A324756 -id:A324756 - cp's OEIS frontend

A324850 Numbers divisible by the product of their prime indices.

1, 2, 4, 6, 8, 12, 16, 24, 28, 30, 32, 36, 48, 56, 60, 64, 72, 96, 112, 120, 128, 144, 152, 156, 168, 180, 192, 216, 224, 240, 256, 288, 304, 312, 330, 336, 360, 384, 432, 448, 476, 480, 512, 576, 608, 624, 660, 672, 720, 768, 784, 828, 840, 848, 864, 888, 896

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, with product A003963(n). For example, the prime indices of 30 are {1,2,3}, with product 6, which divides 30, so 30 is in the sequence.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  12: {1,1,2}
  16: {1,1,1,1}
  24: {1,1,1,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  48: {1,1,1,1,2}
  56: {1,1,1,4}
  60: {1,1,2,3}
  64: {1,1,1,1,1,1}
  72: {1,1,1,2,2}
  96: {1,1,1,1,1,2}

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

Cf. A003963, A036844, A120383, A324847, A324756, A324758, A324847, A324848, A324849, A324852, A324853, A324856, A324923, A324924, A324925, A324931.

Select[Range[100],Divisible[#,Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]]&]

isok(n) = my(f=factor(n)); !(n % prod(k=1, #f~, primepi(f[k,1])^f[k,2])); \\ Michel Marcus, Mar 22 2019

n/A003963(n) = A324933(n)/A324934(n).

A327476 Heinz numbers of integer partitions whose mean A326567/A326568 is not a part.

Original entry on oeis.org

1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 106

Offset: 1

Gus Wiseman, Sep 13 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
    1: {}
    6: {1,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   40: {1,1,1,3}

Complement of A327473.
The enumeration of these partitions by sum is given by A327472.
Subsets whose mean is not an element are A327471.
Cf. A056239, A067538, A112798, A114639, A237984, A240851, A316413, A324756, A324758, A326567/A326568, A327477.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!MemberQ[primeMS[#],Mean[primeMS[#]]]&]

A327472 Number of integer partitions of n not containing their mean.

Original entry on oeis.org

1, 0, 0, 1, 2, 5, 6, 13, 16, 25, 34, 54, 56, 99, 121, 154, 201, 295, 324, 488, 541, 725, 957, 1253, 1292, 1892, 2356, 2813, 3378, 4563, 4838, 6840, 7686, 9600, 12076, 14180, 15445, 21635, 25627, 29790, 33309, 44581, 48486, 63259, 70699, 82102, 104553, 124752

Offset: 0

Gus Wiseman, Sep 13 2019

nonn

			The a(3) = 1 through a(8) = 16 partitions not containing their mean:
  (21)  (31)   (32)    (42)     (43)      (53)
        (211)  (41)    (51)     (52)      (62)
               (221)   (411)    (61)      (71)
               (311)   (2211)   (322)     (332)
               (2111)  (3111)   (331)     (422)
                       (21111)  (421)     (431)
                                (511)     (521)
                                (2221)    (611)
                                (3211)    (3311)
                                (4111)    (5111)
                                (22111)   (22211)
                                (31111)   (32111)
                                (211111)  (41111)
                                          (221111)
                                          (311111)
                                          (2111111)

The Heinz numbers of these partitions are A327476.
Partitions with their mean are A237984.
Subsets without their mean are A327471.
Subsets with n but without their mean are A327477.
Strict partitions without their mean are A240851.
Cf. A007865, A067538, A067538, A102627, A114639, A324756, A327473.

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,Mean[#]]&]],{n,0,20}]

from sympy.utilities.iterables import partitions
def A327472(n): return sum(1 for s,p in partitions(n,size=True) if n%s or n//s not in p) if n else 1 # Chai Wah Wu, Sep 21 2023

A306844 Number of anti-transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 36, 83, 212, 532, 1379, 3577, 9444, 25019, 66943, 179994, 487031, 1323706, 3614622, 9907911

Offset: 1

Gus Wiseman, Mar 13 2019

A rooted tree is anti-transitive if the subbranches are disjoint from the branches, i.e., no branch of a branch is a branch.

			The a(1) = 1 through a(6) = 14 anti-transitive rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (((o)))  (((oo)))   (((ooo)))
                          ((o)(o))   ((o)(oo))
                          ((o(o)))   ((o(oo)))
                          (o((o)))   ((oo(o)))
                          ((((o))))  (o((oo)))
                                     (oo((o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     (((o(o))))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

Gus Wiseman, The a(7) = 36 anti-transitive rooted trees.
Gus Wiseman, The a(10) = 532 anti-transitive rooted trees.

Cf. A276625, A279861, A279861, A290689, A290760, A304360.

Cf. A324694, A324695, A324738, A324741, A324743, A324751, A324754, A324756, A324758, A324759, A324764.

rtall[n_]:=Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])];
Table[Length[Select[rtall[n],Intersection[Union@@#,#]=={}&]],{n,10}]

a(16)-a(20) from Jinyuan Wang, Jun 20 2020

A324758 Heinz numbers of integer partitions containing no prime indices of the parts.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 31, 32, 33, 34, 35, 37, 40, 41, 43, 44, 46, 47, 49, 50, 51, 53, 57, 59, 61, 62, 63, 64, 65, 67, 68, 71, 73, 77, 79, 80, 81, 82, 83, 85, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 100, 101

Offset: 1

Gus Wiseman, Mar 17 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).
These could be described as anti-transitive numbers (cf. A290822), as they are numbers x such that if prime(y) divides x and prime(z) divides y, then prime(z) does not divide x.
Also numbers n such that A003963(n) is coprime to n.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  27: {2,2,2}

The subset version is A324741, with maximal case A324743. The strict integer partition version is A324751. The integer partition version is A324756. An infinite version is A324695.

Cf. A000720, A001221, A001462, A007097, A056239, A112798, A276625, A289509, A290822, A304360, A306844, A324742, A324753, A324764.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[primeMS[#],Union@@primeMS/@primeMS[#]]=={}&]

A324851 Numbers > 1 divisible by the sum of their prime indices.

Original entry on oeis.org

2, 4, 6, 12, 15, 16, 20, 30, 35, 36, 42, 48, 56, 88, 99, 112, 120, 126, 130, 135, 143, 144, 160, 162, 180, 192, 210, 216, 220, 221, 228, 231, 242, 250, 256, 270, 275, 280, 288, 297, 300, 308, 322, 330, 338, 360, 396, 400, 408, 429, 435, 440, 455, 468, 480, 493

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. The sum of prime indices of n is A056239(n). For example, the prime indices of 99 are {2,2,5}, with sum 9, a divisor of 99, so 99 is in the sequence.

For any k>=2, let d be a divisor of k such that d > A056239(k). Then 2^(d-A056239(k))*k is in the sequence. Similarly if k is in the sequence with d = A056239(k), then 2^d*k is in the sequence. - Robert Israel, Mar 19 2019

			The sequence of terms together with their prime indices begins:
    2: {1}
    4: {1,1}
    6: {1,2}
   12: {1,1,2}
   15: {2,3}
   16: {1,1,1,1}
   20: {1,1,3}
   30: {1,2,3}
   35: {3,4}
   36: {1,1,2,2}
   42: {1,2,4}
   48: {1,1,1,1,2}
   56: {1,1,1,4}
   88: {1,1,1,5}
   99: {2,2,5}
  112: {1,1,1,1,4}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  130: {1,3,6}
  135: {2,2,2,3}

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003963, A036844, A056239, A112798, A120383, A290822, A306844.

Cf. A324695, A324741, A324743, A324847, A324756, A324758, A324765, A324847, A324848, A324849, A324850, A324852.

filter:= proc(n) local t; n mod add(numtheory:-pi(t[1])*t[2],t=ifactors(n)[2]) = 0 end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 19 2019

Select[Range[2,100],Divisible[#,Plus@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]*k]]&]

isok(n) = {my(f = factor(n)); (n!=1) && !(n % sum(k=1, #f~, primepi(f[k,1])*f[k,2]));} \\ Michel Marcus, Mar 19 2019

A324846 Positive integers divisible by none of their prime indices.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 121, 123, 125, 127, 129, 131, 133, 137

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

A prime index of n is a number m such that prime(m) divides n. For example, the prime indices of 5673 are {2,11,18}, none of which divides 5673, so 5673 belongs to the sequence.

			The sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  33: {2,5}
  35: {3,4}
  37: {12}
  39: {2,6}

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Complement of A324847.
Cf. A000720, A001222, A003963, A056239, A112798, A120383, A289509, A290822, A304360, A306844.
Cf. A324695, A324741, A324743, A324756, A324758, A324765, A324848, A324849, A324850, A324852, A324853.

q:= n-> ormap(i-> irem(n, numtheory[pi](i[1]))=0, ifactors(n)[2]):
remove(q, [$1..200])[];  # Alois P. Heinz, Mar 19 2019

Select[Range[100],!Or@@Cases[If[#==1,{},FactorInteger[#]],{p_,_}:>Divisible[#,PrimePi[p]]]&]

isok(n) = {my(f = factor(n)[,1]); for (k=1, #f, if (!(n % primepi(f[k])), return (0));); return (1);} \\ Michel Marcus, Mar 19 2019

A324751 Number of strict integer partitions of n containing no prime indices of the parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 3, 2, 4, 5, 5, 6, 8, 8, 12, 10, 14, 13, 18, 19, 26, 25, 30, 34, 39, 40, 51, 55, 60, 71, 77, 90, 97, 111, 123, 136, 153, 170, 179, 216, 230, 264, 282, 322, 345, 385, 423, 470, 513, 573, 629, 686, 755, 834, 910, 1005, 1095, 1194, 1303, 1433

Offset: 0

Gus Wiseman, Mar 16 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 a(1) = 1 through a(13) = 8 strict integer partitions (A...D = 10...13):
  1   2   3   4    5   6    7    8    9    A    B     C     D
              31       42   43   71   54   64   65    75    76
                       51   52        63   73   83    84    85
                                      72   82   542   93    94
                                           91   731   A2    B2
                                                      B1    643
                                                            751
                                                            931

The subset version is A324741, with maximal case A324743. The non-strict version is A324756. The Heinz number version is A324758. An infinite version is A304360.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A305713, A306844, A324764.
Cf. A324695, A324742, A324748.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,0,30}]

A324764 Number of anti-transitive rooted identity trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 9, 20, 41, 89, 196, 443, 987, 2246, 5114, 11757, 27122, 62898, 146392, 342204, 802429, 1887882

Offset: 1

Gus Wiseman, Mar 17 2019

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. It is anti-transitive if the branches of the branches of the root are disjoint from the branches of the root.
Also the number of finitary sets S with n brackets where no element of an element of S is also an element of S. For example, the a(8) = 20 finitary sets are (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}}}}
  {{o},{o,{{o}}}}
  {{{o}},{o,{o}}}

			The a(1) = 1 through a(7) = 9 anti-transitive rooted identity 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))))))

Gus Wiseman, The a(9) = 41 anti-transitive rooted identity trees.

Cf. A000081, A004111, A276625, A279861, A290689, A304360, A306844 (non-identity version), A316500, A317787, A318185.

Cf. A324694, A324751, A324756, A324758, A324765, A324767, A324768, A324770, A324839, A324840, A324844.

idall[n_]:=If[n==1,{{}},Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&]];
Table[Length[Select[idall[n],Intersection[Union@@#,#]=={}&]],{n,10}]

a(21)-a(22) from Jinyuan Wang, Jun 20 2020

A324849 Positive integers divisible by none of their prime indices > 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 76, 77, 79, 80, 81, 82, 83, 85, 86, 87

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

A prime index of n is a number m such that prime(m) divides n.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}

Robert Israel, Table of n, a(n) for n = 1..10000

Complement of A324771.
Cf. A003963, A056239, A112798, A120383, A289509, A304360, A306844.
Cf. A324695, A324741, A324743, A324756, A324758, A324765, A324846, A324847, A324848, A324850, A324852, A324853, A324856.

filter:= proc(n) andmap(t -> not ((n/numtheory:-pi(t))::integer), numtheory:-factorset(n) minus {2}) end proc:
select(filter, [$1..200]); # Robert Israel, Mar 20 2019

Select[Range[100],!Or@@Cases[If[#==1,{},FactorInteger[#]],{p_,_}:>If[p==2,False,Divisible[#,PrimePi[p]]]]&]

is(n) = my(f=factor(n)[, 1]~, idc=[]); for(k=1, #f, idc=concat(idc, [primepi(f[k])])); for(t=1, #idc, if(idc[t]==1, next); if(n%idc[t]==0, return(0))); 1 \\ Felix Fröhlich, Mar 21 2019

cp's OEIS Frontend

A324850 Numbers divisible by the product of their prime indices.

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A327476 Heinz numbers of integer partitions whose mean A326567/A326568 is not a part.

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A327472 Number of integer partitions of n not containing their mean.

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A306844 Number of anti-transitive rooted trees with n nodes.

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A324758 Heinz numbers of integer partitions containing no prime indices of the parts.

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A324851 Numbers > 1 divisible by the sum of their prime indices.

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A324846 Positive integers divisible by none of their prime indices.

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A324751 Number of strict integer partitions of n containing no prime indices of the parts.

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