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.

Previous Showing 11-17 of 17 results.

A361204 Positive integers k such that 2*omega(k) <= bigomega(k).

Original entry on oeis.org

1, 4, 8, 9, 16, 24, 25, 27, 32, 36, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 100, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 196, 200, 208, 216, 224, 225, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304
Offset: 1

Views

Author

Gus Wiseman, Mar 14 2023

Keywords

Examples

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     8: {1,1,1}
     9: {2,2}
    16: {1,1,1,1}
    24: {1,1,1,2}
    25: {3,3}
    27: {2,2,2}
    32: {1,1,1,1,1}
    36: {1,1,2,2}
    40: {1,1,1,3}
    48: {1,1,1,1,2}
    49: {4,4}
    54: {1,2,2,2}
    56: {1,1,1,4}
    64: {1,1,1,1,1,1}

Links

Crossrefs

These partitions are counted by A237363.
The complement is A361393.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors.
A112798 lists prime indices, sum A056239.
A360005 gives median of prime indices (times 2), distinct A360457.
Comparing twice the number of distinct parts to the number of parts:
less: A360254, ranks A360558
equal: A239959, ranks A067801
greater: A237365, ranks A361393
less or equal: A237363, ranks A361204
greater or equal: A361394, ranks A361395
Cf. A046660, A061395, A067340, A111907.
Cf. A324517, A324521, A324522, A324560, A324562.

Programs

  • Maple
    filter:= proc(n) local F,t;
  F:= ifactors(n)[2];
  add(t[2],t=F) >= 2*nops(F)
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 22 2023
  • Mathematica
    Select[Range[100],2*PrimeNu[#]<=PrimeOmega[#]&]

Formula

A001222(a(n)) >= 2*A001221(a(n)).

A361395 Positive integers k such that 2*omega(k) >= bigomega(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74
Offset: 1

Views

Author

Gus Wiseman, Mar 16 2023

Keywords

Comments

Differs from A068938 in having 1 and 4 and lacking 80.
Includes all squarefree numbers.

Examples

			The prime indices of 80 are {1,1,1,1,3}, with 5 parts and 2 distinct parts, and 2*2 < 5, so 80 is not in the sequence.

Crossrefs

Complement of A360558.
Positions of nonnegative terms in A361205.
These partitions are counted by A361394.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A112798 lists prime indices, sum A056239.
A360005 gives median of prime indices (times 2), distinct A360457.
Comparing twice the number of distinct parts to the number of parts:
less: A360254, ranks A360558
equal: A239959, ranks A067801
greater: A237365, ranks A361393
less or equal: A237363, ranks A361204
greater or equal: A361394, ranks A361395
Cf. A046660, A061395, A067340, A111907.
Cf. A324515, A324517, A324521, A324560, A324562.
Cf. A360248, A360249, A360454.

Programs

  • Mathematica
    Select[Range[100],2*PrimeNu[#]>=PrimeOmega[#]&]

Formula

A001222(a(n)) <= 2*A001221(a(n)).

A361393 Positive integers k such that 2*omega(k) > bigomega(k).

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85
Offset: 1

Views

Author

Gus Wiseman, Mar 16 2023

Keywords

Comments

First differs from A317090 in having 120 and lacking 360.
There are numbers like 1, 120, 168, 180, 252,... which are not in A179983 but in here, and others like 360, 504, 540, 600,... which are in A179983 but not in here. - R. J. Mathar, Mar 21 2023

Examples

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   10: {1,3}
   11: {5}
   12: {1,1,2}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
The prime indices of 120 are {1,1,1,2,3}, with 3 distinct parts and 5 parts, and 2*3 > 5, so 120 is in the sequence.
The prime indices of 360 are {1,1,1,2,2,3}, with 3 distinct parts and 6 parts, and 2*3 is not greater than 6, so 360 is not in the sequence.

Crossrefs

These partitions are counted by A237365.
The complement is A361204.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors.
A112798 lists prime indices, sum A056239.
A326567/A326568 gives mean of prime indices.
A360005 gives median of prime indices (times 2), distinct A360457.
Comparing twice the number of distinct parts to the number of parts:
less: A360254, ranks A360558
equal: A239959, ranks A067801
greater: A237365, ranks A361393
less or equal: A237363, ranks A361204
greater or equal: A361394, ranks A361395
Cf. A046660, A061395, A067340, A111907.
Cf. A324515, A324517, A324519, A324521, A324522, A324560, A324562.

Programs

  • Maple
    isA361393 := proc(n)
    if 2*A001221(n) > numtheory[bigomega](n) then
        true;
    else
        false ;
    end if:
end proc:
for n from 1 to 100 do
    if isA361393(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Mar 21 2023
  • Mathematica
    Select[Range[1000],2*PrimeNu[#]>PrimeOmega[#]&]

Formula

{k: 2*A001221(k) > A001222(k)}. - R. J. Mathar, Mar 21 2023

A362558 Number of integer partitions of n without a nonempty initial consecutive subsequence summing to n/2.

Original entry on oeis.org

1, 1, 1, 3, 2, 7, 6, 15, 11, 30, 27, 56, 44, 101, 93, 176, 149, 297, 271, 490, 432, 792, 744, 1255, 1109, 1958, 1849, 3010, 2764, 4565, 4287, 6842, 6328, 10143, 9673, 14883, 13853, 21637, 20717, 31185, 29343, 44583, 42609, 63261, 60100, 89134, 85893, 124754
Offset: 0

Views

Author

Gus Wiseman, Apr 24 2023

Keywords

Comments

Also the number of n-multisets of positive integers that (1) have integer median, (2) cover an initial interval, and (3) have weakly decreasing multiplicities.

Examples

			The a(1) = 1 through a(7) = 15 partitions:
  (1)  (2)  (3)    (4)   (5)      (6)     (7)
            (21)   (31)  (32)     (42)    (43)
            (111)        (41)     (51)    (52)
                         (221)    (222)   (61)
                         (311)    (411)   (322)
                         (2111)   (2211)  (331)
                         (11111)          (421)
                                          (511)
                                          (2221)
                                          (3211)
                                          (4111)
                                          (22111)
                                          (31111)
                                          (211111)
                                          (1111111)
The partition y = (3,2,1,1,1) has nonempty initial consecutive subsequences (3,2,1,1,1), (3,2,1,1), (3,2,1), (3,2), (3), with sums 8, 7, 6, 5, 3. Since 4 is missing, y is counted under a(8).

Crossrefs

The odd bisection is A058695.
The version for compositions is A213173.
The complement is counted by A322439 aerated.
The even bisection is A362051.
For mean instead of median we have A362559.
A000041 counts integer partitions, strict A000009.
A325347 counts partitions with integer median, complement A307683.
A359893/A359901/A359902 count partitions by median.
Cf. A058398, A108917, A169942, A325676, A353864, A360254, A360672, A360675, A360686, A360687, A362560.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!MemberQ[Accumulate[#],n/2]&]],{n,0,15}]

A362051 Number of integer partitions of 2n without a nonempty initial consecutive subsequence summing to n.

Original entry on oeis.org

1, 1, 2, 6, 11, 27, 44, 93, 149, 271, 432, 744, 1109, 1849, 2764, 4287, 6328, 9673, 13853, 20717, 29343, 42609, 60100, 85893, 118475, 167453, 230080, 318654, 433763, 595921, 800878, 1090189, 1456095, 1957032, 2600199, 3465459, 4558785, 6041381, 7908681
Offset: 0

Views

Author

Gus Wiseman, Apr 24 2023

Keywords

Comments

Even bisection of A362558.
a(0) = 1; a(n) = A000041(2n) - A322439(n). - Alois P. Heinz, Apr 27 2023

Examples

			The a(1) = 1 through a(4) = 11 partitions:
  (2)  (4)   (6)     (8)
       (31)  (42)    (53)
             (51)    (62)
             (222)   (71)
             (411)   (332)
             (2211)  (521)
                     (611)
                     (3221)
                     (3311)
                     (5111)
                     (32111)
The partition y = (3,2,1,1,1) has nonempty initial consecutive subsequences (3,2,1,1,1), (3,2,1,1), (3,2,1), (3,2), (3), with sums 8, 7, 6, 5, 3. Since 4 is missing, y is counted under a(4).

Crossrefs

The version for compositions is A000302, bisection of A213173.
The complement is counted by A322439.
Even bisection of A362558.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with all equal run-sums.
A325347 counts partitions with integer median, complement A307683.
A353836 counts partitions by number of distinct run-sums.
A359893/A359901/A359902 count partitions by median.
Cf. A108917, A169942, A237363, A325676, A353864, A360254, A360672, A360675, A360686, A360952, A362560.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[2n],!MemberQ[Accumulate[#],n]&]],{n,0,15}]

A360682 Number of integer partitions of n of length > 2 whose second differences have median 0.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 5, 4, 10, 13, 18, 23, 44, 44, 72, 98, 132, 162, 241, 277, 394, 497, 643, 800, 1076, 1287, 1660, 2078, 2604, 3192, 4065, 4892, 6113, 7490, 9166, 11110, 13717, 16429, 20033, 24201, 29143, 34945, 42251, 50219, 60253, 71852, 85503, 101501, 120899
Offset: 0

Views

Author

Gus Wiseman, Feb 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).

Examples

			The a(3) = 1 through a(9) = 13 partitions:
  (111)  (1111)  (11111)  (222)     (22111)    (2222)      (333)
                          (321)     (31111)    (3221)      (432)
                          (2211)    (211111)   (3311)      (531)
                          (21111)   (1111111)  (22211)     (22221)
                          (111111)             (32111)     (33111)
                                               (41111)     (51111)
                                               (221111)    (222111)
                                               (311111)    (321111)
                                               (2111111)   (411111)
                                               (11111111)  (2211111)
                                                           (3111111)
                                                           (21111111)
                                                           (111111111)

Crossrefs

For first differences we have A237363.
For sum instead of median we have A360683.
For mean instead of median we have A360683 - A008619.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A325347 counts partitions with integer median, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives median of prime indices (times two).
Cf. A000975, A027193, A067538, A114638, A307683, A359908, A360245, A360254, A360687, A360688.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Median[Differences[#,2]]==0&]],{n,0,30}]

A360691 Number of integer partitions of n with non-integer median of 0-prepended first differences.

Original entry on oeis.org

0, 1, 0, 1, 2, 4, 3, 4, 5, 10, 10, 15, 22, 26, 34, 42, 57, 63, 85, 105, 121, 149, 202, 230, 305, 355, 459, 544, 687, 778, 991, 1130, 1396, 1598, 1947, 2258, 2761, 3143, 3820, 4412, 5330, 6104, 7404, 8499, 10105, 11694, 13922, 15917, 18904, 21646, 25462, 29213
Offset: 1

Views

Author

Gus Wiseman, Feb 22 2023

Keywords

Comments

All of these partitions have even length.
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 a(1) = 0 through a(10) = 10 partitions:
  .  (11)  .  (31)  (32)    (33)    (52)    (53)    (54)      (55)
                    (2111)  (51)    (2221)  (71)    (72)      (73)
                            (2211)  (4111)  (3311)  (3222)    (91)
                            (3111)          (5111)  (6111)    (3322)
                                                    (321111)  (3331)
                                                              (4411)
                                                              (5311)
                                                              (7111)
                                                              (322111)
                                                              (421111)

Crossrefs

For median 0 we have A360254, ranks A360558.
These partitions have ranks A360557, complement A360556.
The complement is counted by A360688.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A027193, A237363, A240219, A359907, A359908, A360455, A360555.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Differences[Prepend[Reverse[#],0]]]]&]],{n,30}]
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