A035544 -id:A035544 - cp's OEIS frontend

A357704 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.

1, 0, 1, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 3, 0, 0, 2, 2, 0, 3, 0, 0, 3, 1, 3, 0, 4, 0, 0, 3, 2, 4, 2, 0, 4, 0, 0, 4, 2, 6, 2, 3, 0, 5, 0, 0, 4, 3, 5, 7, 3, 3, 0, 5, 0, 0, 5, 3, 8, 4, 10, 2, 4, 0, 6, 0, 0, 5, 4, 8, 6, 11, 9, 3, 4, 0, 6, 0, 0, 6, 4, 11, 5, 15, 8, 13, 3, 5, 0, 7

Offset: 0

Gus Wiseman, Oct 10 2022

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 - ...

			Triangle begins:
  1
  0  1
  0  0  2
  0  0  1  2
  0  0  2  0  3
  0  0  2  2  0  3
  0  0  3  1  3  0  4
  0  0  3  2  4  2  0  4
  0  0  4  2  6  2  3  0  5
  0  0  4  3  5  7  3  3  0  5
  0  0  5  3  8  4 10  2  4  0  6
  0  0  5  4  8  6 11  9  3  4  0  6
  0  0  6  4 11  5 15  8 13  3  5  0  7
  0  0  6  5 11  8 13 19 10 13  4  5  0  7
  0  0  7  5 14  8 19 13 25  9 17  4  6  0  8
  0  0  7  6 14 11 19 17 29 23 13 18  5  6  0  8
Row n = 7 counts the following reversed partitions:
  .  .  (115)   (124)   (133)      (11113)   .  (7)
        (1114)  (1222)  (223)      (111112)     (16)
        (1123)          (11122)                 (25)
                        (1111111)               (34)

Row sums are A000041.
Last entry of row n is A008619(n).
The central column in the non-reverse case is A035363, skew A035544.
For original reverse-alternating sum we have A344612.
For original alternating sum we have A344651, ordered A097805.
The non-reverse version is A357637, skew A357638.
The central column is A357639, skew A357640.
The non-reverse ordered version (compositions) is A357645, skew A357646.
The skew-alternating version is A357705.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357621 gives half-alternating sum of standard compositions, skew A357623.
A357629 gives half-alternating sum of prime indices, skew A357630.
A357633 gives half-alternating sum of Heinz partition, skew  A357634.
Cf. A029862, A053251, A357136, A357189, A357487, A357488, A357631, A357632, A357641.

halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
Table[Length[Select[Reverse/@IntegerPartitions[n],halfats[#]==k&]],{n,0,15},{k,-n,n,2}]

A357707 Numbers whose prime indices have equal number of parts congruent to each of 1 and 3 (mod 4).

Original entry on oeis.org

1, 3, 7, 9, 10, 13, 19, 21, 27, 29, 30, 34, 37, 39, 43, 49, 53, 55, 57, 61, 62, 63, 70, 71, 79, 81, 87, 89, 90, 91, 94, 100, 101, 102, 107, 111, 113, 115, 117, 129, 130, 131, 133, 134, 139, 147, 151, 159, 163, 165, 166, 169, 171, 173, 181, 183, 186, 187, 189

Offset: 1

Gus Wiseman, Oct 12 2022

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 terms together with their prime indices begin:
     1: {}
     3: {2}
     7: {4}
     9: {2,2}
    10: {1,3}
    13: {6}
    19: {8}
    21: {2,4}
    27: {2,2,2}
    29: {10}
    30: {1,2,3}

These partitions are counted by A035544.
Includes A066207 = products of primes of even index.
The conjugate partitions are ranked by A357636, reverse A357632.
The conjugate reverse version is A357640 (aerated).
A056239 adds up prime indices, row sums of A112798.
A316524 gives alternating sum of prime indices, reverse A344616.
A344651 counts partitions by alternating sum, ordered A097805.
A357705 counts reversed partitions by skew-alternating sum, half A357704.
Cf. A035363, A035550, A035594, A053251, A298311, A357486, A357623, A357638.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[primeMS[#],?(Mod[#,4]==1&)]==Count[primeMS[#],?(Mod[#,4]==3&)]&]

A357851 Numbers k such that the half-alternating sum of the prime indices of k is 1.

Original entry on oeis.org

2, 8, 18, 32, 45, 50, 72, 98, 105, 128, 162, 180, 200, 231, 242, 275, 288, 338, 392, 420, 429, 450, 455, 512, 578, 648, 663, 720, 722, 800, 833, 882, 924, 935, 968, 969, 1050, 1058, 1100, 1125, 1152, 1235, 1250, 1311, 1352, 1458, 1463, 1568, 1680, 1682, 1716

Offset: 1

Gus Wiseman, Oct 28 2022

nonn

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 - ...

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:
     2: {1}
     8: {1,1,1}
    18: {1,2,2}
    32: {1,1,1,1,1}
    45: {2,2,3}
    50: {1,3,3}
    72: {1,1,1,2,2}
    98: {1,4,4}
   105: {2,3,4}
   128: {1,1,1,1,1,1,1}
   162: {1,2,2,2,2}
   180: {1,1,2,2,3}
   200: {1,1,1,3,3}

The version for k = 0 is A357631, standard compositions A357625-A357626.
The version for original alternating sum is A001105.
Positions of ones in A357629, reverse A357633.
The skew version for k = 0 is A357632, reverse A357636.
Partitions with these Heinz numbers are counted by A035444, skew A035544.
The reverse version is A357635, k = 0 version A000583.
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-length A357642.
Cf. A003963, A053251, A055932, A345958, A357621-A357624, A357630, A357634, A357637, A357639, A357640.

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]}];
Select[Range[1000],halfats[primeMS[#]]==1&]

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A357704 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.

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A357707 Numbers whose prime indices have equal number of parts congruent to each of 1 and 3 (mod 4).

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A357851 Numbers k such that the half-alternating sum of the prime indices of k is 1.

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Mathematica