiplies together the prime indices of n.
A289509 lists numbers with relatively prime prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Numbers of which it is not possible to choose a different prime factor of each prime index (with multiplicity).
+10
78
2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106
COMMENTS
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. Includes all even numbers.
EXAMPLE
The terms together with their prime indices begin:
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
9: {2,2}
10: {1,3}
12: {1,1,2}
14: {1,4}
16: {1,1,1,1}
18: {1,2,2}
20: {1,1,3}
21: {2,4}
22: {1,5}
24: {1,1,1,2}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Select[Tuples[primeMS/@primeMS[#]], UnsameQ@@#&]=={}&]
CROSSREFS
The case of all divisors (not just primes) is A355740, zeros of A355739. A003963 multiplies together the prime indices of n.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Numbers of which it is not possible to choose a different divisor of each prime index.
+10
74
4, 8, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 48, 50, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 100, 104, 108, 112, 116, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188
COMMENTS
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. By Hall's marriage theorem, k is a term if and only if there is a sub-multiset S of the prime indices of k such that fewer than |S| numbers are divisors of a member of S. Equivalently, k is divisible by a member of A370348. - Robert Israel, Feb 15 2024
EXAMPLE
The terms together with their prime indices begin:
4: {1,1}
8: {1,1,1}
12: {1,1,2}
16: {1,1,1,1}
18: {1,2,2}
20: {1,1,3}
24: {1,1,1,2}
27: {2,2,2}
28: {1,1,4}
32: {1,1,1,1,1}
36: {1,1,2,2}
40: {1,1,1,3}
44: {1,1,5}
48: {1,1,1,1,2}
For example, the choices of a divisor of each prime index of 90 are: (1,1,1,1), (1,1,1,3), (1,1,2,1), (1,1,2,3), (1,2,1,1), (1,2,1,3), (1,2,2,1), (1,2,2,3). But none of these has all distinct elements, so 90 is in the sequence.
MAPLE
filter:= proc(n) uses numtheory, GraphTheory; local B, S, F, D, E, G, t, d;
F:= ifactors(n)[2];
F:= map(t -> [pi(t[1]), t[2]], F);
D:= `union`(seq(divisors(t[1]), t = F));
F:= map(proc(t) local i; seq([t[1], i], i=1..t[2]) end proc, F);
if nops(D) < nops(F) then return false fi;
E:= {seq(seq({t, d}, d=divisors(t[1])), t = F)};
S:= map(t -> convert(t, name), [op(F), op(D)]);
E:= map(e -> map(convert, e, name), E);
G:= Graph(S, E);
B:= BipartiteMatching(G);
B[1] = nops(F);
end proc:
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Select[Tuples[Divisors/@primeMS[#]], UnsameQ@@#&]=={}&]
CROSSREFS
The case of just prime factors (not all divisors) is A355529, odd A355535. A003963 multiplies together the prime indices of n.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Number of condensed integer partitions of n.
+10
53
1, 1, 1, 2, 3, 3, 5, 6, 9, 10, 14, 16, 23, 27, 33, 41, 51, 62, 75, 93, 111, 134, 159, 189, 226, 271, 317, 376, 445, 520, 609, 714, 832, 972, 1129, 1304, 1520, 1753, 2023, 2326, 2692, 3077, 3540, 4050, 4642, 5298, 6054, 6887, 7854, 8926, 10133, 11501, 13044
COMMENTS
Suppose that p is a partition of n. Let x(1), x(2), ..., x(k) be the distinct parts of p, and let m(i) be the multiplicity of x(i) in p. Let c(p) be the partition {m(1)*x(1), m(2)*x(2), ..., x(k)*m(k)} of n. Call a partition q of n a condensed partition of n if q = c(p) for some partition p of n. Then a(n) is the number of distinct condensed partitions of n. Note that c(p) = p if and only if p has distinct parts and that condensed partitions can have repeated parts.
Also the number of integer partitions of n such that it is possible to choose a different divisor of each part. For example, the partition (6,4,4,1) has choices (3,2,4,1), (3,4,2,1), (6,2,4,1), (6,4,2,1) so is counted under a(15). - Gus Wiseman, Mar 12 2024
EXAMPLE
a(5) = 3 gives the number of partitions of 5 that result from condensations as shown here: 5 -> 5, 41 -> 41, 32 -> 32, 311 -> 32, 221 -> 41, 2111 -> 32, 11111 -> 5.
The a(1) = 1 through a(9) = 10 condensed partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(2,1) (2,2) (3,2) (3,3) (4,3) (4,4) (5,4)
(3,1) (4,1) (4,2) (5,2) (5,3) (6,3)
(5,1) (6,1) (6,2) (7,2)
(3,2,1) (3,2,2) (7,1) (8,1)
(4,2,1) (3,3,2) (4,3,2)
(4,2,2) (4,4,1)
(4,3,1) (5,2,2)
(5,2,1) (5,3,1)
(6,2,1)
(End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0, {[]},
`if`(i=1, {[n]}, {seq(map(x-> `if`(j=0, x,
sort([x[], i*j])), b(n-i*j, i-1))[], j=0..n/i)}))
end:
a:= n-> nops(b(n$2)):
MATHEMATICA
u[n_, k_] := u[n, k] = Map[Total, Split[IntegerPartitions[n][[k]]]]; t[n_] := t[n] = DeleteDuplicates[Table[Sort[u[n, k]], {k, 1, PartitionsP[n]}]]; Table[Length[t[n]], {n, 0, 30}]
Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[Divisors/@#], UnsameQ@@#&]]>0&]], {n, 0, 30}] (* Gus Wiseman, Mar 12 2024 *)
CROSSREFS
The complement is counted by A370320. The version for prime factors (not all divisors) is A370592, ranks A368100. A237685 counts partitions of depth 1, or A353837 if we include depth 0.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A355535, A355733, A355739, A367867, A368097, A368414, A370583, A370584, A370594, A370806, A370808.
Number of multisets that can be obtained by choosing a prime factor of each prime index of n.
+10
49
1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2
COMMENTS
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.
EXAMPLE
The a(169) = 3 multisets are: {2,2}, {2,3}, {3,3}.
The a(507) = 3 multisets are: {2,2,2}, {2,2,3}, {2,3,3}.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Union[Sort/@Tuples[primeMS/@primeMS[n]]]], {n, 100}]
CROSSREFS
Counting sequences instead of multisets gives A355741. Choosing weakly increasing sequences of divisors gives A355745. A003963 multiplies together the prime indices of n.
A324850 lists numbers divisible by the product of their prime indices.
A344606 counts alternating permutations of prime indices.
Cf. A000720, A076610, A120383, A289509, A335433, A340852, A355731, A355735, A355737, A355739, A355740.
Least k such that there are exactly n ways to choose a sequence of divisors, one of each element of the multiset of prime indices of k (with multiplicity).
+10
48
1, 3, 7, 9, 53, 21, 311, 27, 49, 159, 8161, 63, 38873, 933, 371, 81, 147, 477, 2177, 24483, 189, 2809, 343, 2799, 1113, 243, 57127, 16483, 441, 1431, 6531, 73449, 2597, 567, 96721, 8427, 1029, 8397, 3339, 15239, 729, 49449, 1323, 19663, 4293, 2401, 19593, 7791
COMMENTS
This is the position of first appearance of n in A355731. 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.
EXAMPLE
The terms together with their prime indices begin:
1: {}
3: {2}
7: {4}
9: {2,2}
53: {16}
21: {2,4}
311: {64}
27: {2,2,2}
49: {4,4}
159: {2,16}
8161: {1024}
63: {2,2,4}
For example, the choices for a(12) = 63 are:
(1,1,1) (1,2,2) (2,1,4)
(1,1,2) (1,2,4) (2,2,1)
(1,1,4) (2,1,1) (2,2,2)
(1,2,1) (2,1,2) (2,2,4)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
mnrm[s_]:=If[Min@@s==1, mnrm[DeleteCases[s-1, 0]]+1, 0];
az=Table[Times@@Length/@Divisors/@primeMS[n], {n, 1000}];
Table[Position[az, k][[1, 1]], {k, mnrm[az]}]
CROSSREFS
Positions of first appearances in A355731. Counting distinct sequences after sorting: A355734, firsts of A355733. Requiring the result to be weakly increasing: A355736, firsts of A355735. Requiring the result to be relatively prime: A355738, firsts of A355737. A003963 multiplies together the prime indices of n.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A076610, A340606, A355739, A355740, A355741, A355742, A355744, A355746, A355747, A355748.
Numbers of which it is possible to choose a different divisor of each prime index.
+10
41
1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97
COMMENTS
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. By Hall's marriage theorem, k is a term if and only if there is no sub-multiset S of the prime indices of k such that fewer than |S| numbers are divisors of a member of S. Equivalently, there is no divisor of k in A370348. - Robert Israel, Feb 15 2024
FORMULA
Heinz numbers of the partitions counted by A239312.
EXAMPLE
The terms together with their prime indices begin:
1: {}
2: {1}
3: {2}
5: {3}
6: {1,2}
7: {4}
9: {2,2}
10: {1,3}
11: {5}
13: {6}
14: {1,4}
15: {2,3}
17: {7}
19: {8}
21: {2,4}
22: {1,5}
23: {9}
25: {3,3}
26: {1,6}
29: {10}
30: {1,2,3}
MAPLE
filter:= proc(n) uses numtheory, GraphTheory; local B, S, F, D, E, G, t, d;
F:= ifactors(n)[2];
F:= map(t -> [pi(t[1]), t[2]], F);
D:= `union`(seq(divisors(t[1]), t = F));
F:= map(proc(t) local i; seq([t[1], i], i=1..t[2]) end proc, F);
if nops(D) < nops(F) then return false fi;
E:= {seq(seq({t, d}, d=divisors(t[1])), t = F)};
S:= map(t -> convert(t, name), [op(F), op(D)]);
E:= map(e -> map(convert, e, name), E);
G:= Graph(S, E);
B:= BipartiteMatching(G);
B[1] = nops(F);
end proc:
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Select[Tuples[Divisors/@prix[#]], UnsameQ@@#&]!={}&]
CROSSREFS
Positions of nonzero terms in A355739. A003963 multiplies together the prime indices of n.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A000720, A076610, A111774, A335433, A335448, A340852, A355733, A355734, A355737, A355749, A370348.
Number of ways to choose a prime factor of each prime index of n (with multiplicity, in weakly increasing order) such that the result is also weakly increasing.
+10
40
1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2
COMMENTS
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.
EXAMPLE
The prime indices of 1469 are {6,30}, and there are five valid choices: (2,2), (2,3), (2,5), (3,3), (3,5), so a(1469) = 5.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Select[Tuples[Union/@primeMS/@primeMS[n]], LessEqual@@#&]], {n, 100}]
CROSSREFS
Not requiring an increasing sequence gives A355741. Choosing a multiset instead of sequence gives A355744. A003963 multiplies together the prime indices of n.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Number of ways to choose a binary index of each binary index of n.
+10
35
1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 4, 4, 4, 4, 8, 8, 8, 8, 3, 3, 3, 3, 6, 6, 6, 6, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6, 6, 6, 12, 12, 12
COMMENTS
A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}. Run-lengths are all 4 or 8.
EXAMPLE
The binary indices of binary indices of 20 are {{1,2},{1,3}}, with choices (1,1), (1,3), (2,1), (2,3), so a(20) = 4.
The binary indices of binary indices of 52 are {{1,2},{1,3},{2,3}}, with choices (1,1,1), (1,1,3), (1,3,2), (1,3,3), (2,1,2), (2,1,3), (2,3,2), (2,3,3), so a(52) = 8.
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Table[Length[Tuples[bpe/@bpe[n]]], {n, 0, 100}]
CROSSREFS
All entries appear to belong to A003586. A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326753, A355731, A355739, A367771, A367905, A367906, A367915.
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