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Search: a289509 -id:a289509
Displaying 201-208 of 208 results found. page 1 ... 20 21
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A316900 Heinz numbers of integer partitions into relatively prime parts whose reciprocal sum is an integer. +10
0
2, 4, 8, 16, 18, 32, 36, 64, 72, 128, 144, 162, 195, 250, 256, 288, 294, 324, 390, 500, 512, 576, 588, 648, 780, 1000, 1024, 1125, 1152, 1176, 1296, 1458, 1560, 1755, 2000, 2048, 2250, 2304, 2352, 2592, 2646, 2916, 3120, 3185, 3510, 4000, 4096, 4500, 4608 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
LINKS
Table of n, a(n) for n=1..49.
Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums
EXAMPLE
The sequence of partitions whose Heinz numbers belong to this sequence begins: (1), (11), (111), (1111), (221), (11111), (2211), (111111), (22111), (1111111), (221111), (22221), (632), (3331), (11111111).
MATHEMATICA
Select[Range[2, 1000], And[GCD@@PrimePi/@FactorInteger[#][[All, 1]]==1, IntegerQ[Sum[m[[2]]/PrimePi[m[[1]]], {m, FactorInteger[#]}]]]&]
CROSSREFS
Cf. A000837, A002966, A051908, A058360, A100953, A289509, A296150, A316854, A316855, A316856, A316857, A316888-A316904.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 16 2018
STATUS
approved
A316901 Heinz numbers of integer partitions into relatively prime parts whose reciprocal sum is the reciprocal of an integer. +10
0
2, 195, 3185, 5467, 6475, 6815, 8455, 10527, 15385, 16401, 17719, 20445, 20535, 21045, 25365, 28897, 40001, 46155, 49841, 50431, 54677, 92449, 101543, 113849, 123469, 137731, 156883, 164255, 171941, 185803, 218855, 228085, 230347, 261457, 267883, 274261 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
LINKS
Table of n, a(n) for n=1..36.
Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums
EXAMPLE
5467 is the Heinz number of (20,5,4) and 1/20 + 1/5 + 1/4 = 1/2, so 5467 belongs to the sequence.
The sequence of partitions whose Heinz numbers belong to this sequence begins: (1), (6,3,2), (6,4,4,3), (20,5,4), (12,4,3,3), (15,10,3), (24,8,3), (10,5,5,2)
MATHEMATICA
Select[Range[2, 100000], And[GCD@@PrimePi/@FactorInteger[#][[All, 1]]==1, IntegerQ[1/Sum[m[[2]]/PrimePi[m[[1]]], {m, FactorInteger[#]}]]]&]
CROSSREFS
Cf. A000837, A002966, A051908, A058360, A100953, A289509, A296150, A316854, A316855, A316856, A316857, A316888-A316904.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 16 2018
STATUS
approved
A319292 Number of series-reduced locally nonintersecting rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n. +10
0
1, 1, 6, 48, 455, 5700, 83138, 1454870 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally nonintersecting if the intersection of all branches directly under any given node with at least two branches is empty.
LINKS
Table of n, a(n) for n=1..8.
EXAMPLE
The a(3) = 6 trees are: (1(12)), (112), (1(23)), (2(13)), (3(12)), (123). Missing from this list but counted by A316651 are: (1(11)), (2(11)), (111).
MATHEMATICA
nonintQ[u_]:=Intersection@@u=={};
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
gro[m_]:=gro[m]=If[Length[m]==1, {m}, Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m], Length[#]>1&])], nonintQ]];
Table[Sum[Length[gro[m]], {m, Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n]}], {n, 5}]
CROSSREFS
Cf. A000081, A007562, A273873, A289509, A301700, A316470, A316473, A316475, A316651, A319271.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 16 2018
STATUS
approved
A327404 Quotient of n over the maximum divisor of n that is 2 or whose prime indices have a common divisor > 1. +10
0
1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 3, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 6, 1, 16, 3, 2, 5, 4, 1, 2, 1, 8, 1, 2, 1, 4, 5, 2, 1, 16, 1, 2, 3, 4, 1, 2, 5, 8, 1, 2, 1, 12, 1, 2, 1, 32, 1, 6, 1, 4, 3, 10, 1, 8, 1, 2, 3, 4, 7, 2, 1, 16, 1, 2, 1, 4, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
First differs from A327395 at a(195) = 65, A327395(195) = 195.
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.
LINKS
Table of n, a(n) for n=1..85.
Gus Wiseman, Sequences counting and encoding certain classes of multisets
EXAMPLE
The divisors of 90 that are 2 or whose prime indices have a common divisor > 1 are {1, 2, 3, 5, 9}, so a(90) = 90/9 = 10.
MATHEMATICA
Table[n/Max[Select[Divisors[n], #==2||GCD@@PrimePi/@First/@FactorInteger[#]!=1&]], {n, 100}]
CROSSREFS
See link for additional cross-references.
Cf. A000005, A006530, A056239, A112798, A289509, A302569, A327407, A327656.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 23 2019
STATUS
approved
A327531 a(n) = 1 if the prime indices of n are relatively prime, otherwise a(n) = n. +10
0
1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 1, 1, 1, 17, 1, 19, 1, 21, 1, 23, 1, 25, 1, 27, 1, 29, 1, 31, 1, 1, 1, 1, 1, 37, 1, 39, 1, 41, 1, 43, 1, 1, 1, 47, 1, 49, 1, 1, 1, 53, 1, 1, 1, 57, 1, 59, 1, 61, 1, 63, 1, 65, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
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. Numbers whose prime indices are relatively prime are A289509. The maximum divisor of n that is 1 or whose prime indices are relatively prime is A327529(n).
Also the quotient of n over the maximum divisor of n that is 1 or whose prime indices are relatively prime.
LINKS
Table of n, a(n) for n=1..78.
Gus Wiseman, Sequences counting and encoding certain classes of multisets
MATHEMATICA
Table[n/Max[Select[Divisors[n], #==1||GCD@@PrimePi/@First/@FactorInteger[#]==1&]], {n, 100}]
CROSSREFS
See link for additional cross-references.
Cf. A000005, A112798, A281116, A289509.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 17 2019
STATUS
approved
A327538 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1, prime, or whose prime indices are relatively prime (A327535, A327537). +10
0
0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,9
COMMENTS
The first index m such that a(m) > 1 but m is not in A322336 is m = 2335.
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. Numbers that are 1, prime, or whose prime indices are relatively prime are A327534. The number of divisors of n satisfying the same conditions is A327536(n).
LINKS
Table of n, a(n) for n=1..87.
Gus Wiseman, Sequences counting and encoding certain classes of multisets
FORMULA
a(1) = 0; if n is prime or has relatively prime prime indices, then a(n) = 1; otherwise a(n) = Omega(n) = A001222(n).
EXAMPLE
We have 441 -> 63 -> 9 -> 3 -> 1, so a(441) = 4.
MATHEMATICA
Table[Length[FixedPointList[#/Max[Select[Divisors[#], #==1||PrimeQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1&]]&, n]]-2, {n, 100}]
CROSSREFS
See link for additional cross-references.
Cf. A000005, A001222, A006530, A056239, A112798, A289509, A327407.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 17 2019
STATUS
approved
A327540 Number of factorizations of A327534(n), the n-th number that is 1, prime, or whose prime indices are relatively prime, into numbers > 1 satisfying the same conditions. +10
0
1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 4, 1, 2, 2, 5, 1, 3, 1, 4, 2, 1, 7, 2, 4, 1, 5, 1, 7, 2, 2, 2, 7, 1, 2, 7, 1, 4, 1, 4, 3, 2, 1, 12, 3, 2, 4, 1, 4, 2, 7, 2, 1, 11, 1, 2, 11, 5, 1, 4, 2, 5, 1, 13, 1, 2, 3, 4, 2, 4, 1, 12, 2, 1, 9, 2, 2, 7, 1, 9, 4, 2, 2, 2, 19, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
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. Numbers that are 1, prime, or whose prime indices are relatively prime are A327534. The number of divisors of n satisfying the same conditions is A327536(n).
LINKS
Table of n, a(n) for n=1..85.
Gus Wiseman, Sequences counting and encoding certain classes of multisets
EXAMPLE
The a(74) = 9 factorizations of 84 together with the corresponding multiset partitions of {1,1,2,4}:
(2*2*3*7) {{1},{1},{2},{4}}
(2*3*14) {{1},{2},{1,4}}
(2*6*7) {{1},{1,2},{4}}
(2*42) {{1},{1,2,4}}
(3*4*7) {{2},{1,1},{4}}
(3*28) {{2},{1,1,4}}
(6*14) {{1,2},{1,4}}
(7*12) {{4},{1,1,2}}
(84) {{1,1,2,4}}
MATHEMATICA
nn=100;
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, #]&]}]];
y=Select[Range[nn], #==1||PrimeQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1&];
Table[Length[facsusing[Rest[y], n]], {n, y}]
CROSSREFS
See link for additional cross-references.
Cf. A006530, A056239, A112798, A281116, A289509, A327407.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 17 2019
STATUS
approved
A327659 Number of factorizations of A318978(n - 1), the n-th number that is 1 or whose prime indices have a common divisor > 1, into numbers > 1 satisfying the same conditions. +10
0
1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 4, 2, 1, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 4, 2, 3, 1, 2, 1, 2, 1, 1, 4, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 7, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 7, 2, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
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. Numbers whose prime indices have a common divisor > 1 are listed in A318978.
LINKS
Table of n, a(n) for n=1..87.
Gus Wiseman, Sequences counting and encoding certain classes of multisets
FORMULA
a(n) = A001055(A318978(n - 1)).
MATHEMATICA
nn=100;
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, #]&]}]];
y=Select[Range[1000], GCD@@PrimePi/@First/@FactorInteger[#]!=1&];
Table[Length[facsusing[Rest[y], n]], {n, y}]
CROSSREFS
See link for additional cross-references.
Cf. A056239, A112798, A281116, A289509, A318721, A327406.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 21 2019
STATUS
approved
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Last modified August 29 09:16 EDT 2024. Contains 375511 sequences. (Running on oeis4.)