A324850 -id:A324850 - cp's OEIS frontend

A324971 Number of rooted identity trees with n vertices whose non-leaf terminal subtrees are not all different.

0, 0, 0, 0, 0, 1, 4, 12, 31, 79, 192, 459, 1082, 2537, 5922, 13816, 32222, 75254, 176034, 412667, 969531, 2283278

Offset: 1

Gus Wiseman, Mar 21 2019

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root.

			The a(6) = 1 through a(8) = 12 trees:
  ((o)((o)))  ((o)(o(o)))   (o(o)(o(o)))
              (o(o)((o)))   (((o))(o(o)))
              (((o)((o))))  (((o)(o(o))))
              ((o)(((o))))  ((o)((o(o))))
                            ((o)(o((o))))
                            ((o(o)((o))))
                            (o((o)((o))))
                            (o(o)(((o))))
                            ((((o)((o)))))
                            (((o))(((o))))
                            (((o)(((o)))))
                            ((o)((((o)))))

The Matula-Goebel numbers of these trees are given by A324970.

Cf. A000081, A004111, A290689, A317713, A324850, A324922, A324923, A324924, A324931, A324935, A324936, A324979.

rits[n_]:=Join@@Table[Select[Union[Sort/@Tuples[rits/@ptn]],UnsameQ@@#&],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[rits[n],!UnsameQ@@Cases[#,{},{0,Infinity}]&]],{n,10}]

A324978 Matula-Goebel numbers of rooted trees that are not identity trees but whose non-leaf terminal subtrees are all different.

Original entry on oeis.org

4, 7, 8, 12, 14, 16, 17, 19, 20, 21, 24, 28, 32, 34, 35, 37, 38, 40, 42, 43, 44, 48, 51, 52, 53, 56, 57, 59, 64, 67, 68, 70, 71, 73, 74, 76, 77, 80, 84, 85, 86, 88, 89, 91, 95, 96, 102, 104, 106, 107, 112, 114, 116, 118, 124, 128, 129, 131, 133, 134, 136, 139

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

An unlabeled rooted tree is an identity tree if there are no repeated branches directly under the same root.

			The sequence of trees together with the Matula-Goebel numbers begins:
   4: (oo)
   7: ((oo))
   8: (ooo)
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  20: (oo((o)))
  21: ((o)(oo))
  24: (ooo(o))
  28: (oo(oo))
  32: (ooooo)
  34: (o((oo)))
  35: (((o))(oo))
  37: ((oo(o)))
  38: (o(ooo))
  40: (ooo((o)))
  42: (o(o)(oo))
  43: ((o(oo)))

Gus Wiseman, The first 36 trees together with their Matula-Goebel numbers.

Cf. A000081, A004111, A007097, A196050, A276625, A317713, A324850, A324923, A324935, A324936, A324968, A324969, A324970, A324971, A324979.

mgtree[n_]:=If[n==1,{},mgtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And[!And@@Cases[mgtree[#],q:{}:>UnsameQ@@q,{0,Infinity}],UnsameQ@@Cases[mgtree[#],{},{0,Infinity}]]&]

Complement of A276625 in A324935.

A379844 Squarefree numbers x such that the product of prime indices of x is a multiple of the sum of prime indices of x.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 154, 157, 163, 165, 167, 173, 179, 181, 190, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

Offset: 1

Gus Wiseman, Jan 19 2025

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 sum and product of prime indices are A056239 and A003963 respectively.

Squarefree case of A326149.
For nonprime instead of squarefree we have A326150.
The non-prime case is A326158.
Partitions of this type are counted by A379733, see A379735.
The even case is A379845, counted by A380221.
A003963 multiplies together prime indices.
A005117 lists the squarefree numbers.
A056239 adds up prime indices.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A000720, A001222, A036844, A112798, A324850, A324851, A324925, A326151, A326153/A326154, A326156, A326157.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],SquareFreeQ[#]&&Divisible[Times@@prix[#],Plus@@prix[#]]&]

Satisfies A056239(a(n))|A003963(a(n)).

A379845 Even squarefree numbers x such that the product of prime indices of x is a multiple of the sum of prime indices of x.

Original entry on oeis.org

2, 30, 154, 190, 390, 442, 506, 658, 714, 874, 1110, 1118, 1254, 1330, 1430, 1786, 1794, 1798, 1958, 2310, 2414, 2442, 2470, 2730, 2958, 3034, 3066, 3266, 3390, 3534, 3710, 3770, 3874, 3914, 4042, 4466, 4526, 4758, 4930, 5106, 5434, 5474, 5642, 6090, 6106

Offset: 1

Gus Wiseman, Jan 20 2025

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 sum and product of prime indices are A056239 and A003963 respectively.

			The terms together with their prime indices begin:
     2: {1}
    30: {1,2,3}
   154: {1,4,5}
   190: {1,3,8}
   390: {1,2,3,6}
   442: {1,6,7}
   506: {1,5,9}
   658: {1,4,15}
   714: {1,2,4,7}
   874: {1,8,9}
  1110: {1,2,3,12}

Even squarefree case of A326149.
For nonprime instead of even we have A326158.
Squarefree case of A379319.
Even case of A379844.
Partitions of this type are counted by A380221, see A379733, A379735.
A003963 multiplies together prime indices.
A005117 lists the squarefree numbers.
A056239 adds up prime indices.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568, ranks A326149
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A000720, A001222, A112798, A175508, A324850, A324851, A326150, A326151, A326153/A326154, A326156, A326157.

Select[Range[2,1000],EvenQ[#]&&SquareFreeQ[#]&&Divisible[Times@@prix[#],Plus@@prix[#]]&]

A380216 Numbers whose prime indices have (product)/(sum) equal to an integer > 1.

Original entry on oeis.org

49, 63, 65, 81, 125, 150, 154, 165, 169, 190, 198, 259, 273, 333, 351, 361, 364, 385, 390, 435, 442, 468, 481, 490, 495, 506, 525, 561, 580, 595, 609, 630, 658, 675, 700, 714, 741, 765, 781, 783, 810, 840, 841, 846, 874, 900, 918, 925, 931, 935, 952, 988

Offset: 1

Gus Wiseman, Jan 23 2025

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 sum and product of prime indices are A056239 and A003963 respectively.

			The terms together with their prime indices begin:
   49: {4,4}
   63: {2,2,4}
   65: {3,6}
   81: {2,2,2,2}
  125: {3,3,3}
  150: {1,2,3,3}
  154: {1,4,5}
  165: {2,3,5}
  169: {6,6}
  190: {1,3,8}
  198: {1,2,2,5}
  259: {4,12}
  273: {2,4,6}
  333: {2,2,12}
  351: {2,2,2,6}
  361: {8,8}
  364: {1,1,4,6}
For example, 198 has prime indices {1,2,2,5}, and 20/10 is an integer > 1, so 198 is in the sequence.

The fraction A003963(n)/A056239(n) reduces to A326153(n)/A326154(n).
The non-proper version is A326149, superset of A326150.
Also a superset of A326151.
The squarefree case is A326158 without first term.
Partitions of this type are counted by A380219.
A379666 counts partitions by sum and product.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568 (strict A379733), ranks A326149, see A379735, A380217.
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A000720, A001222, A028422, A036844, A112798, A301988, A319000, A324850, A324851, A326156, A379319, A379844.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],Divisible[Times@@prix[#],Total[prix[#]]]&&!SameQ[Times@@prix[#],Total[prix[#]]]&]

A324854 Lexicographically earliest sequence containing 1 and all positive integers > 2 whose prime indices already belong to the sequence.

Original entry on oeis.org

1, 4, 7, 8, 14, 16, 17, 19, 28, 32, 34, 38, 43, 49, 53, 56, 59, 64, 67, 68, 76, 86, 98, 106, 107, 112, 118, 119, 128, 131, 133, 134, 136, 139, 152, 163, 172, 191, 196, 212, 214, 224, 227, 236, 238, 241, 256, 262, 263, 266, 268, 272, 277, 278, 289, 301, 304

Offset: 1

Gus Wiseman, Mar 18 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.

A multiplicative semigroup: if x and y are in the sequence then so is x*y. - Robert Israel, Mar 19 2019

			The sequence of terms together with their prime indices begins:
   1: {}
   4: {1,1}
   7: {4}
   8: {1,1,1}
  14: {1,4}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  28: {1,1,4}
  32: {1,1,1,1,1}
  34: {1,7}
  38: {1,8}
  43: {14}
  49: {4,4}
  53: {16}
  56: {1,1,1,4}
  59: {17}
  64: {1,1,1,1,1,1}
  67: {19}
  68: {1,1,7}

Robert Israel, Table of n, a(n) for n = 1..10000
Gus Wiseman, The rooted trees whose Matula-Goebel numbers are the first 64 terms.

Cf. A000002, A000720, A001462, A079254, A112798, A276625, A290822, A304360.

Cf. A324697, A324698, A324736, A324748, A324753, A324843, A324850, A324855.

S:= {1}:
for n from 3 to 400 do
  if map(numtheory:-pi, numtheory:-factorset(n)) subset S then
    S:= S union {n}
  fi
od:
sort(convert(S,list)); # Robert Israel, Mar 19 2019

aQ[n_]:=Switch[n,1,True,2,False,,And@@Cases[FactorInteger[n],{p,k_}:>aQ[PrimePi[p]]]];
Select[Range[100],aQ]

A324933 Denominator in the division of n by the product of prime indices of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 4, 3, 5, 1, 6, 2, 2, 1, 7, 2, 8, 3, 8, 5, 9, 1, 9, 3, 8, 1, 10, 1, 11, 1, 10, 7, 12, 1, 12, 4, 4, 3, 13, 4, 14, 5, 4, 9, 15, 1, 16, 9, 14, 3, 16, 4, 3, 1, 16, 5, 17, 1, 18, 11, 16, 1, 18, 5, 19, 7, 6, 6, 20, 1, 21, 6, 6, 2, 20, 2, 22, 3

Offset: 1

Gus Wiseman, Mar 21 2019

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 sequence of quotients n/A003963(n) begins: 1, 2, 3/2, 4, 5/3, 3, 7/4, 8, 9/4, 10/3, 11/5, 6, 13/6, 7/2, 5/2, 16, ...

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

The numerators are A324932.

Cf. A003963, A109129, A120383, A196050, A317713, A324846, A324849, A324850, A324922, A324923, A324924, A324925, A324931, A324934.

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

A353396 Number of integer partitions of n whose Heinz number has prime shadow equal to the product of prime shadows of its parts.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 3, 1, 3, 4, 3, 7, 5, 9, 8, 12, 15, 15, 20, 21, 25, 31, 33, 38, 42, 46, 56, 61, 67, 78, 76, 96, 100, 114, 131, 130, 157, 157, 185, 200, 214, 236, 253, 275, 302, 333, 351, 386, 408, 440, 486, 515, 564, 596, 633, 691, 734, 800, 854, 899, 964

Offset: 0

Gus Wiseman, May 15 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			The a(8) = 1 through a(14) = 9 partitions (A..D = 10..13):
  (53)  (72)    (73)    (B)     (75)     (D)      (B3)
        (621)   (532)   (A1)    (651)    (B2)     (752)
        (4221)  (631)   (4331)  (732)    (A21)    (761)
                (4411)          (6321)   (43321)  (A31)
                                (6411)   (44311)  (C11)
                                (43221)           (6521)
                                (44211)           (9221)
                                                  (54221)
                                                  (64211)

The LHS (prime shadow) is A181819, with an inverse A181821.
The RHS (product of prime shadows) is A353394, first appearances A353397.
These partitions are ranked by A353395.
A related comparison is A353398, ranked by A353399.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914, product A005361.
A239455 counts Look-and-Say partitions, ranked by A351294.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000005, A002033, A143773, A182850, A316428, A325131, A325702, A325755, A353389, A353426.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Table[Length[Select[IntegerPartitions[n],Times@@red/@#==red[Times@@Prime/@#]&]],{n,0,15}]

A355738 Least k such that there are exactly n ways to choose a sequence of divisors, one of each prime index of k (with multiplicity), such that the result has no common divisor > 1.

Original entry on oeis.org

1, 2, 6, 9, 15, 49, 35, 27, 45, 98, 63, 105, 171, 117, 81, 135, 245, 343, 273, 549, 189, 1083, 315, 5618, 741, 686, 507, 513, 351, 243, 405, 7467, 6419, 5575, 735, 6859, 1813, 3231, 1183, 1197, 3537, 819, 1647, 567, 945, 2197, 8397, 3211, 1715, 3249, 3367

Offset: 1

Gus Wiseman, Jul 21 2022

nonn

This is the position of first appearance of n in A355737.

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 terms together with their prime indices begin:
     1: {}
     2: {1}
     6: {1,2}
     9: {2,2}
    15: {2,3}
    49: {4,4}
    35: {3,4}
    27: {2,2,2}
    45: {2,2,3}
    98: {1,4,4}
    63: {2,2,4}
   105: {2,3,4}
   171: {2,2,8}
   117: {2,2,6}
    81: {2,2,2,2}
   135: {2,2,2,3}
For example, the choices for a(12) = 105 are:
  (1,1,1)  (1,3,2)  (2,1,4)
  (1,1,2)  (1,3,4)  (2,3,1)
  (1,1,4)  (2,1,1)  (2,3,2)
  (1,3,1)  (2,1,2)  (2,3,4)

Wikipedia, Coprime integers.

Not requiring coprimality gives A355732, firsts of A355731.
Positions of first appearances in A355737.
A000005 counts divisors.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A120383 lists numbers divisible by all of their prime indices.
A289508 gives GCD of prime indices.
A289509 ranks relatively prime partitions, odd A302697, squarefree A302796.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A007359, A051424, A076610, A302696, A302698, A355733, A355735, A355739, A355741, A355748.

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[Length[Select[Tuples[Divisors/@primeMS[n]],GCD@@#==1&]],{n,100}];
Table[Position[az+1,k][[1,1]],{k,mnrm[az+1]}]

A324932 Numerator in the division of n by the product of prime indices of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 10, 11, 6, 13, 7, 5, 16, 17, 9, 19, 20, 21, 22, 23, 12, 25, 13, 27, 7, 29, 5, 31, 32, 33, 34, 35, 9, 37, 19, 13, 40, 41, 21, 43, 44, 15, 46, 47, 24, 49, 50, 51, 26, 53, 27, 11, 14, 57, 29, 59, 10, 61, 62, 63, 64, 65, 33, 67, 68, 23

Offset: 1

Gus Wiseman, Mar 21 2019

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 sequence of quotients n/A003963(n) begins: 1, 2, 3/2, 4, 5/3, 3, 7/4, 8, 9/4, 10/3, 11/5, 6, 13/6, 7/2, 5/2, 16, ...

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

The denominators are A324933.

Cf. A003963, A109129, A120383, A196050, A324846, A324849, A324850, A324922, A324923, A324924, A324925, A324931, A324934.

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

cp's OEIS Frontend

A324971 Number of rooted identity trees with n vertices whose non-leaf terminal subtrees are not all different.

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A324978 Matula-Goebel numbers of rooted trees that are not identity trees but whose non-leaf terminal subtrees are all different.

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A379844 Squarefree numbers x such that the product of prime indices of x is a multiple of the sum of prime indices of x.

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A379845 Even squarefree numbers x such that the product of prime indices of x is a multiple of the sum of prime indices of x.

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A380216 Numbers whose prime indices have (product)/(sum) equal to an integer > 1.

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A324854 Lexicographically earliest sequence containing 1 and all positive integers > 2 whose prime indices already belong to the sequence.

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A324933 Denominator in the division of n by the product of prime indices of n.

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A353396 Number of integer partitions of n whose Heinz number has prime shadow equal to the product of prime shadows of its parts.

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A355738 Least k such that there are exactly n ways to choose a sequence of divisors, one of each prime index of k (with multiplicity), such that the result has no common divisor > 1.

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