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.

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A357630 Skew-alternating sum of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 0, 3, -1, 4, -1, 0, -2, 5, -2, 6, -3, -1, 0, 7, -3, 8, -3, -2, -4, 9, 1, 0, -5, -2, -4, 10, -4, 11, 1, -3, -6, -1, 0, 12, -7, -4, 2, 13, -5, 14, -5, -3, -8, 15, 2, 0, -5, -5, -6, 16, -1, -2, 3, -6, -9, 17, 1, 18, -10, -4, 0, -3, -6, 19, -7, -7, -6, 20
Offset: 1

Views

Author

Gus Wiseman, Oct 09 2022

Keywords

Comments

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....
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.

Examples

			The prime indices of 525 are {2,3,3,4} so a(525) = 2 - 3 - 3 + 4 = 0.

Crossrefs

The original alternating sum is A316524, reverse A344616.
The reverse version is A357634.
The half-alternating form is A357629, reverse A357633.
Positions of zeros are A357632, reverse A357636.
The version for standard compositions is A357623, reverse A357624.
These partitions are counted by A357638, half A357637.
A056239 adds up prime indices, row sums of A112798.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357621, A357622, A357625, A357626, A357635, A357640.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Table[skats[primeMS[n]],{n,30}]

A357634 Skew-alternating sum of the partition having Heinz number n.

Original entry on oeis.org

0, 1, 2, 0, 3, 1, 4, -1, 0, 2, 5, 0, 6, 3, 1, 0, 7, -1, 8, 1, 2, 4, 9, 1, 0, 5, -2, 2, 10, 0, 11, 1, 3, 6, 1, 0, 12, 7, 4, 2, 13, 1, 14, 3, -1, 8, 15, 2, 0, -1, 5, 4, 16, -1, 2, 3, 6, 9, 17, 1, 18, 10, 0, 0, 3, 2, 19, 5, 7, 0, 20, 1, 21, 11, -2, 6, 1, 3, 22, 3
Offset: 1

Views

Author

Gus Wiseman, Oct 09 2022

Keywords

Comments

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....
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.

Examples

			The partition with Heinz number 525 is (4,3,3,2) so a(525) = 4 - 3 - 3 + 2 = 0.

Crossrefs

The original alternating sum is A316524, reverse A344616.
The non-reverse version is A357630.
The half-alternating form is A357633, non-reverse A357629.
Positions of zeros are A357636, non-reverse A357632.
The version for standard compositions is A357624, non-reverse A357623.
These partitions are counted by A357638, half A357637.
A056239 adds up prime indices, row sums of A112798.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357621, A357622, A357625, A357626, A357635, A357640.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Table[skats[Reverse[primeMS[n]]],{n,30}]

A357632 Numbers k such that the skew-alternating sum of the prime indices of k is 0.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 90, 100, 121, 144, 169, 196, 210, 225, 256, 289, 324, 360, 361, 400, 441, 462, 484, 525, 529, 550, 576, 625, 676, 729, 784, 840, 841, 858, 900, 910, 961, 1024, 1089, 1155, 1156, 1225, 1296, 1326, 1369, 1440, 1444, 1521, 1600
Offset: 1

Views

Author

Gus Wiseman, Oct 09 2022

Keywords

Comments

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....
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.

Examples

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     9: {2,2}
    16: {1,1,1,1}
    25: {3,3}
    36: {1,1,2,2}
    49: {4,4}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
    90: {1,2,2,3}
   100: {1,1,3,3}
   121: {5,5}
   144: {1,1,1,1,2,2}

Crossrefs

The version for original alternating sum is A000290.
The version for standard compositions is A357627, reverse A357628.
Positions of zeros in A357630, reverse A357634.
The half-alternating form is A357631, reverse A357635.
The reverse version is A357636.
These partitions are counted by A357640, half A357639.
A056239 adds up prime indices, row sums of A112798.
A316524 gives alternating sum of prime indices, reverse A344616.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357641 counts comps w/ half-alt sum 0, even A357642.
Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357621-A357626, A357629, A357637, A357638.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Select[Range[1000],skats[primeMS[#]]==0&]

A357633 Half-alternating sum of the partition having Heinz number n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 1, 4, 4, 5, 2, 6, 5, 5, 0, 7, 3, 8, 3, 6, 6, 9, 1, 6, 7, 2, 4, 10, 4, 11, 1, 7, 8, 7, 2, 12, 9, 8, 2, 13, 5, 14, 5, 3, 10, 15, 2, 8, 5, 9, 6, 16, 1, 8, 3, 10, 11, 17, 3, 18, 12, 4, 2, 9, 6, 19, 7, 11, 6, 20, 3, 21, 13, 4, 8, 9, 7, 22, 3, 0
Offset: 1

Views

Author

Gus Wiseman, Oct 09 2022

Keywords

Comments

We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...
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.

Examples

			The partition with Heinz number 525 is (4,3,3,2) so a(525) = 4 + 3 - 3 - 2 = 2.

Crossrefs

The original alternating sum is A316524, reverse A344616.
The version for standard compositions is A357622, non-reverse A357621.
The skew-alternating form is A357634, non-reverse A357630.
Positions of zeros are A000583, non-reverse A357631.
The reverse version is A357629.
These partitions are counted by A357637, skew A357638.
A056239 adds up prime indices, row sums of A112798.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357623-A357626, A357632, A357636, A357640.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
Table[halfats[Reverse[primeMS[n]]],{n,30}]

A357643 Number of integer compositions of n into parts that are alternately equal and unequal.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 5, 5, 9, 7, 17, 14, 28, 25, 49, 42, 87, 75, 150, 132, 266, 226, 466, 399, 810, 704, 1421, 1223, 2488, 2143, 4352, 3759, 7621, 6564, 13339, 11495, 23339, 20135, 40852, 35215, 71512, 61639, 125148, 107912, 219040, 188839, 383391, 330515, 670998
Offset: 0

Views

Author

Gus Wiseman, Oct 12 2022

Keywords

Examples

			The a(1) = 1 through a(8) = 9 compositions:
  (1)  (2)   (3)  (4)    (5)    (6)     (7)      (8)
       (11)       (22)   (113)  (33)    (115)    (44)
                  (112)  (221)  (114)   (223)    (116)
                                (1122)  (331)    (224)
                                (2211)  (11221)  (332)
                                                 (1133)
                                                 (3311)
                                                 (22112)
                                                 (112211)

Links

Crossrefs

The even-length version is A003242, ranked by A351010, partitions A035457.
Without equal relations we have A016116, equal only A001590 (apparently).
The version for partitions is A351005.
The opposite version is A357644, partitions A351006.
A011782 counts compositions.
A357621 gives half-alternating sum of standard compositions, skew A357623.
A357645 counts compositions by half-alternating sum, skew A357646.
Cf. A029862, A035544, A097805, A122129, A122134, A122135, A351003, A351004, A351007, A357136, A357641.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]==#[[i+1]],{i,1,Length[#]-1,2}]&&And@@Table[#[[i]]!=#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,15}]
  • PARI
    C_x(N) = {my(x='x+O('x^N), h=(1+sum(k=1,N, (x^k)/(1+x^(2*k))))/(1-sum(k=1,N, (x^(2*k))/(1+x^(2*k))))); Vec(h)}
C_x(50) \\ John Tyler Rascoe, May 28 2024

Formula

G.f.: (1 + Sum_{k>0} (x^k)/(1 + x^(2*k)))/(1 - Sum_{k>0} (x^(2*k))/(1 + x^(2*k))). - John Tyler Rascoe, May 28 2024

Extensions

More terms from Alois P. Heinz, Oct 12 2022

A357644 Number of integer compositions of n into parts that are alternately unequal and equal.

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 8, 13, 17, 25, 30, 44, 58, 77, 98, 142, 176, 245, 311, 426, 548, 758, 952, 1319, 1682, 2308, 2934, 4059, 5132, 7087, 9008, 12395, 15757, 21728, 27552, 38019, 48272, 66515, 84462, 116467, 147812, 203825, 258772, 356686, 452876, 624399, 792578
Offset: 0

Views

Author

Gus Wiseman, Oct 14 2022

Keywords

Examples

			The a(1) = 1 through a(7) = 13 compositions:
  (1)  (2)  (3)   (4)    (5)    (6)     (7)
            (12)  (13)   (14)   (15)    (16)
            (21)  (31)   (23)   (24)    (25)
                  (211)  (32)   (42)    (34)
                         (41)   (51)    (43)
                         (122)  (411)   (52)
                         (311)  (1221)  (61)
                                (2112)  (133)
                                        (322)
                                        (511)
                                        (2113)
                                        (3112)
                                        (12211)

Links

Crossrefs

Without equal relations we have A000213, equal only A027383.
Even-length opposite: A003242, ranked by A351010, partitions A035457.
The version for partitions is A351006.
The opposite version is A357643, partitions A351005.
A011782 counts compositions.
A357621 gives half-alternating sum of standard compositions, skew A357623.
A357645 counts compositions by half-alternating sum, skew A357646.
Cf. A001590, A029862, A035544, A097805, A122129, A122134, A122135, A351003, A351004, A351007, A357136, A357641.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]==#[[i+1]],{i,2,Length[#]-1,2}]&&And@@Table[#[[i]]!=#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,10}]

Extensions

More terms from Alois P. Heinz, Oct 19 2022

A351003 Number of integer partitions y of n such that y_i = y_{i+1} for all even i.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 28, 36, 42, 51, 62, 75, 88, 106, 124, 147, 173, 202, 236, 278, 320, 371, 431, 497, 572, 661, 756, 867, 993, 1132, 1291, 1474, 1672, 1898, 2155, 2439, 2756, 3117, 3512, 3957, 4458, 5008, 5624, 6316, 7072, 7919, 8862, 9899
Offset: 0

Views

Author

Gus Wiseman, Jan 31 2022

Keywords

Examples

			The a(1) = 1 through a(7) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (311)    (51)      (61)
                    (1111)  (2111)   (222)     (322)
                            (11111)  (411)     (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)

Crossrefs

The ordered version (compositions) is A027383.
The version for unequal instead of equal is A122135, even-length A351008.
For odd instead of even indices we have A351004, even-length A035363.
Requiring inequalities at odd positions gives A351006, even-length A351007.
The even-length case is A351012.
Cf. A000070, A018819, A088218, A101417, A122129, A350837, A351005.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And@@Table[#[[i]]==#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,10}]

A351007 Number of even-length integer partitions of n into parts that are alternately unequal and equal.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 9, 10, 13, 14, 16, 18, 20, 23, 27, 28, 32, 37, 40, 44, 51, 54, 60, 67, 73, 81, 90, 96, 107, 118, 127, 139, 154, 166, 181, 198, 213, 232, 256, 273, 297, 325, 348, 377, 411, 440, 476, 516, 555, 598, 647, 692, 746, 807
Offset: 0

Views

Author

Gus Wiseman, Jan 31 2022

Keywords

Comments

These are partitions whose multiplicities begin with a 1, are followed by any number of 2's, and end with another 1.

Examples

			The a(3) = 1 through a(15) = 13 partitions (A..E = 10..14):
  21  31  32  42  43  53    54    64    65    75    76    86    87
          41  51  52  62    63    73    74    84    85    95    96
                  61  71    72    82    83    93    94    A4    A5
                      3221  81    91    92    A2    A3    B3    B4
                            4221  5221  A1    B1    B2    C2    C3
                                        4331  4332  C1    D1    D2
                                        6221  5331  5332  5441  E1
                                              7221  6331  6332  5442
                                                    8221  7331  6441
                                                          9221  7332
                                                                8331
                                                                A221
                                                                433221

Crossrefs

The alternately equal and unequal version is A035457, any length A351005.
This is the even-length case of A351006, odd-length A053251.
Without equalities we have A351008, any length A122129, opposite A122135.
Without inequalities we have A351012, any length A351003, opposite A351004.
Cf. A000070, A003242, A018819, A027383, A035363, A088218, A122134, A350842, A350844, A351011.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]],{i,2,Length[#]-1,2}]&&And@@Table[#[[i]]!=#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]

A357636 Numbers k such that the skew-alternating sum of the partition having Heinz number k is 0.

Original entry on oeis.org

1, 4, 9, 12, 16, 25, 30, 36, 49, 63, 64, 70, 81, 90, 100, 108, 121, 144, 154, 165, 169, 192, 196, 210, 225, 256, 273, 286, 289, 300, 324, 325, 360, 361, 400, 441, 442, 462, 480, 484, 525, 529, 550, 561, 576, 588, 595, 625, 646, 676, 700, 729, 741, 750, 784
Offset: 1

Views

Author

Gus Wiseman, Oct 09 2022

Keywords

Comments

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....
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.

Examples

			The terms together with their prime indices begin:
    1: {}
    4: {1,1}
    9: {2,2}
   12: {1,1,2}
   16: {1,1,1,1}
   25: {3,3}
   30: {1,2,3}
   36: {1,1,2,2}
   49: {4,4}
   63: {2,2,4}
   64: {1,1,1,1,1,1}
   70: {1,3,4}
   81: {2,2,2,2}
   90: {1,2,2,3}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  121: {5,5}
  144: {1,1,1,1,2,2}

Crossrefs

The version for original alternating sum is A000290.
The half-alternating form is A000583, non-reverse A357631.
The version for standard compositions is A357628, non-reverse A357627.
The non-reverse version is A357632.
Positions of zeros in A357634, non-reverse A357630.
These partitions are counted by A357640, half A357639.
A056239 adds up prime indices, row sums of A112798.
A316524 gives alternating sum of prime indices, reverse A344616.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357641 counts comps w/ half-alt sum 0, even A357642.
Cf. A003963, A035594, A053251, A055932, A357189, A357485-A357488, A357621-A357626, A357629, A357637, A357638.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Select[Range[1000],skats[Reverse[primeMS[#]]]==0&]

A351004 Alternately constant partitions. Number of integer partitions y of n such that y_i = y_{i+1} for all odd i.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 4, 7, 7, 10, 9, 15, 13, 21, 19, 28, 26, 40, 35, 54, 49, 72, 64, 97, 87, 128, 115, 167, 151, 220, 195, 284, 256, 366, 328, 469, 421, 598, 537, 757, 682, 959, 859, 1204, 1085, 1507, 1354, 1880, 1694, 2338, 2104, 2892, 2609, 3574, 3218, 4394
Offset: 0

Views

Author

Gus Wiseman, Jan 31 2022

Keywords

Comments

These are partitions of n with all even multiplicities (or run-lengths), except possibly the last.

Examples

			The a(1) = 1 through a(9) = 7 partitions:
  1  2   3    4     5      6       7        8         9
     11  111  22    221    33      331      44        333
              1111  11111  222     22111    332       441
                           2211    1111111  2222      22221
                           111111           3311      33111
                                            221111    2211111
                                            11111111  111111111

Crossrefs

The ordered version (compositions) is A016116.
The even-length case is A035363.
A reverse version is A096441, both A349060.
The version for unequal instead of equal is A122129, even-length A351008.
The version for even instead of odd indices is A351003, even-length A351012.
The strict version is A351005, opposite A351006, even-length A035457.
Cf. A000070, A018819, A027383, A088218, A067661, A101417, A122134, A122135, A350842, A351007.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And@@Table[#[[i]]==#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]
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