cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A363727 Numbers whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 90, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199
Offset: 1

Views

Author

Gus Wiseman, Jun 23 2023

Keywords

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.
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The terms together with their prime indices begin:
     2: {1}          29: {10}              79: {22}
     3: {2}          31: {11}              81: {2,2,2,2}
     4: {1,1}        32: {1,1,1,1,1}       83: {23}
     5: {3}          37: {12}              89: {24}
     7: {4}          41: {13}              90: {1,2,2,3}
     8: {1,1,1}      43: {14}              97: {25}
     9: {2,2}        47: {15}             101: {26}
    11: {5}          49: {4,4}            103: {27}
    13: {6}          53: {16}             107: {28}
    16: {1,1,1,1}    59: {17}             109: {29}
    17: {7}          61: {18}             113: {30}
    19: {8}          64: {1,1,1,1,1,1}    121: {5,5}
    23: {9}          67: {19}             125: {3,3,3}
    25: {3,3}        71: {20}             127: {31}
    27: {2,2,2}      73: {21}             128: {1,1,1,1,1,1,1}

Crossrefs

These partitions are counted by A363719, factorizations A363741.
For unequal instead of equal we have A363730, counted by A363720.
Excluding primes gives A363722.
Excluding prime-powers gives A363729, counted by A363728.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with multiple modes, counted by A362610.
A360005 gives twice the median of prime indices.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A363486 gives least mode in prime indices, A363487 greatest.
Just two statistics:
- (mean) = (median): A359889, counted by A240219.
- (mean) != (median): A359890, counted by A359894.
- (mean) = (mode): counted by A363723, see A363724, A363731.
- (median) = (mode): counted by A363740.
Cf. A215366, A327473, A327476, A359893, A359908, A360009, A360248, A360550, A363721, A363725.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Select[Range[100],{Mean[prix[#]]}=={Median[prix[#]]}==modes[prix[#]]&]

Formula

Assuming there is a unique mode, we have A326567(a(n))/A326568(a(n)) = A360005(a(n))/2 = A363486(a(n)) = A363487(a(n)).

A363719 Number of integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 5, 2, 10, 2, 7, 7, 12, 2, 18, 2, 24, 16, 13, 2, 58, 15, 18, 37, 60, 2, 123, 2, 98, 79, 35, 103, 332, 2, 49, 166, 451, 2, 515, 2, 473, 738, 92, 2, 1561, 277, 839, 631, 1234, 2, 2043, 1560, 2867, 1156, 225, 2, 9020
Offset: 1

Views

Author

Gus Wiseman, Jun 19 2023

Keywords

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.
Without loss of generality, we may assume there is a unique middle-part (A238478).
Includes all constant partitions.

Examples

			The a(n) partitions for n = 1, 2, 4, 6, 8, 12, 14, 16 (A..G = 10..16):
  1  2   4     6       8         C             E               G
     11  22    33      44        66            77              88
         1111  222     2222      444           2222222         4444
               111111  3221      3333          3222221         5443
                       11111111  4332          3322211         6442
                                 5331          4222211         7441
                                 222222        11111111111111  22222222
                                 322221                        32222221
                                 422211                        33222211
                                 111111111111                  42222211
                                                               52222111
                                                               1^16

Crossrefs

For unequal instead of equal: A363720, ranks A363730, unique mode A363725.
The odd-length case is A363721.
These partitions have ranks A363727, nonprime A363722.
The case of non-constant partitions is A363728, ranks A363729.
The version for factorizations is A363741, see A359909, A359910.
Just two statistics:
- (mean) = (median) gives A240219, also A359889, A359895, A359897, A359899.
- (mean) != (median) gives A359894, also A359890, A359896, A359898, A359900.
- (mean) = (mode) gives A363723, see A363724, A363731.
- (median) = (mode) gives A363740.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or negative mean), strict A008289.
A359893 and A359901 count partitions by median, odd-length A359902.
A362608 counts partitions with a unique mode.
Cf. A027193, A237984, A325347, A326567/A326568, A327472, A363726, A363742.

Programs

  • Mathematica
    modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], {Mean[#]}=={Median[#]}==modes[#]&]],{n,30}]

A363730 Numbers whose prime indices have different mean, median, and mode.

Original entry on oeis.org

42, 60, 66, 70, 78, 84, 102, 114, 130, 132, 138, 140, 150, 154, 156, 165, 170, 174, 180, 182, 186, 190, 195, 204, 220, 222, 228, 230, 231, 246, 255, 258, 260, 266, 276, 282, 285, 286, 290, 294, 308, 310, 315, 318, 322, 330, 340, 345, 348, 354, 357, 360, 364
Offset: 1

Views

Author

Gus Wiseman, Jun 24 2023

Keywords

Comments

If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.
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 mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The prime indices of 180 are {1,1,2,2,3}, with mean 9/5, median 2, modes {1,2}, so 180 is in the sequence.
The prime indices of 108 are {1,1,2,2,2}, with mean 8/5, median 2, modes {2}, so 108 is not in the sequence.
The terms together with their prime indices begin:
   42: {1,2,4}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
  102: {1,2,7}
  114: {1,2,8}
  130: {1,3,6}
  132: {1,1,2,5}
  138: {1,2,9}
  140: {1,1,3,4}
  150: {1,2,3,3}

Crossrefs

These partitions are counted by A363720
For equal instead of unequal we have A363727, counted by A363719.
The version for factorizations is A363742, equal A363741.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with multiple modes, counted by A362610.
A360005 gives twice the median of prime indices.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A363486 gives least mode in prime indices, A363487 greatest.
Just two statistics:
- (mean) = (median): A359889, counted by A240219.
- (mean) != (median): A359890, counted by A359894.
- (mean) = (mode): counted by A363723, see A363724, A363731.
- (median) = (mode): counted by A363740.
Cf. A000961, A327473, A327476, A359908, A363722, A363725, A363729.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Select[Range[100],{Mean[prix[#]]}!={Median[prix[#]]}!=modes[prix[#]]&]

Formula

All three of A326567(a(n))/A326568(a(n)), A360005(a(n))/2, and A363486(a(n)) = A363487(a(n)) are different.

A364061 Numbers whose exponent of 2 in their canonical prime factorization is smaller than all the other exponents.

Original entry on oeis.org

2, 4, 8, 16, 18, 32, 50, 54, 64, 98, 108, 128, 162, 242, 250, 256, 324, 338, 450, 486, 500, 512, 578, 648, 686, 722, 882, 972, 1024, 1058, 1250, 1350, 1372, 1458, 1682, 1922, 1944, 2048, 2178, 2250, 2450, 2500, 2646, 2662, 2738, 2916, 3042, 3362, 3698, 3888
Offset: 1

Views

Author

Gus Wiseman, Jul 12 2023

Keywords

Comments

Also numbers whose multiset of prime factors has unique co-mode 2. Here, a co-mode in a multiset is an element that appears at most as many times as each of the other elements. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

Examples

			The terms together with their prime factors begin:
    2 = 2
    4 = 2*2
    8 = 2*2*2
   16 = 2*2*2*2
   18 = 2*3*3
   32 = 2*2*2*2*2
   50 = 2*5*5
   54 = 2*3*3*3
   64 = 2*2*2*2*2*2
   98 = 2*7*7
  108 = 2*2*3*3*3
  128 = 2*2*2*2*2*2*2

Links

Crossrefs

For any unique co-mode: A359178, counted by A362610, complement A362606.
For high mode: A360013, positions of 1's in A363487, counted by A241131.
For low mode: A360015, positions of 1's in A363486, counted by A241131.
Partitions of this type are counted by A364062.
For low co-mode: A364158, positions of 1's in A364192, counted by A364159.
Positions of 1's in A364191, high A364192.
A112798 lists prime indices, length A001222, sum A056239.
A356862 ranks partitions w/ unique mode, count A362608, complement A362605.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
Cf. A000265, A007814, A327473, A327476, A362616, A360014, A363722, A363723, A363725, A363727, A363730.

Programs

  • Maple
    filter:= proc(n) local F,F2,Fo;
  F:= ifactors(n)[2];
  F2,Fo:= selectremove(t -> t[1]=2, F);
  Fo = [] or F2[1,2] < min(Fo[..,2])
end proc:
select(filter, 2*[$1..5000]); # Robert Israel, Apr 22 2024
  • Mathematica
    prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Select[Range[100],comodes[prifacs[#]]=={2}&]
  • Python
    from sympy import factorint
from itertools import count, islice
def A364061_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:(l:=(~n&n-1).bit_length()) < min(factorint(m:=n>>l).values(),default=0) or m==1, count(max(startvalue+startvalue&1,2),2))
A364061_list = list(islice(A364061_gen(),30)) # Chai Wah Wu, Jul 14 2023

Formula

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=2} (1-1/2^(k-1))*(s(k)-s(k+1)) = 1.16896822653093929144..., where s(k) = Product_{primes p >= 3} (1 + 1/(p^(k-1)*(p-1))) is the sum of reciprocals of the odd k-full numbers (numbers whose prime factorization has no exponent that is smaller than k). - Amiram Eldar, Aug 30 2024

A363729 Numbers that are not a power of a prime but whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

90, 270, 525, 550, 756, 810, 1666, 1911, 1950, 2268, 2430, 2625, 2695, 2700, 2750, 5566, 6762, 6804, 6897, 7128, 7290, 8100, 8500, 9310, 9750, 10285, 10478, 11011, 11550, 11662, 12250, 12375, 12495, 13125, 13377, 13750, 14014, 14703, 18865, 19435, 20412, 21384
Offset: 1

Views

Author

Gus Wiseman, Jun 24 2023

Keywords

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.
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The prime indices of 6897 are {2,5,5,8}, with mean 5, median 5, and modes {5}, so 6897 is in the sequence.
The terms together with their prime indices begin:
     90: {1,2,2,3}
    270: {1,2,2,2,3}
    525: {2,3,3,4}
    550: {1,3,3,5}
    756: {1,1,2,2,2,4}
    810: {1,2,2,2,2,3}
   1666: {1,4,4,7}
   1911: {2,4,4,6}
   1950: {1,2,3,3,6}
   2268: {1,1,2,2,2,2,4}
   2430: {1,2,2,2,2,2,3}

Crossrefs

For just primes instead of prime powers we have A363722.
Including prime-powers gives A363727, counted by A363719.
These partitions are counted by A363728.
For unequal instead of equal we have A363730, counted by A363720.
A000961 lists the prime powers, complement A024619.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with multiple modes, counted by A362610.
A360005 gives twice the median of prime indices.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A363486 gives least mode in prime indices, A363487 greatest.
Just two statistics:
- (mean) = (median): A359889, counted by A240219.
- (mean) != (median): A359890, counted by A359894.
- (mean) = (mode): counted by A363723, see A363724, A363731.
- (median) = (mode): counted by A363740.
Cf. A215366, A327473, A327476, A359893, A359908, A360009, A360248, A360550, A363721, A363725, A363741.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Select[Range[1000],!PrimePowerQ[#]&&{Mean[prix[#]]}=={Median[prix[#]]}==modes[prix[#]]&]
Showing 1-5 of 5 results.

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